{"id":18,"date":"2015-11-13T10:11:41","date_gmt":"2015-11-13T15:11:41","guid":{"rendered":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/?page_id=18"},"modified":"2016-12-13T16:52:50","modified_gmt":"2016-12-13T21:52:50","slug":"seminars","status":"publish","type":"page","link":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/seminars\/","title":{"rendered":"Seminars"},"content":{"rendered":"<p>The weekly Computational Analysis Seminar is attended by faculty, students, and visiting researchers working in one or more of the following areas of mathematics: constructive approximation theory, splines, wavelets, signal processing, image<br \/>\ncompression, shift-invariant spaces, constrained approximation and interpolation, computer-aided geometric design, and a few other related areas. If you need more information and\/or want to be included on our mailing list, please email us at <a href=\"mailto:cca@vanderbilt.edu\">cca@vanderbilt.edu<\/a> or <a href=\"mailto:aleksandr.b.reznikov@vanderbilt.edu\">aleksandr.b.reznikov@vanderbilt.edu<\/a>.<\/p>\n<p><!-- 2017 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2017<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2017\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>:<\/p>\n<p>\t        <b>Speaker<\/b>:<\/p>\n<p>\t        <b>Title<\/b>:<\/p>\n<p>\t\t\t<b>Abstract<\/b>:\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2016 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2016<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2016\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 30, 2016, 4:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Enrico Au-Yeung, DePaul University<\/p>\n<p>\t        <b>Title<\/b>: Tensor Networks and the Blessing of Dimensionality<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  Have you ever looked at your thumb and admire how smart is your thumb? The protein molecules in your body can perform computation hundreds of times faster than a cluster of computers. There are three short stories that I want to tell. Tensor networks are tools that can be used to solve a wide class of data intensive problems in machine learning, physics, and signals processing. The basic idea is to turn a long vector or a large matrix into a tensor, then draw some cute diagrams. Each such diagram actually represents a formidable equation. Another story here is Optimization beyond Grandma&#8217;s Lagrange Multiplier. The term Compressed Sensing means recovering a long vector by making a small number of measurements. Until a few years ago, to do compressed sensing, you need a matrix to satisfy RIP (restricted isometry property). What if your matrix does not satisfy RIP, but you have a good toolbox for solving optimization problems? For the third story, you will have to hear it at the talk. Most of this talk will be accessible to graduate students in mathematics<\/p>\n<\/td>\n<\/tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 16, 2016, 4:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Luc Vinet, University of Montreal<\/p>\n<p>\t        <b>Title<\/b>: Quantum state transport, entanglement generation and orthogonal polynomials<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  TBA<\/p>\n<\/td>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 19, 2016, 4:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  John Jasper, University of Cincinatti<\/p>\n<p>\t        <b>Title<\/b>: Equiangular tight frames from association schemes<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound. Though they arise in many applications, there are only a few known methods for constructing ETFs. One of the most popular classes of ETFs, called harmonic ETFs, is constructed using the structure of finite abelian groups. In this talk we will discuss a broad generalization of harmonic ETFs. This generalization allows us to construct ETFs using many different structures in the place of abelian groups, including nonabelian groups, Gelfand pairs of finite groups, and more. We apply this theory to construct an infinite family of ETFs using the group schemes associated with certain Suzuki 2-groups. Notably, this is the first known infinite family of equiangular lines arising from nonabelian groups.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 16, 2016, 3:10 pm, SC 1310<\/p>\n<p>\t        <b>Speaker<\/b>:  David Benko, University of South Alabama<\/p>\n<p>\t        <b>Title<\/b>: Estimating the probability of heads of a fake coin.<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  We tossed a biased coin 10 times and we got 3 heads. What is the probability of heads? The maximum-likelihood method claims it is 0.3 but we are unhappy with that method. Using game theory we answer the question explicitly for 1 and 2 tosses.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 6, 2016, 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Alexander Volberg, Michigan State University<\/p>\n<p>\t        <b>Title<\/b>: Monge&#8211;Amp&#232;re equations with drift and end-point estimates in harmonic analysis.<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  We will review a couple of end-point estimates in harmonic analysis that can be sort of equivalently reduced to understanding of the behavior of solutions of certain MA equations with drift, where the behavior of the drift seems to be curial. As a result, some new end-point estimates for singular integrals will be proved.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: March 16, 2016, 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Oleksandra Beznosova, University of Alabama<\/p>\n<p>\t        <b>Title<\/b>: On the star discrepancy conjecture.<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  The L<sub>&#8734;<\/sub>-star discrepancy measures how well a discrete measure supported on a given set of N points approximates a uniform measure on the multidimensional unit cube, the smaller the discrepancy the better the approximation. Therefore, we are interested in the sharp lower bound on discrepancy as a function of N and optimal sets on which it is achieved.<br \/>\nBounds on the discrepancy are used, for example, in the error bounds for quasi-Monte Carlo methods.<br \/>\nIt is somewhat intuitive that a discrete measure cannot approximate continuous measure too well. In dimension 2 we know (up to a numerical constant) lower bounds on the discrepancy as function of N, and some examples of sets on which lower bounds are achieved. In higher dimension d&gt;2, it is an open conjecture that optimal L<sub>&#8734;<\/sub>-star discrepancy is of the order N<sup>-1<\/sup>(log N)<sup>d\/2 <\/sup>\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2015 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2015<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2015\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 11, 2015, 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Laura De Carli, Florida International University<\/p>\n<p>\t        <b>Title<\/b>: Constructing new bases from old.<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  available <a href=\"decarli.pdf\">here<\/a>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 11, 2015, 4:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Grady Wright, Boise State University<\/p>\n<p>\t        <b>Title<\/b>: Computing with functions on the sphere using low rank approximations<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  A collection of algorithms for computing with functions on the surface of the unit two-sphere is presented.  Central to these algorithms is a new scheme for approximating functions to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method.  The scheme gives a low rank representation of the approximants that reduces oversampling issues near the poles, converges for certain analytic functions, and allows for stable differentiation.  The low rank representation also makes operations such as function evaluation, differentiation, and integration particularly efficient.   A demonstration of the algorithms, which are implemented in Chebfun, will be given.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 4, 2015, 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Yujian Su, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Dissertation defense<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  TBA<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 7, 2015, 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Tim Michaels, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Point sets on the sphere and their Riesz energies<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  Generating suitable point sets and meshes on the sphere is a problem spanning many areas in numerical analysis. We present a survey of quickly generated point sets on S^2 which have been created for a variety of purposes, examine their equidistribution properties, separation, covering, and mesh ratio constants and derive a new point set, equal area icosahedral points, with low mesh ratio. We analyze numerically the leading order asymptotics for the Riesz and logarithmic potential energy for these configurations with total points up to 10,000.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 30, 2015, 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Bubacarr Bah, University of Texas at Austin<\/p>\n<p>\t        <b>Title<\/b>: Structured sparse recovery with sparse sampling matrices<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  Compressed sensing seeks to exploit the simplicity (sparsity) of a<br \/>\nsignal to under sample the signal significantly. Sparsity is a first<br \/>\norder prior information on the signal. In many applications signals<br \/>\nexhibit an additional structure beyond sparsity. Exploiting this<br \/>\nsecond order prior information about the signal not only enables<br \/>\nfurther sub-sampling but also improves accuracy of reconstruction. On<br \/>\nthe other hand, a lot of the sampling matrices, for which we are able<br \/>\nto prove optimal recovery guarantees, are dense and hence do not scale<br \/>\nwell with the dimension of the signal. Sparse matrices scale better<br \/>\nthan their dense counterparts but they are more difficult to give<br \/>\nprovable guarantees on. The sparse sampling operators we consider are<br \/>\nadjacency matrices of lossless expander graphs. They are non-mean zero<br \/>\nand they reflect more some of the applications of compressed sensing<br \/>\nlike the single pixel camera. We also propose two reconstruction<br \/>\nalgorithms. A non-convex algorithm that converges linearly with the<br \/>\nsignal dimension and a convex algorithm that is comparable and<br \/>\nsometimes outperforms existing popular algorithms. We also derived<br \/>\nsharp sample complexity bounds.<br \/>\nThis talk will give a general overview of results on structured<br \/>\nsparsity in compressed sensing (model-based<br \/>\ncompressed sensing). It will discuss sampling and recovery in<br \/>\nmodel-based compressed sensing generally<br \/>\nbut will narrow down to give latest results our work on model-based<br \/>\ncompressed sensing with sparse<br \/>\nsensing matrices from expanders.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 9, 2015, 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Keaton Hamm, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Sampling and Interpolation with Radial Basis Functions<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  For some time, there have been connections between interpolation schemes involving radial basis functions and classical sampling theory.  This talk will explore some of these connections both in the uniform and nonuniform settings.  In the former case, the technique of cardinal interpolation seeks to approximate a given smooth function by integer translates of a single function, for example, the Gaussian kernel or Hardy multiquadric.  This is similar to the classical sampling theorem which provides exact recovery of a bandlimited function via translates of sinc.  However, in the nonuniform case, the problem becomes somewhat more functional analytic, and so far there are some restrictions on what type of point sets one may use in the interpolation schemes; in particular, so-called complete interpolating sequences for Paley-Wiener spaces are such an admissible set.  Time permitting, we may also discuss some ways in which one may obtain approximation rates for the schemes discussed before.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 29, 2015, 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>:  Liao Wenjing, Duke University<\/p>\n<p>\t        <b>Title<\/b>: Gridding error and super-resolution in spectral estimation<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  The problem of spectral estimation, namely &#8211; recovering the frequency<br \/>\ncontents of a signal &#8211; arises in various applications, including array imaging and remote sensing. In these<br \/>\nfields, the spectrum of natural signals is composed of a few spikes on the continuum of a bounded domain. After<br \/>\nthe emergence of compressive sensing, spectral estimation has been widely studied with an emphasis on sparse<br \/>\nmeasurements. However, with few exceptions, the spectrum considered in the compressive sensing community is<br \/>\nassumed to be located on a DFT grid, which results in a significant gridding error.<\/p>\n<p>In this talk, I will present the MUltiple SIgnal Classification (MUSIC)  algorithm and some modified greedy<br \/>\nalgorithms, and show how the problem of gridding error can be resolved by these methods. Our work focuses on a<br \/>\nstability analysis as well as  numerical studies on the performance of these algorithms. Moreover, MUSIC features<br \/>\nits super-resolution effect, i.e.,  the  capability of resolving closely spaced frequencies. We will provide<br \/>\nnumerical experiments and theoretical justifications to show that the noise tolerance of MUSIC follows a power<br \/>\nlaw with respect to the minimum separation of frequencies.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: March 18, 2015, 3:10 pm, SC 1431<\/p>\n<p>\t        <b>Speaker<\/b>: Qiang Wu, Middle Tennessee State University<\/p>\n<p>\t        <b>Title<\/b>: Mathematical Foundation of the Minimum Error Entropy Algorithm<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  Information theoretical learning (ITL) is an important research<br \/>\narea in signal processing and machine learning. It uses concepts of entropies and divergences from<br \/>\ninformation theory to substitute the conventional statistical descriptors of<br \/>\nvariances and covariances. The empirical minimum error entropy (MEE) algorithm<br \/>\nis a typical approach falling into this this framework and has been<br \/>\nsuccessfully used in both regression and classification problems.<\/p>\n<p> In this talk, I will discuss the consistency analysis of the MEE algorithm. For this purpose,<br \/>\nwe introduce two types of consistency. The error entropy consistency, which requires the error<br \/>\nentropy of the learned function to approximate the minimum error entropy, is proven<br \/>\nwhen the bandwidth parameter tends to 0 at an appropriate rate. The regression<br \/>\nconsistency, which requires the learned function to approximate the<br \/>\nregression function, however, is a complicated issue. We prove that the error entropy<br \/>\nconsistency implies the regression consistency for homoskedastic models where<br \/>\nthe noise is independent of the input variable. But for heteroskedastic models,<br \/>\na counterexample is constructed to show that the two types of consistency are<br \/>\nnot necessarily coincident. A surprising result is that the regression<br \/>\nconsistency holds when the bandwidth parameter is sufficiently large.<br \/>\nRegression consistency of two classes of special models is shown to hold with<br \/>\nfixed bandwidth parameter. These results illustrate the complication of the MEE algorithm.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: January 23, 2015 (Friday). 3:10 pm, SC 1431<\/p>\n<p>\t        <b>Speaker<\/b>: Shahaf Nitzan, Kent State University<\/p>\n<p>\t        <b>Title<\/b>: Exponential frames on unbounded sets<\/p>\n<p>\t\t\t<b>Abstract<\/b>: In contrast to orthonormal and Riesz bases, exponential frames (i.e.,<br \/>\n&#8216;over complete bases&#8217;) are in many cases easy to come by. In particular,<br \/>\nit is not difficult to show that every bounded set of positive measure<br \/>\nadmits an exponential frame.<\/p>\n<p>When unbounded sets (of finite measure) are considered, the problem<br \/>\nbecomes more delicate. In this talk I will discuss a joint work with<br \/>\nA. Olevskii and A. Ulanovskii, where we prove that every such set admits<br \/>\nan exponential frame. To obtain this result we apply one of the outcomes<br \/>\nof Marcus, Spielman and Srivastava&#8217;s recent solution of the<br \/>\nKadison-Singer conjecture.<\/p>\n<p>This talk is part of the Shanks Workshop on &#8220;Uncertainty Principles in Time Frequency Analysis&#8221;\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2014 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2014<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2014\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 12, 2014. 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>: Maryke van der Walt, University of Missouri, St. Louis<\/p>\n<p>\t        <b>Title<\/b>: Signal analysis via instantaneous frequency estimation of signal components<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  The empirical mode decomposition (EMD) algorithm, introduced by<br \/>\nN.E. Huang et al in 1998, is arguably the most popular mathematical scheme for non-stationary signal<br \/>\ndecomposition and analysis. The objective of EMD is to separate a given signal into a number of<br \/>\ncomponents, called intrinsic mode functions (IMF&#8217;s), after which the instantaneous frequency (IF) and amplitude<br \/>\nof each IMF are computed through Hilbert spectral analysis (HSA). On the other hand, the synchrosqueezed wavelet<br \/>\ntransform (SST), introduced by I. Daubechies and S. Maes in 1996 and further developed by I. Daubechies, J. Lu<br \/>\nand H.-T. Wu in 2011, is applied to estimate the IF&#8217;s of all signal components of the given signal, based on one<br \/>\nsingle reference &ldquo;IF function,&rdquo; under the assumption that the signal components satisfy certain strict properties<br \/>\nof a so-called adaptive harmonic model (AHM), before the signal components of the model are recovered. The<br \/>\nobjective of our paper is to develop a hybrid EMD-SST computational scheme by applying a &ldquo;modified SST&rdquo; to<br \/>\neach IMF of the EMD, as an alternative approach to the original EMD-HSA method. While our modified SST<br \/>\nassures non-negative instantaneous frequencies of the IMF&#8217;s, application of the EMD scheme eliminates the<br \/>\ndependence of a single reference IF value as well as the guessing work of the number of signal components in<br \/>\nthe original SST approach. Our modification of the SST consists of applying vanishing moment<br \/>\nwavelets (introduced in a recent paper by C.K. Chui and H.-T. Wu) with stacked knots to process signals on<br \/>\nbounded or half-infinite time intervals, and spline curve fitting with optimal smoothing parameter selection<br \/>\nthrough generalized cross-validation. In addition, we formulate a local cubic spline interpolation scheme for<br \/>\nreal-time realization of the EMD sifting process that improves over the standard global cubic spline<br \/>\ninterpolation, both in quality and computational cost, particularly when applied to bounded and half-infinite<br \/>\ntime intervals. This is a joint work with C.K. Chui.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 5, 2014. 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>: Guilherme de Silva, KU Leuven<\/p>\n<p>\t        <b>Title<\/b>: Breaking the Symmetry in the Normal Matrix Model<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  We consider the normal matrix model with cubic plus linear potential.<br \/>\nThe model is ill-defined, and to regualrize it, Elbau and Felder proposed to make a cut-off on the complex<br \/>\nplane, leading to a system of orthogonal polynomials with respect to a certain 2D measure.  When studying this<br \/>\nmodel with a monic cubic weight, Bleher and Kuijlaars associated to this model a system of non-hermitian multiple<br \/>\northogonal polynomials, which are expected to be asymptotically the same as the 2D orthogonal polynomials<\/p>\n<p>In this talk, we will focus on the non-hermitian MOP&#8217;s in the spirit of Bleher and Kuijlaars, but now adding a<br \/>\nlinear term to the cubic potential.  It will be shown how some quantities of the normal matrix model are<br \/>\nrelated to those orthogonal polynomials.  At the technical level, the linear term breaks the symmetry of the model,<br \/>\nand in order to deal with it, we introduce a quadratic differential on the spectral curve and describe<br \/>\nglobally its trajectories.  The trajectories of the quadratic differential play a fundamental role in the<br \/>\nasymptotic analysis of the MOP&#8217;s.<\/p>\n<p>This is an ongoing project with Pavel Bleher (Indiana University &#8211; Purdue University Indianapolis).<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 1, 2014. 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>: Andrei Martinez-Finkelshtein, University of Almeria (visiting Vanderbilt)<\/p>\n<p>\t        <b>Title<\/b>: Two approximation problems in ophthalmology, or how Gauss can beat Zernike<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  Modern corneal topographers or videokeratometers based on the<br \/>\nprinciple of Placido disks collect the data (either corneal altimetry or corneal power) in a discrete set of<br \/>\npoints on the disk organized in a nearly concentric pattern. A reliable reconstruction of the cornea from this<br \/>\ninformation is essential for a correct early diagnosis of several ophthalmological diseases. A standard<br \/>\nprocedure used in clinical practice is based on a least squares fit by Zernike polynomials (an orthonormal<br \/>\nfamily with respect to the plane measure on the disk). However well this method works for regular corneas, it<br \/>\nhas several drawbacks and lacks precision in more complex (and thus, clinically relevant) cases. <\/p>\n<p>On the other hand, the point-spread-function (PSF) of an eye carries important information about the eye as<br \/>\nan optical instrument. PSF can be found from non-invasive objective measurements, e.g. from the wavefront<br \/>\naberrations of the eye. However, the actual calculation of the PSF (which boils down to computing 2D Fourier<br \/>\ntransforms of functions on a disk for different parameters) is costly. Here also the Zernike polynomials play a<br \/>\npredominant role, laying the groundwork for the so-called Extended Nijboer-Zernike analysis.<\/p>\n<p>It turns out that in both problems the gaussian functions can be used as an alternative to Zernike<br \/>\npolynomials. For the first problem, we devise an adaptive and multi-scale algorithm that fits the corneal<br \/>\ndata by means of anisotropic Gaussian radial basis functions. The shape parameters of these functions, chosen<br \/>\ndynamically in dependence of the data, constitute an important additional source of information about the corneal<br \/>\nirregularity. <\/p>\n<p>For the second problem, an approximation of the wavefront aberrations by gaussian functions results in a fast<br \/>\nand reliable method of parallel computation of these 2D Fourier integrals and of the through-focus<br \/>\ncharacteristics of a human eye.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 24, 2014. 3:10 pm, SC 1432<\/p>\n<p>\t        <b>Speaker<\/b>: Dustin Mixon, Air Force Institute of Technology<\/p>\n<p>\t        <b>Title<\/b>: Phase retrieval: Approaching the theoretical limits in practice<\/p>\n<p>\t\t\t<b>Abstract<\/b>: In many areas of imaging science, it is difficult to measure the phase<br \/>\nof linear measurements. As such, one often wishes to reconstruct a<br \/>\nsignal from intensity measurements, that is, perform phase retrieval.<br \/>\nVery little is known about how to design injective intensity<br \/>\nmeasurements, let alone stable measurements with efficient<br \/>\nreconstruction algorithms. This talk helps to fill the void &#8211; I will<br \/>\ndiscuss a wide variety of recent results in phase retrieval, including<br \/>\nvarious conditions for injectivity and stability (joint work with<br \/>\nAfonso S. Bandeira (Princeton), Jameson Cahill (Duke) and Aaron A.<br \/>\nNelson (AFIT)) as well as measurement designs based on spectral graph<br \/>\ntheory which allow for efficient reconstruction (joint work with Boris<br \/>\nAlexeev (Princeton), Afonso S. Bandeira (Princeton) and Matthew Fickus<br \/>\n(AFIT)). In particular, I will show how Fourier-type tricks can be<br \/>\nleveraged in concert with this graph-theoretic design to produce<br \/>\npseudorandom aperatures for X-ray crystallography and related<br \/>\ndisciplines (joint work with Afonso S. Bandeira (Princeton) and Yutong<br \/>\nChen (Princeton)).<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 2, 2014. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Anne Gelb, Arizona State University<\/p>\n<p>\t        <b>Title<\/b>: Numerical Approximation Methods for Non-Uniform Fourier Data<\/p>\n<p>\t\t\t<b>Abstract<\/b>: <\/p>\n<p>In this talk I discuss the reconstruction of compactly supported<br \/>\npiecewise smooth functions from non-uniform samples of their Fourier transform. This problem is relevant in<br \/>\napplications such as magnetic resonance imaging (MRI) and synthetic aperture radar (SAR).<\/p>\n<p>Two standard<br \/>\nreconstruction techniques, convolutional gridding (the non-uniform FFT) and uniform resampling, are<br \/>\nsummarized, and some of the difficulties are discussed. It is then demonstrated how spectral reprojection can be<br \/>\nused to mollify both the Gibbs phenomenon and the error due to the non-uniform sampling. It is further shown that<br \/>\nincorporating prior information, such as the internal edges of the underlying function, can greatly improve the<br \/>\nreconstruction quality. Finally, an alternative approach to the problem that uses Fourier frames is proposed.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 12, 2014. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Charles Martin, Vandebilt University<\/p>\n<p>\t        <b>Title<\/b>: Perturbations of Green Functions and the Dirichlet Problem<\/p>\n<p>\t\t\t<b>Abstract<\/b>: The Dirichlet problem for the Laplacian on a domain is better understood<br \/>\nand more easily computed than it is for that of a more general elliptic operator. If an elliptic operator is<br \/>\nsomehow a small perturbation from the Laplacian, what corrections can we make to the solutions to the Dirichlet<br \/>\nproblem? In this talk we&#8217;ll address this question by first considering perturbation of Green functions. With<br \/>\nvarious perturbative formulas (and a few series expansions) in hand, we turn to the problem of bounding the<br \/>\nresulting error terms. <\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: January 22, 2014. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Stefano de Marchi, University of Padua<\/p>\n<p>\t        <b>Title<\/b>: Padua points: theory, computation, applications and open problems.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: The so called &#8220;Padua points&#8221; are the first set of unisolvent<br \/>\npoints in the square that give a simple, geometric, and explicit construction of bivariate polynomial interpolation.<br \/>\nTheir associated Lebesgue constant, which measures the goodness of approximation, has minimal order of growth,<br \/>\ni.e. O(log^2(n)) with n the polynomial degree.<br \/>\nIn the talk we present a stable and efficient implementation of the corresponding Lagrange interpolation and<br \/>\nquadrature formulas. We also discuss extensions of (non-polynomial) Padua-like interpolation to<br \/>\nother domains, such as triangles and ellipses.  Applications to finding approximate Fekete points on<br \/>\ntensor-product domains are also discussed. We conclude with some open problems. <\/p>\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2013 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2013<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2013\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 20, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Igor Pritsker, Oklahoma State University<\/p>\n<p>\t        <b>Title<\/b>: Riesz decomposition for the farthest distance functions<br \/>\nvia logarithmic, Green and Riesz potentials.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: We discuss several versions of the Riesz Decomposition Theorem for<br \/>\nsuperharmonic functions. This theorem is usually stated for Newtonian and logarithmic potentials in the<br \/>\nliterature, but it isalso true for some Riesz kernels.  However, no full version for Riesz potentials<br \/>\nis available. We mention related topics on $\\alpha$-superharmonic and polyharmonic functions, and on fractional<br \/>\nLaplacian.  We apply Riesz decompositions to obtain integral representations of the farthest distance functions<br \/>\nfor compact sets  as logarithmic, Green and Riesz potentials of positive measures with unbounded<br \/>\nsupport. The representing  measures encode many geometric properties of compact sets via their<br \/>\ndistance functions. <\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 6, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Koushik Ramachandran, Purdue University<\/p>\n<p>\t        <b>Title<\/b>: Asymptotic behavior of positive harmonic functions in certain unbounded domains<\/p>\n<p>\t\t\t<b>Abstract<\/b>: We derive asymptotic estimates at infinity for positive harmonic<br \/>\nfunctions in large class of non-smooth unbounded domains. These include<br \/>\ndomains whose sections, after rescaling, resemble a Lipschitz cylinder or<br \/>\na Lipschitz cone. Examples of such domains are various paraboloids and,<br \/>\nhorn domains.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 30, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Mark Iwen, Michigan State University<\/p>\n<p>\t        <b>Title<\/b>: Fast Algorithms for Fitting High-Dimensional Data with Hyperplanes<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  I will discuss computational methods for fitting large sets of points in<br \/>\nhigh dimensional Euclidean space with low-dimensional subspaces that are &#8220;near-optimal&#8221;.  Several different<br \/>\nmeasures of optimality will be considered, including one closely related to kolmogorov n-widths.  In this last<br \/>\nsetting we will present a fast (i.e., linear time in the number of points) algorithm with rigorous approximation<br \/>\nguarantees.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 9, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Jorge Roman, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>:  An Introduction to Markov Chain Monte Carlo Methods<\/p>\n<p>\t\t\t<b>Abstract<\/b>: The need to approximate an intractable integral with respect to a<br \/>\nprobability distribution P is a problem that frequently arises across many different disciplines. A popular<br \/>\nalternative to numerical integration and analytical approximation methods is the Monte Carlo (MC) method which<br \/>\nuses computer simulations to estimate the integral. In the MC method, one generates independent and identically<br \/>\ndistributed (iid) samples from P and then uses sample averages to estimate the integral. However, in many<br \/>\nsituations, P is a complex high-dimensional probability distribution and obtaining iid samples from it is either<br \/>\nimpossible or impractical. When this happens, one may still be able to use the increasingly popular Markov<br \/>\nchain Monte Carlo (MCMC) method in which the iid draws are replaced by a Markov chain that has P as its<br \/>\nstationary distribution.   In this talk, I will give a brief introduction to the MC and MCMC methods. The focus<br \/>\nwill be on the MCMC method and its applications to Bayesian statistics. <\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 2, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Ding-Xuan Zhou, City University Hong Kong<\/p>\n<p>\t        <b>Title<\/b>: Learning Theory and Minimum Error Entropy Principle<\/p>\n<p>\t\t\t<b>Abstract<\/b>:<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 25, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Jean-Luc Bouchot, Drexel University<\/p>\n<p>\t        <b>Title<\/b>: Progress on Hard Thresholding Pursuit<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: The Hard Thresholding Pursuit algorithm for sparse recovery is revisited<br \/>\nusing a new theoretical analysis. The main result states that all sparse vectors can be exactly recovered from<br \/>\nincomplete linear measurements in a number of iterations at most proportional to the sparsity level as soon as<br \/>\nthe measurement matrix obeys a restricted isometry condition. The recovery is also robust to measurement error<br \/>\nThe same conclusions are derived for a variation of Hard Thresholding Pursuit, called Graded Hard Thresholding<br \/>\nPursuit, which is a natural companion to Orthogonal Matching Pursuit and runs without a prior estimation of the<br \/>\nsparsity level. In two extreme cases of the vector shape, it is also shown that, with high probability on the<br \/>\ndraw of random measurements, a fixed sparse vector is robustly recovered in a number of iterations precisely<br \/>\nequal to the sparsity level. These theoretical findings are experimentally validated, too.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 18, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Matt Fickus, Air Force Institute of Technology<\/p>\n<p>\t        <b>Title<\/b>: Compressed Sensing with Equiangular Tight Frames<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>:  Compressed sensing (CS) is changing the way we think about measuring<br \/>\nhigh-dimensional signals and images.  In particular, CS promises to revolutionize hyperspectral imaging.  Indeed,<br \/>\nemerging camera prototypes are exploiting random masks in order to greatly reduce the exposure times needed to<br \/>\nform hyperspectral images.  Here, the randomness of the masks is due to the crucial role that random matrices<br \/>\nplay in CS.  In short, in terms of CS&#8217;s restricted isometry property (RIP), random matrices far outshine all<br \/>\nknown deterministic matrix constructions.  To be clear, for most deterministic constructions, it is unknown<br \/>\nwhether this performance shortfall (known as the &#8220;square-root bottleneck&#8221;) is simply a consequence of poor proof<br \/>\ntechniques or, more seriously, a flaw in the matrix design itself.  In the remainder of this talk, we focus on<br \/>\nthis particular question in the special case of matrices formed from equiangular tight frames (ETFs).  ETFs are<br \/>\novercomplete collections of unit vectors with minimal coherence, namely optimal packings of a given number of<br \/>\nlines in a Euclidean space of a given dimension.  We discuss the degree to which the recently-introduced Steiner<br \/>\nand Kirkman ETFs satisfy the RIP.  We further discuss how a popular family of ETFs, namely harmonic ETFs arising<br \/>\nfrom McFarland difference sets, are particular examples of Kirkman ETFs.  Overall, we find that many families of<br \/>\nETFs are shockingly bad when it comes to RIP, being provably incapable of exceeding the square-root bottleneck.<br \/>\nSuch ETFs are nevertheless useful in variety of other real-world applications, including waveform design for<br \/>\nwireless communication and algebraic coding theory.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: August 28, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Oleg Davydov, Strathclyde University (Scotland)<\/p>\n<p>\t        <b>Title<\/b>: Error bounds for kernel-based numerical differentiation<\/p>\n<p>\t\t\t<b>Abstract<\/b>:  The literature on meshless methods observed that kernel-based numerical<br \/>\ndifferentiation formulae are highly accurate and robust. We present error bounds for such formulas, using the new<br \/>\ntechnique of growth functions. It allows to bypass certain technical assumptions that were needed to prove the<br \/>\nstandard error bounds for kernel-based interpolation but are not applicable in this setting. Since differentiation<br \/>\nformulas based on polynomials also have error bounds in terms of growth functions, we show that kernel-based<br \/>\nformulas are comparable in accuracy to the best possible polynomial-based formulas. The talk is based on joint<br \/>\nresearch with Robert Schaback.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 10, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Maria Navascues, University of Zaragoza<\/p>\n<p>\t        <b>Title<\/b>: Some historical precedents of fractal functions<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: In this talk, we wish to pay tribute to the scientists of older generations,<br \/>\nwho, through their reseatch, lead to the current state of knowledge of the fractal functions. We review the fundamental<br \/>\nmilestones of the origin and evolution of the self-similar curves that, in some cases, agree with continuous and<br \/>\nnowhere differentiable functions, but they are not exhausted by them.  Our main interest is to emphasize the lesser<br \/>\nknown examples, due to a deficient or late publication (Bolzano&#8217;s map for instance).<\/p>\n<p>\tWe will review different ways of defining self-similar curves.  We will recall the first functions without<br \/>\ntangent, but also some fractal functions having derivative almost everywhere.  Most of the models studied may seem quite<br \/>\nparadoxical (&#8220;monsters&#8221; in the words of Poincare) as, for instance, curves with a fractal dimension of two and<br \/>\nhaving a tangent at every point.  These instances suggest that the classification and even the definition of fractal<br \/>\nfunctions are far from being established.  The strategies of definition of each example compose a toolbox that<br \/>\nwill provide the audience with a selection of procedures for the construction of its own fractal function.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 3, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Keri Kornelson, University of Oklahoma<\/p>\n<p>\t        <b>Title<\/b>: Fourier bases on fractals<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: The study of Bernoulli convolution measures dates<br \/>\nback to the 1930&#8217;s,  yet there has been a recent resurgence in the theory prompted by the<br \/>\nconnection between convolution measures and iterated function systems (IFSs).  The<br \/>\nmeasures are supported on fractal Cantor subsets of the real line, and exhibit their own<br \/>\nsort of self-similarity.  We will use the IFS connection to discover Fourier bases on the<br \/>\nL^2 Hilbert spaces with respect to Bernoulli convolution measures.<\/p>\n<p>There are some interesting phenomena that arise in this setting.  We find that some Cantor<br \/>\nsets support Fourier bases while others do not.   In cases where a Fourier basis does<br \/>\nexist, we can sometimes scale or shift the Fourier frequencies by an integer to obtain<br \/>\nanother ONB.  We also discover properties of the unitary operator mapping between two such<br \/>\nbases.  The self-similarity of the measure and the support space can, in some cases, carry<br \/>\nover into a self-similarity of the operator. <\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: March 27, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Johan De Villiers, Stellenbosch University<\/p>\n<p>\t        <b>Title<\/b>: Wavelet Analysis Based on Algebraic Polynomial Identities<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>:  By starting out from a given refinable function,<br \/>\nand relying on a corresponding space decomposition which is not necessarily<br \/>\northogonal, we present a general wavelet construction method based on<br \/>\nthe solution of a system of algebraic polynomial identities. The<br \/>\nresulting decomposition sequences are finite, and, for any given<br \/>\nvanishing moment order, the wavelets thus constructed are minimally<br \/>\nsupported, and possess robust- stable integer shifts. The special case<br \/>\nof cardinal B-splines are given special attention.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 20, 2013. 3:10 pm, SC 1307 (cancelled)<\/p>\n<p>\t        <b>Speaker<\/b>: Kamen Ivanov, University of South Carolina<\/p>\n<p>\t        <b>Title<\/b>: TBA<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: TBA<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 13, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Roza Aceska, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Gabor frames, Wilson bases and multi-systems<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Frames can be seen as generalized bases, that is, over-complete<br \/>\ncollections, which are used for stable representations of signals as linear combinations of basic building<br \/>\natoms.  It is very useful when we can use locally adapted atoms, which in addition behave as elements of local<br \/>\nbases.  We explore the possibility of using localized parts of frames and bases when building a customized frame.<br \/>\nAfter a review on Gabor frames and Wilson bases, we consider the question of combining parts of these collections<br \/>\ninto a multi-frame set and look at its properties.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 7, 2013. 4:10 pm, SC 1425 (also a Colloquium)<\/p>\n<p>\t        <b>Speaker<\/b>: Barry Simon, Caltech<\/p>\n<p>\t        <b>Title<\/b>: Tales of Our Forefathers<\/p>\n<p>\t\t\t<b>Abstract<\/b>: This is not a mathematics talk but it is a talk<br \/>\n                        for mathematicians. Too often, we think of historical<br \/>\n                        mathematicians as only names assigned to theorems. With vignettes<br \/>\n                        and anecdotes, I&#8217;ll convince you they were also human beings and<br \/>\n                        that, as the Chinese say, &#8220;May you live in interesting times&#8221;<br \/>\n                        really is a curse.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: January 30, 2013. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Eduardo Lima (MIT) and Laurent Baratchart (INRIA)<\/p>\n<p>\t        <b>Title<\/b>: Overview of Inverse Problems in Planar Magnetization<\/p>\n<p>\t\t\t<b>Abstract<\/b>: TBA<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2012 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2012<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2012\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 28, 2012. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Manos Papadakis, University of Houston<\/p>\n<p>\t        <b>Title<\/b>: Texture Analysis in 3D for the Detection of Liver Cancer in X-ray CT Scans<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: We propose a method for the 3D-rigid motion invariant texture<br \/>\ndiscrimination for discrete 3D-textures that are spatially homogeneous. We model these  textures as stationary<br \/>\nGaussian random fields. We formally develop the concept of 3D-texture rotations in the 3D-digital domain. We use<br \/>\nthis novel concept to define a `distance&#8217; between 3D-textures that remains invariant under all 3D-rigid motions<br \/>\nof the texture. This concept of `distance&#8217; can be used for a monoscale or a multiscale setting to test the<br \/>\n3D-rigid motion invariant statistical similarity of  stochastic 3D-textures. To extract this novel<br \/>\ntexture `distance&#8217; we use the Isotropic Mutliresolution Analysis. We also show how to construct wavelets<br \/>\nassociated with this structure by means of extension principles and we discuss some very recent results by<br \/>\nAtreas, Melas and Stavropoulos on the geometric structure underlying the various extension principles.<\/p>\n<p>The  3D-texture `distance&#8217; is used to define  a set of<br \/>\n3D-rigid motion invariant texture features. We experimentally establish that when they are combined with<br \/>\nmean attenuation intensity  differences the new augmented features are capable of discriminating normal from<br \/>\nabnormal liver tissue in arterial phase contrast enhanced X-ray CT-scans with high sensitivity and<br \/>\nspecificity.  To extract these features CT-scans are processed in their native dimensionality. We<br \/>\nexperimentally  observe that the 3D-rotational invariance of the proposed features improves the clustering<br \/>\nof the feature vectors extracted from normal liver tissue samples.  This work is joint with R.<br \/>\nAzencott, S. Jain, S. Upadhyay, I.A. Kakadiaris and G. Gladish, MD.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 14, 2012. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Ben Adcock, Purdue University<\/p>\n<p>\t        <b>Title<\/b>: Breaking the coherence barrier: semi-random sampling in compressed sensing<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>:  Compressed sensing is a recent development in the field of sampling<br \/>\nBased on the notion of sparsity, it provides a theory and techniques for the recovery of images and signals from<br \/>\nonly a relatively small number of measurements.  The key ingredients that permit this so-called subsampling are<br \/>\n(i) sparsity of the signal in a particular basis and (ii) mutual incoherence between such basis and the sampling<br \/>\nsystem.  Provided the corresponding coherence parameter is sufficiently small, one can recover a sparse signal<br \/>\nusing a number of measurements that is, up to a log factor, on the order of the sparsity.<\/p>\n<p>Unfortunately, many problems that one encounters in practice are not incoherent.  For example, Fourier<br \/>\nsampling, the type of sampling encountered in Magnetic Resonance Imaging (MRI), is typically not incoherent<br \/>\nwith wavelet or polynomials bases.  To overcome this `coherence barrier&#8217; we introduce a new theory of compressed<br \/>\nsensing, based on so-called asymptotic incoherence and asymptotic sparsity.  When combined with a semi-random<br \/>\nsampling strategy, this allows for significant subsampling in problems for which standard compressed sensing<br \/>\ntools are limited by the lack of incoherence.  Moreover, we demonstrate how the amount of subsampling possible<br \/>\nwith this new approach actually increases with resolution.  In other words, this technique is particularly well<br \/>\nsuited to higher resolution problems.<\/p>\n<p>This is joint work with Anders Hansen and Bogdan Roman (University of Cambridge)<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: TBA (postponed from October 31)<\/p>\n<p>\t        <b>Speaker<\/b>: Doron Lubinsky, Georgia Institute of Technology<\/p>\n<p>\t        <b>Title<\/b>: L<sup>p<\/sup> Christoffel functions and Paley-Wiener spaces<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Let &omega; be a finite positive Borel measure on the unit circle. Let p&gt;0 and <\/p>\n<p>&lambda; <sub>n,p<\/sub>(&omega;,z) =inf<sub>deg P &le; n-1<\/sub><br \/>\n<font size=\"4\">(<\/font>&int;<sub>-&pi;<\/sub><sup>&pi;<\/sup>|P(e<sup>i&theta;<\/sup>)|<br \/>\n<sup>p<\/sup>d&omega;(&theta;)<font size=\"4\">)<\/font><font size=\"4\">(<\/font>|P(z)|<br \/>\n<sup>p<\/sup><font size=\"4\">)<\/font><sup>-1<\/sup><\/p>\n<p>denote the corresponding L<sub>p<\/sub> Christoffel function. The asymptotic<br \/>\nbehavior of &lambda;<sub>n,p<\/sub>(&omega;,z) as n&rarr;&infin; is well understood for<br \/>\n|z|&lt;1, falling naturally<br \/>\nwithin the ambit of Szego theory. We provide asymptotics on the unit<br \/>\ncircle, for all p&gt;0. These involve an extremal problem for L<sub>&pi;<\/sub><sup>p<\/sup>,<br \/>\nthe Paley-Wiener space of entire functions f of exponential type at most &pi;, with<br \/>\n&int;<sub>-&infin;<\/sub><sup>&infin;<\/sup>|f|<sup>p<\/sup>&lt; &infin;.<br \/>\nLet<\/p>\n<p>E<sub>p<\/sub>=inf <font size=\"4\">{<\/font>&int;<sub>-&infin;<\/sub><sup>&infin;<\/sup>|<br \/>\nf|<sup>p<\/sup> : f&isin; L<sub>&pi;<\/sub><sup>p<\/sup>  with f(0) =1<font size=\"4\">}<\/font>.<\/p>\n<p>We show that for all p&gt;0, <\/p>\n<p>lim<sub>n&rarr;&infin;<\/sub>n&lambda;<sub>n,p<\/sub>(&omega;,z)=2&pi;<br \/>\nE<sub>p<\/sub>&omega;<sup>&#8216;<\/sup>(z) , <\/p>\n<p>when &omega; is a regular measure on the unit circle, and z is a<br \/>\nLebesgue point of &omega;, while &omega;<sup>&#8216;<\/sup> is lower<br \/>\nsemi-continuous at z. For p&ne;2, they seem to be new even for Lebesgue<br \/>\nmeasure on the unit circle.<\/p>\n<p>In addition, for p&gt;1, we establish universality type limits. Let<br \/>\nP<sub>n,p,z<\/sub> be a polynomial of degree at most n-1 with P<sub>n,p,z<\/sub>(<br \/>\nz)=1, attaining the infimum above. We prove that uniformly for u in<br \/>\ncompact subsets of the plane,<\/p>\n<p>lim<sub>n&rarr;&infin;<\/sub>P<sub>n,p,z<\/sub>(ze<sup>2&pi;iu\/n<\/sup>)=e<sup>iu&pi;<br \/>\n<\/sup>f<sub>p<\/sub>(u) <\/p>\n<p>where f<sub>p<\/sub>&isin; L<sub>&pi;<\/sub><sup>p<\/sup> satisfies f<sub>p<\/sub>(0)=1 and<br \/>\nattains the second infimum in above. When p=2, this reduces to a special case of the<br \/>\nuniversality limit associated with random matrices. Analogous results are<br \/>\npresented for measures on [-1,1].<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 17, 2012. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Matt Hirn, Yale University<\/p>\n<p>\t        <b>Title<\/b>: Diffusion maps for changing data<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Much of the data collected today is massive and high dimensional,<br \/>\nyet hidden within is a low dimensional structure that is key to understanding it. As such, recently there has<br \/>\nbeen a large class of research that utilizes nonlinear mappings into low dimensional spaces in order to organize<br \/>\nhigh dimensional data according to its intrinsic geometry. Examples include, but are not limited to, locally<br \/>\nlinear embedding (LLE), ISOMAP, Hessian LLE, Laplacian eigenmaps, and diffusion maps. The type of question we<br \/>\nshall ask in this talk is the following: if my data is in some way dynamic, either evolving over time or changing<br \/>\ndepending on some set of input parameters, how do these low dimensional embeddings behave? Is there a way to go<br \/>\nbetween the embeddings, or better still, track the evolution of the data in its intrinsic geometry? Can we<br \/>\nunderstand the global behavior of the data in a concise way? Focusing on the diffusion maps framework, we shall<br \/>\naddress these questions and a few others. We will begin with a review the original work on diffusion maps by<br \/>\nCoifman and Lafon, and then present some current theoretical results. Various synthetic and real world examples<br \/>\nwill be presented to illustrate these ideas in practice, including examples taken from image analysis and<br \/>\ndynamical systems. Parts of this talk are based on joint work with Ronald Coifman, Simon Adar, Yoel Shkolnisky,<br \/>\nEyal Ben Dor, and Roy Lederman.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 10, 2012. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Yaniv Plan, University of Michigan<\/p>\n<p>\t        <b>Title<\/b>: One-bit matrix completion<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Let Y be a matrix representing voting results in which each entry is<br \/>\neither 1 or -1.  For example, we may take Y<sub>ij<\/sub> = 1 if senator i votes &ldquo;yes&rdquo; on bill j, and -1 otherwise.<br \/>\nNow suppose that a number of entries are missing from Y (for example, senators may be out of town during a vote).<br \/>\nCould you guess how to fill in the missing entries (how would senator i have voted on bill j)?  Similar questions<br \/>\narise in many other applications such as recommender systems or binary survey completion.<\/p>\n<p>In this talk, we assume that the binary data is generated according a probability distribution which is<br \/>\nparameterized by an underlying matrix M.  Further, we assume that M has low rank &ndash; loosely, this means that<br \/>\nthe voting preferences of each senator may be defined by just a few characteristics (Democrat, Republican, etc.),<br \/>\nalthough these characteristics need not be known.  We show that the probability distribution of the missing<br \/>\nentries of Y may be well approximated using maximum likelihood estimation under a nuclear-norm constraint.  Under<br \/>\nappropriate assumptions, we demonstrate that the approximation error is nearly minimax.  The upper bounds are<br \/>\nproven using techniques from probability in Banach spaces.  The lower bounds are proven using information<br \/>\ntheoretic techniques, in particular Fano&rsquo;s inequality.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 26, 2012. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Hau-tieng Wu, University of California Berkeley<\/p>\n<p>\t        <b>Title<\/b>: Instantaneous frequency, shape functions, Synchrosqueezing transform, and some applications<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: <a href=\"http:\/\/sitemason.vanderbilt.edu\/a\/ag9nc47\/WuAbstract.pdf\">PDF<\/a><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 5, 2012. 3:10 pm, SC 1307<\/p>\n<p>\t        <b>Speaker<\/b>: Maxim Yattselev, University of Oregon<\/p>\n<p>\t        <b>Title<\/b>: Bernstein-Szego Theorem on Algebraic S-Contours<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: <a href=\"http:\/\/sitemason.vanderbilt.edu\/a\/ag9nc47\/Yatselev.pdf\">PDF<\/a><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 25, 2012. 3:10 pm, SC1310<\/p>\n<p>\t        <b>Speaker<\/b>: Antoine Ayache, Laboratoire Paul Painlev&eacute;<\/p>\n<p>\t        <b>Title<\/b>: Optimal Series Representations of Continuous Gaussian Random Fields<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Any continuous Gaussian random field X(t) can<br \/>\nbe represented as a weighted combination (with weights a sequence of independent standard<br \/>\nGaussian random variables) of a sequence of deterministic continuous functions that is<br \/>\nalmost surely convergent in a Banach space of continuous functions. A representation of<br \/>\nthis type is said to be optimal when the norm of the tail of the series converges to zero<br \/>\nas fast as possible. X(t) is associated to a sequence of &#8220;l-numbers&#8221;, which determine this<br \/>\nfastest possible rate, and the asymptotic behavior of the latter sequence can be estimated<br \/>\nby using operator theory; also, it is worth noticing that the latter behavior is closely<br \/>\nconnected with the behavior of small ball probabilities of {X(t)}t?[0,1]N. The main three<br \/>\ngoals of our talk are the following: (a) to connect the Holder regularity<br \/>\nof {X(t)}t?[0,1]N with the rate of convergence of its l-numbers; (b) to show that<br \/>\nthe Meyer, Sellan and Taqqu wavelet series representations of fractional Brownian<br \/>\nmotion are optimal; (c) to investigate, for the Riemann-Liouville process<br \/>\n(that is the high frequency part of fractional Brownian motion), the optimality of the<br \/>\nseries representations obtained via the Haar and the trigonometric systems.<\/p>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 18, 2012. 3:10 pm, SC1310<\/p>\n<p>\t        <b>Speaker<\/b>: Rayan Saab, Duke University<\/p>\n<p>\t        <b>Title<\/b>: High Accuracy Finite Frame Quantization Using Sigma-Delta Schemes<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: In this talk, we address the digitization of<br \/>\noversampled signals in the finite-dimensional setting. In particular, we show that by<br \/>\nquantizing the $N$-dimensional frame coefficients of signals in $\\R^d$ using Sigma-Delta<br \/>\nquantization schemes, it is possible to achieve root- exponential accuracy in the<br \/>\noversampling rate $\\lambda:= N\/d$ (even when one bit per measurement is used). These are<br \/>\ncurrently the best known error rates in this context. To that end, we construct a family<br \/>\nof finite frames tailored specifically for Sigma-Delta quantization. Our construction<br \/>\nallows for error guarantees that behave as $e^{-c\\sqrt{\\lambda}}$, where under a mild<br \/>\nrestriction on the oversampling rate, the constants are absolute. Moreover, we show that<br \/>\nharmonic frames can be used to achieve the same guarantees, but with the constants now<br \/>\ndepending on d. Finally, we show a somewhat surprising result whereby random frames<br \/>\nachieve similar, albeit slightly weaker guarantees (with high probability). Finally, we<br \/>\ndiscuss connections to quantization of compressed sensing measurements. This is joint<br \/>\nwork, in part with F. Krahmer and R. Ward, and in part with O. Yilmaz.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 11, 2012. 3:10 pm, SC1310<\/p>\n<p>\t        <b>Speaker<\/b>: Pete Casazza, University of Missouri<\/p>\n<p>\t        <b>Title<\/b>: Algorithms for Threat Detection<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Fusion frames are a recent development in<br \/>\nHilbert space theory which have broad application to modeling problems in distributed<br \/>\nprocessing, visual\/hearing systems, geophones in geophysics, forest fire detection and<br \/>\nmuch more. We will look at recent applications of fusion frames to wireless sensor<br \/>\nnetworks for detecting and intercepting chemical\/biological\/nuclear materials which are<br \/>\nbeing transported. This is a totally new subject and so we will present many more problems<br \/>\nthan solutions. <\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: January 25, 2012. 3:10 pm, SC1310<\/p>\n<p>\t        <b>Speaker<\/b>: Anthony Mays, University of Melbourne<\/p>\n<p>\t        <b>Title<\/b>: A Geometrical Triumvirate of Random Matrices<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: A random matrix is, broadly speaking, a matrix with entries<br \/>\nrandomlychosen from some distribution. In the non-random case eigenvalues<br \/>\ncanoccur anywhere in the complex plane, but, remarkably, random elements<br \/>\nimply predictable behaviour, albeit in a probabilistic sense.<\/p>\n<p>Correlation functions are one measure of a probabilistic characterisation<br \/>\nand we discuss a 5-part scheme, based upon orthogonal polynomials, to<br \/>\ncalculate the eigenvalue correlation functions. We apply this scheme to<br \/>\nthree ensembles of random matrices, each of which can be identified with<br \/>\none of the surfaces of constant Gaussian curvature: the plane, the sphere<br \/>\nand the anti- or pseudo-sphere. We will be using real random matrices,<br \/>\nwhich possess the added complication of having a finite probability of<br \/>\nreal eigenvalues.<\/p>\n<p>This talk aims to be accessible, and to that end we will start with a<br \/>\ngeneral overview of random matrices and then discuss the 5-step method,<br \/>\nhopefully keeping technicalities to a minimum, and with plenty of<br \/>\npictures.<\/p>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2011 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2011<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2011\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 26, 2011. 3:10 pm, SC1310<\/p>\n<p>\t        <b>Speaker<\/b>: Xuemei Chen, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Almost Sure Convergence for the Kaczmarz Algorithm with Random<br \/>\nMeasurements<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: The Kaczmarz algorithm is an iterative method for<br \/>\nreconstructing a signal $x\\in\\R^d$ from an overcomplete collection of<br \/>\nlinear measurements $y_n = \\langle x, \\varphi_n \\rangle$, $n \\geq 1$.<br \/>\nWe prove quantitative bounds on the rate of almost sure exponential<br \/>\nconvergence in the Kaczmarz algorithm for suitable classes of random<br \/>\nmeasurement vectors $\\{\\varphi_n\\}_{n=1}^{\\infty} \\subset \\R^d$.<br \/>\nRefined convergence results are given for the special case when each<br \/>\n$\\varphi_n$ has i.i.d. Gaussian entries and,  more generally, when<br \/>\neach $\\varphi_n\/\\|\\varphi_n\\|$ is uniformly distributed on<br \/>\n$\\mathbb{S}^{d-1}$. This work on almost sure convergence complements<br \/>\nthe mean squared error analysis of Strohmer and Vershynin for<br \/>\nrandomized versions of the Kaczmarz algorithm.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 12, 2011. 3:10 pm, SC1310<\/p>\n<p>\t        <b>Speaker<\/b>: Baili Min, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Approach Regions for Domains in $\\CC^2$ of Finite Type<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Recall the Fatou theorem for the unit disc in $\\CC$. In this talk we<br \/>\nwill see the generalization to the domain in $\\CC^2$. More exactly, we<br \/>\nwill see strongly pseudoconvex domains and those of finite type.<br \/>\nWe are also going to show that the approach regions studied by Nagel,<br \/>\nStein, Wainger and Neff are the best possible ones for the boundary<br \/>\nbehavior of bounded analytic functions, and there is no Fatou theorem<br \/>\nfor complex tangentially broader approach regions.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 5, 2011. 3:10 pm, SC1310<\/p>\n<p>\t        <b>Speaker<\/b>: J. Tyler Whitehouse, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Consistent Reconstruction and Random Polytopes<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 14, 2011. 3:10 pm, SC1310<\/p>\n<p>\t        <b>Speaker<\/b>: Aleks Ignjatovic, University of New South Wales<\/p>\n<p>\t        <b>Title<\/b>: Chromatic Derivatives and Approximations<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Chromatic derivatives are special, numerically robust linear differential<br \/>\noperators which provide a unification framework for a broad class of<br \/>\northogonal polynomials with a broad class of special functions.<br \/>\nThey are used to define chromatic expansions which generalize the Neumann<br \/>\nseries of Bessel functions. Such expansions are motivated by signal processing;<br \/>\nthey provide local signal representation complementary to the global signal<br \/>\nrepresentation given by the Shannon sampling expansion. They were<br \/>\nintroduced for the purpose of designing a switch mode amplifier.<br \/>\nUnlike the Taylor expansion which they are intended to replace, they share<br \/>\nall the properties of the Shannon expansion which are crucial for<br \/>\nsignal processing. Besides being a promissing new tool for signal processing, chromatic<br \/>\nderivatives and expansions have intriguing mathematical properties related to harmonic<br \/>\nanalysis. For example, they naturaly introduce spaces of almost<br \/>\nperiodic functions which corespond to orthogonal polynomials of a very broad class,<br \/>\ncontaining classical<br \/>\nfamilies of orthogonal polynomials. We will alo present an open<br \/>\nconjecture related<br \/>\nto a possible generalization of the Paley Wiener Theorem.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 21, 2011. 3:10 pm, SC1310<\/p>\n<p>\t        <b>Speaker<\/b>: Aleks Ignjatovic, University of New South Wales<\/p>\n<p>\t        <b>Title<\/b>: Chromatic Derivatives and Approximations (Continued)<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Chromatic derivatives are special, numerically robust linear differential<br \/>\noperators which provide a unification framework for a broad class of<br \/>\northogonal polynomials with a broad class of special functions.<br \/>\nThey are used to define chromatic expansions which generalize the Neumann<br \/>\nseries of Bessel functions. Such expansions are motivated by signal processing;<br \/>\nthey provide local signal representation complementary to the global signal<br \/>\nrepresentation given by the Shannon sampling expansion. They were<br \/>\nintroduced for the purpose of designing a switch mode amplifier.<br \/>\nUnlike the Taylor expansion which they are intended to replace, they share<br \/>\nall the properties of the Shannon expansion which are crucial for<br \/>\nsignal processing. Besides being a promissing new tool for signal processing, chromatic<br \/>\nderivatives and expansions have intriguing mathematical properties related to harmonic<br \/>\nanalysis. For example, they naturaly introduce spaces of almost<br \/>\nperiodic functions which corespond to orthogonal polynomials of a very broad class,<br \/>\ncontaining classical<br \/>\nfamilies of orthogonal polynomials. We will alo present an open<br \/>\nconjecture related<br \/>\nto a possible generalization of the Paley Wiener Theorem.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 13, 2011. 4:10 pm, SC1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Hans-Peter Blatt, Katholische University Eichstatt<\/p>\n<p>\t        <b>Title<\/b>: Growth behavior and value distibution of rational approximants<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: We investigate the growth and the distribution of zeros of rational<br \/>\nuniform approximations with numerator degree n and<br \/>\ndenominator degree m(n) for meromorphic functions f on a<br \/>\ncompact set E of the complex plane, where m(n) = o(n\/log n) as n tends to<br \/>\ninfinity. We obtain a Jentzsch-Szeg\ufffd type result, i. e., the zero<br \/>\ndistribution converges weakly to the equilibrium distribution of the<br \/>\nmaximal Green domain of meromorphy of f if the function f has a<br \/>\nsingularity of multivalued character on the boundary of this domain. In the case that f has an essential singularity on this domain, we<br \/>\ncan prove that such a point is an accumulation point of zeros or poles of<br \/>\nbest uniform rational approximants. An outlook is given for the<br \/>\napproximation of f on an interval, where the function is not holomorphic.<br \/>\nApplications for Pad\ufffd approximation are discussed.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 23, 2011. 4:10 pm, SC1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Thomas Hangelbroek, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Boundary effects in kernel approximation and the polyharmonic Dirichlet problem<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: In this talk I will discuss boundary effects in kernel approximation &#8212;<br \/>\nspecifically the pathology of the boundary as it  relates to convergence rates.<br \/>\nAccompanying this I will introduce a new approximation scheme, one<br \/>\nthat delivers theoretically optimal and previously unrealized<br \/>\nconvergence rates by isolating the boundary effects in easily managed integrals.<br \/>\nDriving this is a potential theoretic integral representation derived from<br \/>\n the boundary layer potential solution of the polyharmonic Dirichlet problem.<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2010<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2010\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 29, 2010. 4:10 pm, SC1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Thomas Hangelbroek, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Approximation and interpolation on Riemannian manifolds with kernels<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: In this talk I will present very recent results for interpolation and approximation<br \/>\non compact Riemannian manifolds using kernels. I will introduce a new family of<br \/>\nkernels and discuss the rapid decay of associated Lagrange functions, the Lp stability<br \/>\nof bases for the underlying kernel spaces, the uniform boundedness of Lebesgue constants, the uniform boundedness of the L2 projector in Lp, and progress on specific problems on spheres and SO(3). If time permits, I&#8217;ll discuss how such kernels can be<br \/>\nused to treat highly non-uniform arrangements of data.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 15, 2010. 4:10 pm, SC1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Dominik<br \/>\nSchmid, Institute of Biomathematics and Biometry<br \/>\nat the German Research Center for Environmental Health<\/p>\n<p>\t        <b>Title<\/b>: Approximation on the rotation group<\/p>\n<p>\n\t\t\t<b>Abstract<\/b>: Scattered data approximation problems on the rotation group naturally arise in various fields in science in engineering. After<br \/>\nintroducing such problems, we briefly present different approaches to handle such questions. By considering one of these approaches in more detail, we will encounter so-called Marcinkiewicz-Zygmund inequalities. These inequalities provide a norm equivalence between the continuous and discrete $L^p$- norm of certain basis functions and<br \/>\nare a very powerful tool in order to answer important questions that come along with the approximation of<br \/>\nscattered data on the underlying structure. We will present the main tools and techniques<br \/>\nthat enable us to establish such inequalities in our setting of the rotation group.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 30, 2010. 4:10 pm, room TBA.<\/p>\n<p>\t        <b>Speaker<\/b>: Hendrik Speleers, Catholic University of Leuven<\/p>\n<p>\t        <b>Title<\/b>: Convex splines over triangulations<\/p>\n<p>\t\t\t<b>Abstract<\/b>: Convexity is often required in the design of surfaces. Typically, a nonlinear optimization problem arises, where the objective function controls the fairness of the surface and the constraints include convexity conditions. We consider convex polynomial spline functions defined on triangulations. In general, convexity conditions on polynomial patches are nonlinear. In order to simplify the<br \/>\noptimization problem, it is advantageous to have linear conditions. We present a simple construction to generate<br \/>\nsets of sufficient linear convexity conditions for polynomials defined on a triangle. This general approach<br \/>\nsubsumes the known sets of linear conditions in the literature. In addition, it allows us to give a geometric interpretation, and we can easily construct sets of linear conditions that are symmetric..\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 27, 2010. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Abey Lopez, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Multiple orthogonal polynomials on star like sets<\/p>\n<p>\t\t\t<b>Abstract<\/b>: I will describe different asymptotic properties of multiple orthogonal polynomials associated with measures supported on a star centered at the origin with equidistant rays. The ratio asymptotic behavior can be described with the help of a certain compact Riemann surface of genus zero. The nth root asymptotic behavior and zero asymptotic distribution are described in terms of the solution to a<br \/>\nvector equilibrium problem for logarithmic potentials. All the necessary definitions will be properly introduced. Some conjectures about the<br \/>\nlimiting behavior of the recurrence coefficients associated with these polynomials will be mentioned. This work complements recent investigations of Aptekarev, Kalyagin and Saff on strong asymptotics of monic polynomials generated by a three-term recurrence relation of arbitrary order..\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 23, 2010. 3:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Radu Balan, University of Maryland<\/p>\n<p>\t        <b>Title<\/b>: Signal Reconstruction From Its Spectrogram<\/p>\n<p>\t\t\t<b>Abstract<\/b>: This paper presents a framework for discrete-time signal<br \/>\nreconstruction from absolute values of its short-time Fourier<br \/>\ncoefficients. Our approach has two steps. In step one we reconstruct a<br \/>\nband-diagonal matrix associated to the rank-one operator $K_x=xx^*$.<br \/>\nIn step two we recover the signal $x$ by solving an optimization<br \/>\nproblem. The two steps are somewhat independent, and one purpose of<br \/>\nthis talk is to present a framework that decouples the two problems.<br \/>\nThe solution to the first step is connected to the problem of<br \/>\nconstructing frames for spaces of Hilbert-Schmidt operators. The<br \/>\nsecond step is somewhat more elusive. Due to inherent redundancy in<br \/>\nrecovering $x$ from its associated rank-one operator $K_x$, the<br \/>\nreconstruction problem allows for imposing supplemental conditions. In<br \/>\nthis paper we make one such choice that yields a fast and robust<br \/>\nreconstruction. However this choice may not necessarily be optimal in<br \/>\nother situations. It is worth mentioning that this second step is<br \/>\nrelated to the problem of finding a rank-one approximation to a matrix<br \/>\nwith missing data.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 20, 2010. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Bernhard Bodmann, University of Houston<\/p>\n<p>\t        <b>Title<\/b>: Combinatorics and complex equiangular tight frames<\/p>\n<p>\t\t\t<b>Abstract<\/b>: Equiangular tight frames combine a curious mix of spectral<br \/>\nand geometric properties that makes them a fascinating topic<br \/>\nin matrix theory. Moreover, these frames turn out to be optimal<br \/>\nfor certain applications in signal communications.<br \/>\nSeidel has pioneered the use of combinatorial constructions<br \/>\nof such frames for real Hilbert spaces. In a recent work with<br \/>\nHelen Elwood, we follow Seidel&#8217;s footsteps into a corresponding<br \/>\ncombinatorial characterization of complex equiangular tight frames.<br \/>\nTo this end, we relate the construction of such frames to Hermitian<br \/>\nmatrices with two eigenvalues which contain $p$th roots of unity.<br \/>\nWe deduce necessary conditions for the existence of complex<br \/>\nSeidel matrices, under the assumption that $p$ is prime.  Explicitly<br \/>\nexamining  the necessary conditions for smallest values of $p$<br \/>\nrules out the existence of many such frames with a number of<br \/>\nvectors less than 50. Nevertheless, there are examples, which<br \/>\nwe confirm by constructing examples.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 13, 2010. 3:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Wojciech Czaja, University of Maryland<\/p>\n<p>\t        <b>Title<\/b>: Multispectral imaging techniques for mapping molecular processes within<br \/>\nthe human retina<\/p>\n<p><b>Abstract<\/b>: We developed multispectral noninvasive fluorescence imaging techniques of<br \/>\nthe human retina. This is done by means of modifying standard fundus<br \/>\ncameras by adding selected interference filter sets. The resulting<br \/>\nmultispectral datasets are processed by novel dimension reduction and<br \/>\nclassification algorithms. These algorithms resulted from a combination of<br \/>\nthe theory of frames with state of the art kernel based dimension<br \/>\nreduction methods. Examples of applications of these techniques include<br \/>\ndetection of photoproducts in early Age-related Macular Degeneration, or<br \/>\nmapping and monitoring macular pigment distributions.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: March 15, 2010. 3:00 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Simon Foucart, University Pierre et Marie Curie<\/p>\n<p>\t        <b>Title<\/b>: Gelfand widths, pre-Gaussian random matrices, and joint sparsity<\/p>\n<p>\t\t\t<b>Abstract<\/b>: In this talk, we explore three topics in Compressive Sensing. For the first topic, we outline the role of Gelfand widths before presenting natural (i.e., based only on ideas from Compressive Sensing) arguments to derive sharp estimates for the Gelfand widths of $\\ell_p$-balls in $\\ell_q$ when $0 &lt; p \\le 1$ and $p &lt; q \\le 2$. For the second topic, we show<br \/>\nhow sparse recovery via $\\ell_1$-minimization can be achieved with pre-Gaussian random matrices using the<br \/>\nminimal (up to constants) number of measurements. For the third topic, we<br \/>\nexplain why joint-sparse recovery by mixed $\\ell_{1,2}$-minimization is not uniformly better than separate recovery by $\\ell_1$-minimization, thus extending the equivalence between real and complex null space properties.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t\t\t<b>Time<\/b>: February 2, 2010.  4:10 pm, room 1312.<\/p>\n<p>\t\t\t<b>Speaker<\/b>: Luis Daniel Abreu, CMUC, University of Coimbra Portugal<\/p>\n<p>\t\t\t<b>Title<\/b>: Time-frequency analysis of Bergman-type spaces<\/p>\n<p>\t\t\t<b>Abstract<\/b>: In this talk we will present a real variable approach to some spaces of area measure (Bergmann-type) in the plane and in the upper-half plane. Underlying this approach is the Gabor transform with Hermite functions and the wavelet transform with Laguerre functions.<\/p>\n<p>We will show how our method leads to new results. Some of them would be out of reach using &#8220;pure&#8221; Complex Analysis and only recent advances in time-frequency analysis (e.g. the structure of Gabor frames) made it possible to prove them<\/p>\n<p>1) New(?) orthogonal functions in two variables with respect to area measure.<\/p>\n<p>2) Sampling and interpolation in Fock spaces of polyanalytic functions (this is connected to recent work of Gr&ouml;chenig and Lyubarskii).<\/p>\n<p>3) Beurling density conditions for sampling and interpolation in Bergmann-type spaces.<\/p>\n<p>4) Necessary density conditions for wavelet frames with Laguerre functions.\n\t\t\t<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2009 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2009<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2009\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 21, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Deanna Needell, University of California at Davis<\/p>\n<p>\t        <b>Title<\/b>: Greedy Algorithms in Compressed Sensing<\/p>\n<p>\t\t\t<b>Abstract<\/b>: Compressed sensing is a new and fast growing field of applied mathematics that addresses the shortcomings of conventional signal compression. Given a signal with few nonzero coordinates relative to its dimension, compressed sensing seeks to reconstruct the signal from few nonadaptive linear measurements. As work in this area developed, two major approaches to the problem emerged, each with its own set of advantages and<br \/>\ndisadvantages. The first approach, L1-Minimization, provided strong results, but lacked the speed of the second, the greedy approach. The greedy approach, while providing a fast runtime, lacked stability and uniform guarantees.  This gap between the approaches led<br \/>\nresearchers to seek an algorithm that could provide the benefits of both.  We bridged this gap and provided a breakthrough algorithm, called Regularized Orthogonal Matching Pursuit (ROMP). ROMP is the first algorithm to provide the stability and uniform guarantees similar to those of L1-Minimization, while providing speed as a greedy approach. After analyzing these results, we developed the algorithm Compressive Sampling Matching Pursuit (CoSaMP), which improved upon the guarantees of ROMP. CoSaMP is the first<br \/>\nalgorithm to have provably optimal guarantees in every important aspect. This talk will provide a brief introduction to<br \/>\nthe area of compressed sensing and a discussion of these two recent developments.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 16, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Johann S. Brauchart, Graz University of Technology<\/p>\n<p>\t        <b>Title<\/b>: On an external field problem on the sphere<\/p>\n<p>\t\t\t<b>Abstract<\/b>: Consider an isolated charged sphere in the presence of an external field exerted by a point charge over the North Pole (or, more generally, a line charge on the polar axis). The model of interaction is that of the Riesz $s$-potential $1 \/ r^s$ with $d-2 &lt; s &lt; d$. (Here, $d+1$ is the dimension of the embedding space.) We present results from joint work with Peter Dragnev (IPFW) and Ed Saff concerning the weighted extremal measure solving this external<br \/>\nfield problem and its properties (support, representation, potential). Interesting phenomena occur in the case $s to d-2$. Essential<br \/>\ningredients are the signed equilibrium on a spherical cap associated with the given external field (i.e. the signed measure whose potential is<br \/>\nconstant everywhere on this spherical cap), the Mhaskar-Saff functional (which yields the aforementioned constant), and the technique of iterated balayage to single out the spherical cap whose signed equilibrium becomes the weighted extremal measure.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 7, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Brody Johnson, St. Louis University<\/p>\n<p>\t        <b>Title<\/b>: Finite-Dimensional Wavelet Systems on the Torus<\/p>\n<p>\t\t\t<b>Abstract<\/b>: The literature is rich with respect to treatments of wavelet bases for the real line.  Early in the development of this wavelet theory some authors also considered wavelet systems for the torus; however, there has been considerably less work in this direction. Here, we consider a notion of finite-dimensional wavelet systems on the torus which, in many ways, adapts the theory of multiresolution<br \/>\nanalysis from the line to the torus.  The orthonormal wavelet systems produced with this approach will be shown to offer arbitrarily close approximation of square-integrable functions on the torus.  The<br \/>\ntalk will include a brief introduction to wavelet theory on the line.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: March 31, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Guillermo Lopez Lagomasino, Universidad Carlos III de Madrid<\/p>\n<p>\t        <b>Title<\/b>: On a class of perfect systems<\/p>\n<p>\t\t\t<b>Abstract<\/b>: In 1873, CH. Hermite published the paper &#8220;On the exponential function&#8221; where he proved the transcendence of the number e. This paper is considered to mark the origin of the analytic theory of numbers. Years later, around 1936, on the basis of the method used by Hermite for systems of exponential functions, K. Mahler introduced the notion of perfect systems of first and second type. These are systems of functions<br \/>\nsatisfying certain algebraic independence for any polynomial combination of them. Until recently, very few special cases of systems of functions<br \/>\nwere known to be perfect. In 1980, E. M. Nikishin introduced what is now called a Nikishin system. These are systems of Markov type functions generated by measures<br \/>\nsupported on the real line. He also proved normality for such systems of functions when the degrees of the polynomials in the polynomial combination are equal (a system is said to be perfect if it is normal for polynomials of arbitrary degree). On the basis of this the question was posed as to whether or not Nikishin systems are perfect. In this talk we give a positive answer to the question.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: March 24, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Peter Massopust, Technical University of Munich<\/p>\n<p>\t        <b>Title<\/b>: Complex B-Splines: Theme and Variations<\/p>\n<p>\t\t\t<b>Abstract<\/b>: The concept of a complex B-spline is introduced and some of its properties are discussed. Particular emphasis is placed on an interesting relation to Dirichlet averages that allows the derivation of a generalized Hermite-Gennochi formula. Using ridge functions, an extension of univariate complex B-splines to the multivariate setting is given. In<br \/>\nthis context, double Dirichlet averages are employed to define  and compute moments of multivariate complex B-splines. Applications of complex B-splines to<br \/>\ncertain statistical processes are presented. This is joint work with Brigitte Forster.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: March 10, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Burcin Oktay, Bahkesir University, Turkey<\/p>\n<p>\t        <b>Title<\/b>: Approximation by Some Extremal Polynomials over Complex Domains<\/p>\n<p>\t\t\t<b><a href=\"http:\/\/www.math.vanderbilt.edu\/~calendar\/dyncal\/attachments\/brc-abstract.pdf\">Download Abstract<\/a><\/b>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 24, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Bradley Currey, Saint Louis University<\/p>\n<p>\t        <b>Title<\/b>: Heisenberg Frame Sets<\/p>\n<p>\t\t\t<b><a href=\"http:\/\/www.math.vanderbilt.edu\/~calendar\/dyncal\/attachments\/currey_abstract.pdf\">Download Abstract<\/a><\/b>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 5, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Alexander I. Aptekarev, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow<\/p>\n<p>\t        <b>Title<\/b>: Rational approximants for vector of analytic functions with branch points<\/p>\n<p>\t\t\t<b>Abstract<\/b>: Given a vector of power series expansions at infinity point which allows<br \/>\nanalytic continuation along any path of complex plane non-intersecting with a finite set of branch points. For<br \/>\nthis set of functions the Hermite-Pade rational approximants are considered. For<br \/>\nthe case of one function ? the conjecture of Nuttall (that poles of the diagonal Pade approximants of function<br \/>\nwith branch points tend to the cut of minimal capacity making the function single-valued) was proven by Stahl.<br \/>\nWe discuss a generalization for the vector case.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: January 20, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Andriy Prymak, University of Manitoba<\/p>\n<p>\t        <b>Title<\/b>: Approximation of dilated averages and K-functionals<\/p>\n<p>\t\t\t<b><a href=\"http:\/\/www.math.vanderbilt.edu\/~calendar\/dyncal\/attachments\/prymak-abstract.pdf\">Download Abstract<\/a><\/b>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t\t\t<b>Time<\/b>: January 13, 2009. 4:10 pm, room 1312.<\/p>\n<p>\t\t\t<b>Speaker<\/b>: Nikos Stylianopoulos, University of Cyprus<\/p>\n<p>\t        <b>Title<\/b>: Fine asymptotics for Bergman orthogonal polynomials over domains with corners<\/p>\n<p>\t\t\t<b><a href=\"http:\/\/www.math.vanderbilt.edu\/~calendar\/dyncal\/attachments\/FineAsympSeminar.pdf\">Download Abstract<\/a><\/b>\n\t\t\t<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2008 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2008<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2008\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: December 9, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Mike Wakin, Colorado School of Mines<\/p>\n<p>\t        <b>Title<\/b>: Compressive Signal Processing using Manifold Models<\/p>\n<p>\t\t\t<b>Abstract<\/b>: Compressive Sensing (CS) is a framework for signal acquisition built on<br \/>\nthe premise that a sparse signal can be recovered from a small number of random linear measurements. CS is robust<br \/>\nin two important ways: (1) the error in recovering any signal is proportional to its proximity to a sparse signal, and (2) the error in recovering a signal is proportional to the amount of added noise in the measurement vector.<br \/>\nIn this talk I will describe how a geometric interpretation of CS leads naturally to an extension of CS beyond<br \/>\nsparse models to incorporate low-dimensional manifold models for signals. I will discuss how small numbers of<br \/>\nrandom measurements can guarantee a stable embedding of a manifold-modeled signal family in the compressive<br \/>\nmeasurement space, how this leads to analogous robustness guarantees to sparsity-based CS, and how this makes<br \/>\npossible new applications in classification, manifold learning, and multi-signal acquisition.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: December 2, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Truong-Thao Nguyen, City University of New York<\/p>\n<p>\t        <b>Title<\/b>: The tiling phenomenon of 1-bit feedback analog-to-digital converters<\/p>\n<p>\t\t\t<b>Abstract<\/b>: The circuit technology of data acquisition has introduced a high performance technique of analog-to-digital conversion based on the use of coarse quantization compensated by feedback, and called Sigma-Delta modulation. However, while this technique enables data conversion of high resolutions in practice, its design has been mostly developed empirically and its rigorous analysis escapes from standard signal theories. The<br \/>\nfundamental difficulty lies in the existence of a nonlinear operation (namely, the quantization) in a recursive<br \/>\nprocess (physically implemented by the feedback). This prevents a tractable and explicit determination<br \/>\nof the output in terms of the input of the system. Partial answers to this difficult problem have been recently<br \/>\nfound as Sigma-Delta modulators have been observed to carry some new interesting mathematical properties. The state vector of the feedback system appears to systematically remain in a *tile* of the state space. This has been the starting point to new research developments involving mathematical tools that are very unusual to the signal processing and communications communities, while simultaneously bringing new results to applied mathematics. This includes<br \/>\nergodic theory, dynamical systems, as well as spectral theory. In this talk, we give an overview on this research, including the<br \/>\nmathematical origin of this tiling phenomenon and its consequence to the rigorous signal analysis of Sigma-delta modulators.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 18, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Jeff Hogan, University of Arkansas<\/p>\n<p>\t        <b>Title<\/b>: Clifford analysis and hypercomplex signal processing<\/p>\n<p>\t\t\t<b>Abstract<\/b>: In this talk we attempt to synthesize recent progress made in the mathematical and electrical engineering communities on topics in Clifford analysis and the processing of color images (for example), in particular the construction and application of Clifford-Fourier transforms designed to treat vector-valued signals. Emphasis<br \/>\nwill be placed on the two-dimensional setting where the<br \/>\nappropriate underlying Clifford algebra is the set of quaternions. We&#8217;ll<br \/>\nconclude with some results and problems in the construction of discrete wavelet bases for color images, and the difficulties encountered<br \/>\nin constructing the correct Fourier kernels in dimensions 3 and higher. (This talk is part of the Shanks workshop &#8216;Nonlinear Models in Sampling Theory&#8217;.)\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 11, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Simon Foucart, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: A Survey on the Mathematics of Compressed Sensing<\/p>\n<p>\t\t\t<b>Abstract<\/b>: This talk will give an overview of some chosen topics in the theory of Compressed Sensing. Mathematically speaking, one aims at finding sparsest solutions of severely underdetermined linear systems of equations via robust and efficient algorithms. I shall especially discuss the advantages and drawbacks of algorithms based<br \/>\non $\\ell_q$-minimization for $0 &lt; q &lt; 1$ compared to the classical $\\ell_1$-minimization. This will be based on results &#8212; both of positive and negative nature &#8212; recently obtained by Chartrand et al., by Gribonval et al., and by Lai and myself.\n\t\t\t<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 4, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Brigitte Forster, Technische Universit&auml;t M&uuml;nchen<\/p>\n<p>\t        <b>Title<\/b>: Shift-invariant spaces from rotation-covariant functions<\/p>\n<p>\t\t\t<b>Abstract<\/b>: We consider shift-invariant multiresolution spaces generated by rotation-covariant functions $\\rho$ in $\\mathbb{R}^2$. To construct corresponding scaling and wavelet functions, $\\rho$ has to be localized with an appropriate multiplier, such that the localized version is an element of $L^2(\\mathbb{R}^2)$. We consider several classes of multipliers and show a new method to<br \/>\nimprove regularity and decay properties of the corresponding scaling functions and wavelets. The<br \/>\nwavelets are complex-valued functions, which are approximately rotation-covariant and therefore behave as Wirtinger differential operators. Moreover, our class of multipliers gives a novel approach for the construction of polyharmonic B-splines with better polynomial reconstruction<br \/>\nproperties. The method works not only on classical lattices, such as the cartesian or the quincunx, but also on hexagonal lattices.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 28, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Rick Chartrand, Los Alamos National Laboratory<\/p>\n<p>\t        <b>Title<\/b>: Nonconvex compressive sensing: getting the most from very little information (and the other way around).<\/p>\n<p>\t\t\t<b>Abstract<\/b>: In this talk we&#8217;ll look at the exciting, recent results showing that most images and other signals can be reconstructed from much less information than previously thought possible, using simple, efficient algorithms. A consequence has been the explosive growth of the new field known as compressive sensing, so called because the results show how a small number of measurements of a signal can be regarded as tantamount<br \/>\nto a compression of that signal. The many potential applications include reducing exposure time in medical imaging, sensing devices that can collect much less data in the first place instead of<br \/>\ncollecting and then compressing, getting reconstructions from what seems like insufficient data (such as EEG), and very simple compression methods that are effective for streaming data<br \/>\nand preserve nonlinear geometry. We&#8217;ll see how replacing the convex optimization problem typically used in this field with a nonconvex variant has the effect of reducing still further the number of measurements needed to reconstruct a signal. A very surprising result is that a simple algorithm, designed only for finding one of the many local minima of the optimization problem, typically finds the global minimum. Understanding this is an interesting and challenging theoretical problem. We&#8217;ll<br \/>\nsee examples, and discuss algorithms, theory, and applications.\n\t  <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 14, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Akram Aldroubi, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Compressive Sampling via Huffman codes.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: Let $x$ be some vector in $\\R^n$ with at most $k$ much less than $n$ nonzero components (i.e., $x$ is a sparse vector). We wish to determine $x$ from inner products $\\{y_i=a_i\\dot x\\}_{i=1}^m$, the samples. How can we determine a set of $m$ vectors $\\{a_i\\}$ such that $x$ can be completely determined from the samples $\\{y_i=a_i\\dot x\\}_{i=1}^m$ by a computationally<br \/>\nefficient, stable algorithm. The recent theory of compressed sampling addresses this problem using two main approaches: the geometric approach<br \/>\nand the combinatorial approach. In this talk I will present a new information theoretic approach and use results<br \/>\nfrom the theory of Huffman codes to construct a sequence of binary sampling vectors to determine a sparse vector $x$. Unlike the standard approaches, this new method is sequential and adaptive in the sense<br \/>\nthat each sampling vector depends on the previous sample value. The number of measurements we need is no more than $O(k\\log n)$ and the reconstruction is $O(k)$ which is better than any other method.\n\t\t\t<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 7, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Andrii Bondarenko, Kyiv National Taras Shevchenko University<\/p>\n<p>\t        <b>Title<\/b>: New asymptotic estimates for spherical designs.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: The equal weight quadrature formula on the sphere S^n which is exact for all polynomials of n+1 variables and of total degree t is called spherical t-design. We will consider two approaches for constructing good spherical designs for large parameters n and t, which improve essentially the previous upper bounds for minimal number of points in spherical t-design and confirm the well known conjecture<br \/>\nof Korevaar and Meyers. We will also show the connection of this area with energy problems, lattices and group theory.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 23, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Akram Aldroubi, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Invariance of shift-invariance spaces.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. We will characterize those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are<br \/>\ncharacterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations.\n\t\t  <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 16, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Hendrik Speleers, Katholieke Universiteit Leuven<\/p>\n<p>\t        <b>Title<\/b>: From PS splines to QHPS splines.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: Powell-Sabin (PS) splines are C<sup>1<\/sup>-continuous quadratic macro-elements defined on conforming triangulations. They can be represented in a compact normalized spline basis with a geometrically intuitive interpretation involving control triangles. These triangles can be used to interactively change the shape of a PS spline in a predictable way. In this talk we discuss a<br \/>\nhierarchical extension of PS splines, the so-called quasi-hierarchical Powell-Sabin (QHPS) splines. They are defined on a hierarchical<br \/>\ntriangulation obtained through (local) triadic refinement. For this spline space a compact normalized quasi-hierarchical basis can be constructed. Such a basis<br \/>\nretains the advantages of the PS spline basis: the basis functions have a local support, they form a convex partition of unity, and control triangles can be defined. In addition, they<br \/>\nadmit local subdivision in a very natural way. These properties of QHPS splines are appropriate for local adaptive approximation and modelling.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 9, 2008. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Larry Schumaker, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Dimension of Spline Spaces on T-Meshes.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: A T-mesh $\\Delta$ is obtained from a tensor-product mesh by removing certain edges to create a partition with one or more T-nodes. Given $0 \\le r_1 \\le d_1$ and $0 \\le r_2 \\le d_2$, we define an associated spline space $S^{r_1,r_2}_{d_1,d_2}(\\Delta)$ as the space of functions in $C^{r_1,r_2}$ whose restrictions to the rectangles of the<br \/>\npartition are tensor polynomials in $P_{d_1,d_2}$. In this talk we discuss the problem of computing the dimension of these spline spaces. In particular, we<br \/>\ngive various lower bounds which lead to exact formulae in some cases. We also discuss extensions to more than two variables, and also some results for more general L-meshes. Finally, we conclude with several enticing open questions.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 29, 2008. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Maxym Yattselev, INRIA Sophia Antipolis<\/p>\n<p>\t        <b>Title<\/b>: Non-Hermitian Orthogonal Polynomials with Varying Weights on an Arc.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: We consider multipoint Pade approximation of Cauchy transforms of complex measures. We show that if the support of a measure is a smooth Jordan arc and the density of this measure is sufficiently smooth, then the diagonal multipoint Pade approximants associated with interpolation schemes that satisfy special symmetry property with respect to this arc converge locally uniformly to the approximated Cauchy transform. The existence of such interpolation schemes is<br \/>\nproved for the case where support is an analytic Jordan arc. The asymptotic behavior of Pade approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 15, 2008. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Doug Hardin, Vanderbilt University<\/p>\n<p>\t        <b>Title<\/b>: Discrete minimum energy problems and minimal Epstein zeta functions.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: We consider  asymptotic  properties (as $N\\to \\infty$) of `ground state&#8217; configurations of $N$ particles restricted to a $d$-dimensional compact set $A\\subset {\\bf R}^p$ that minimize the Riesz $s$-energy functional $$ \\sum_{i\\neqj}\\frac{1}{|x_{i}-x_{j}|^{s}} $$ for $s&gt;0$. The first part<br \/>\nof this talk will consist of an overview of recent results obtained by the `Vanderbilt minimum energy group&#8217; (aka, the &#8216;couch potatoes&#8217;);  in the second half I will present related<br \/>\nresults and conjectures of Cohn, Elkies and Kumar and to recent results of Sarnak and Str&ouml;mbergsson concerning minimal zeta functions in dimensions 8 and 24.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 8, 2008. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Razvan Teodorescu, Los Alamos National Laboratory.<\/p>\n<p>\t        <b>Title<\/b>: Planar Harmonic Growth with Orthogonal Polynomials.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: This talk will cover recent connections between the theory of orthogonal polynomials with deformed Bargmann kernel and harmonic growth of bounded domains. Singular limits and refined asymptotics will also be discussed.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 26, 2008. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Qiang Wu, Duke University.<\/p>\n<p>\t        <b>Title<\/b>: Dimension Reduction in Supervised Learning.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: Dimension reduction in supervised setting aims at inferring the data structure that are most relevant to the prediction of the labels. It can be motivated from either predictive models or descriptive models. Starting from a predictive model, we showed the gradient outer product matrix contains the information of relevant features and predictive dimensions. Several well known feature selection and dimension reduction methods follow this criterion either<br \/>\nimplicitly or explicitly. We designed an algorithm of learning gradients specifically for the small sample size setting using kernel regularization. The asymptotic analysis shows the<br \/>\nconvergence depends only on the intrinsic dimension of the data and can be fast if the underlying data concentrate on a low dimensional manifold. The gradient estimate was successfully applied to feature selection, dimension reduction, estimation<br \/>\nof conditional dependency and task similarity in high dimensional data analysis. Sliced inverse regression (SIR) is a well known and widely used dimension reduction methods in statistics community. It is motivated from a descriptive model. We studied the relation between the gradient out product matrix and covariance matrix of the inverse regression function and found they are locally equivalent in certain sense. This observation not only helps clarify the theoretical comparison<br \/>\nbetween these two methods but also motivates a new algorithm. We developed localized sliced inverse<br \/>\nregression (LSIR) for dimension reduction which overcomes the degeneracy problem of original SIR and has the<br \/>\nadvantage of finding clustering structure in classification problems.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 19, 2008. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Abey Lopez, Vanderbilt University.<\/p>\n<p>\t        <b>Title<\/b>: Asymptotic Behavior of Greedy Energy Configurations.<\/p>\n<p>\t\t\t<b>Abstract<\/b>: In this talk we will discuss some results about the asymptotic behavior of certain point configurations called Greedy Energy (GE) points. These points form a sequence which is generated by means of a greedy algorithm, which is an energy minimizing construction. The notion of energy that we consider comes from the Riesz potentials V=1\/r^{s} in R^{p}, where s&gt;0 and r denotes the Euclidian distance. It turns out that for certain values of the<br \/>\nparameter s, these configurations behave asymptotically like Minimal<br \/>\nEnergy (ME) configurations. This property will also be discussed in more<br \/>\nabstract contexts like locally compact Hausdorff spaces. For other values of s, GE and ME configurations<br \/>\nexhibit different asymptotic properties, for example for s&gt;1 on the unit circle. We will discuss other questions<br \/>\nlike second order asymptotics on the unit circle and weighted Riesz potentials on unit spheres.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 12, 2008. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Justin Romberg, Georgia Tech.<\/p>\n<p>\t        <b>Title<\/b>: Compressed Sensing for Next-Generation Signal Acquisition.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: From decades of research in signal processing, we have learned that<br \/>\nhaving a good signal representation can be key for tasks such as<br \/>\ncompression, denoising, and restoration. The new theory of Compressed<br \/>\nSensing (CS) shows us how a good representation can fundamentally aid<br \/>\nus in the acquisition (or sampling) process as well. In this talk will<br \/>\noutline the main theoretical results in CS and discuss how the ideas<br \/>\ncan be applied in next-generation acquisition devices. The CS paradigm<br \/>\ncan be summarized neatly: the number of measurements (e.g., samples)<br \/>\nneeded to acquire a signal or image depends more on its inherent<br \/>\ninformation content than on the desired resolution (e.g., number of<br \/>\npixels). The CS theory typically requires a novel measurement scheme<br \/>\nthat generalizes the conventional signal acquisition process: instead<br \/>\nof making direct observations of the signal, for example, an<br \/>\nacquisition device encodes it as a series of random linear projections. The theory of CS, while still in its developing stages, is far-<br \/>\nreaching and draws on subjects as varied as sampling theory, convex<br \/>\noptimization, source and channel coding, statistical estimation,<br \/>\nuncertainty principles, and harmonic analysis. The applications of CS<br \/>\nrange from the familiar (imaging in medicine and radar, high-speed<br \/>\nanalog-to-digital conversion, and super-resolution) to truly novel<br \/>\nimage acquisition and encoding techniques.\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2007 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2007<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2007\">\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: December 5, 2007. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Tom Lyche, University of Oslo.<\/p>\n<p>\t        <b>Title<\/b>: New Formulas for Divided Differences and Partitions of a Convex Polygon.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: Divided differences are a basic tool in approximation theory and numerical<br \/>\nanalysis: they play an important role in interpolation and approximation by polynomials and in spline theory. So<br \/>\nit is worthwhile to look for identities that are analogous to identities for derivatives. An example is the<br \/>\nLeibniz rule for differentiating products of functions. This rule was generalized to divided differences by Popoviciu and Steffensen 70 years ago. To our surprise it was<br \/>\ndiscovered that there were no analog of a 150 year old formula for differentiating composite functions (Faa di<br \/>\nBruno&#8217;s formula) and for differentiating the inverse of a function. In this talk I will discuss chain rules and<br \/>\ninverse rules for divided differences. The inverse rule turns out to have a surprising and beautiful<br \/>\nstructure: it is a sum over partitions of a convex polygon into smaller polygons using only nonintersecting<br \/>\ndiagonals. This provides a new way of enumerating all partitions of a convex polygon with a specified number of<br \/>\ntriangles, quadrilaterals, and so on. The talk is based on joint work with Michael Floater.f new infinite product<br \/>\nrepresentations for trigonometric and hyperbolic functions that have not been known before.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 27, 2007. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Yu. A. Melnikov, Middle Tennessee State University.<\/p>\n<p>\t        <b>Title<\/b>: An innovative approach to the derivation of infinite product representations of elementary functions.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: We will report on a curious outcome from the classical method for the<br \/>\nconstruction of Green&#8217;s functions for Laplace equation. An innovative technique is developed for obtaining<br \/>\ninfinite product representations of elementary functions. Some standard boundary value problems are considered posed for two-dimensional Laplace equation on regions of regular configuration. Classical<br \/>\nanalytic forms of Green&#8217;s functions for such problems are compared against those obtained by the method of images. This<br \/>\nyields a number of new infinite product representations for trigonometric and hyperbolic functions that have not<br \/>\nbeen known before.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: November 13, 2007. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Minh N. Do, University of Illinois at Urbana-Champaign.<\/p>\n<p>\t        <b>Title<\/b>: Sampling Signals from a Union of Subspaces.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: One of the fundamental assumptions in traditional sampling theorems is that the signals to be sampled come from a single vector space (e.g. bandlimited functions). However, in many cases of practical interest the sampled signals actually live in a union of subspaces. Examples include piecewise polynomials, sparse approximations, nonuniform splines, signals with unknown spectral support, overlapping echoes with unknown delay and amplitude, and<br \/>\nso on. For these signals, traditional sampling schemes are either inapplicable or highly inefficient. In this paper, we study a general sampling<br \/>\nframework where sampled signals come from a known union of subspaces and the sampling operator is linear. Geometrically, the<br \/>\nsampling operator can be viewed as projecting sampled signals into a lower dimensional space, while still preserves all the information. We<br \/>\nderive necessary and sufficient conditions for invertible and stable sampling operators in this framework and show that these conditions are applicable in many cases. Furthermore, we find the minimum sampling requirements for several classes of signals, which indicates the power of the framework. The results in this paper can serve as a guideline for designing new algorithms for many applications in signal processing and inverse problems.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 16, 2007. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Kourosh Zarringhalam, Vanderbilt University.<\/p>\n<p>\t        <b>Title<\/b>: Chaotic Unstable Periodic Orbits, Theory and Applications.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: We will present a control scheme for stabilizing the unstable periodic orbits of chaotic systems and investigate the properties of these orbits. These approximated chaotic unstable periodic orbits are called cupolets (Chaotic Unstable Periodic Orbit-lets). The cupolet transformation can be regarded as an alternative to Fourier and wavelet transformations and can be used in variety of applications such<br \/>\nas data and music compression, as well as image and video processing. We will also investigate<br \/>\nthe shadowability of cupolets and present a shadowing theorem, suitable for computational purposes, that<br \/>\nprovides a way to establish the existence of true periodic and non-periodic orbits near the approximated ones.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 9, 2007. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Simon Foucart, Vanderbilt University.<\/p>\n<p>\t        <b>Title<\/b>: Condition numbers of finite-dimensional frames.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: First, motivated by some problems in spline theory, we will introduce the<br \/>\nnotion of condition number of a basis. We will then review some results on best conditioned bases, and examine<br \/>\nhow they relate to minimal projections. Finally, the notion of condition number will be extended &#8212; in finite<br \/>\ndimension &#8212; to frames. This work is in progress and highlights some intriguing questions in connection with the<br \/>\ngeometry of Banach spaces.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: October 2, 2007. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Carolina Beccari, University of Bologna.<\/p>\n<p>\t        <b>Title<\/b>: Tension-controlled interpolatory subdivision.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: Subdivision generates a smooth curve\/surface as the limit of a sequence of successive refinements applied to an initial polyline\/mesh. Although subdivision curves and surfaces can be generated either through interpolation or approximation of the initial control net, interpolatory refinements have been traditionally considered less attractive than approximatory methods, due to the poor visual quality of their limit shapes. This problem will be addressed taking into account the<br \/>\nnovel notions of non-stationarity and non-uniformity in order to include in subdivision models the important capability of tension control together with the capacity of reproducing<br \/>\nprescribed curves and conic sections, that is peculiar to the NURBS representation. To this aim we will explore the definition of subdivision schemes featured<br \/>\nby the presence of tension parameters associated with the edges in the initial control polygon\/net.Since these parameters give us the possibility of locally adjusting the shape of the limit curve, they can be used both to produce a nice-looking interpolation of the initial control points and to achieve the exact modeling of circular arcs, surfaces of revolution and quadrics.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 25, 2007. 3:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Rene Vidal, Johns Hopkins University.<\/p>\n<p>\t        <b>Title<\/b>: Generalized Principal Components Analysis.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: Over the past two decades, we have seen tremendous advances on the simultaneous segmentation and estimation of a collection of models from sample data points, without knowing which points correspond to which model. Most existing segmentation methods treat this problem as &#8220;chicken-and-egg&#8221;, and iterate between model estimation and data segmentation. This lecture will show that for a wide variety of data segmentation problems (e.g. mixtures of subspaces), the &#8220;chicken-and-egg&#8221; dilemma can be tackled using an<br \/>\nalgebraic geometric technique called Generalized Principal Component Analysis (GPCA). This technique is a<br \/>\nnatural extension of classical PCA from one to multiple subspaces. The lecture will touch upon a few motivating<br \/>\napplications of GPCA in computer vision, such as image\/video segmentation, 3-D motion segmentation or dynamic texture segmentation, but will mainly emphasize the basic theory and algorithmic aspects of GPCA.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 18, 2007. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Romain Tessera, Vanderbilt University.<\/p>\n<p>\t        <b>Title<\/b>: Finding left inverses for a class of operators on l^p(Z^d) with concentrated support.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: We will expose various generalizations of the following recent theorem<br \/>\n(due to Aldroubi, Baskarov, Krishtal): Let A=(a_{x,y}) be a matrix indexed by Z^d x Z^d such that a_{x,y}=0<br \/>\nwhenever |x-y|&gt;m for some m. Assume that A has bounded coefficients and is bounded below as an operator on l^p for some p in [1,infty]. Then it has a left-inverse B which is bounded on l^q for all q in [1,infty]. The proof that we propose is quite different from the one of Aldroubi, Baskarov, Krishtal. It<br \/>\nessentially relies on a basic geometric property of Z^d, and hence works in a more general setting.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: September 11, 2007. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Larry Schumaker, Vanderbilt University.<\/p>\n<p>\t        <b>Title<\/b>: Computing Bivariate Splines in Scattered Data Fitting and the FEM Method.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: A number of useful bivariate spline methods are global in nature, i.e., all of the coefficients of an approximating spline must be computed at the same time. Typically this involves solving a (possible large) system of linear equations. Examples include several well-known methods for fitting scattered data, such as the minimal energy, least-squares, and penalized<br \/>\nleast-squares methods. Finite-element methods for solving boundary-value problems are also of this type. We<br \/>\nshow how these types of globally-defined splines can be<br \/>\nefficiently computed, provided we work with spline spaces with stable local bases.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 19, 2007. 2:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>: Laurent Baratchart, INRIA, Sophia Antipolis.<\/p>\n<p>\t        <b>Title<\/b>: Dirichlet problems and Hardy spaces for the real Beltrami equation.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: Motivated by extremal problems connected with locating the plasma boundary in a Tokamak vessel, we consider Dirichlet problems for the real Beltrami equation: \\partial f\/\\partial{\\bar z}=\\nu\\overline{\\partial f\/\\partial z} on the disk or the annulus. We show the existence of a unique solution with given real part in certain Sobolev spaces of the boundary for bounded measurable nu bounded away from below, the<br \/>\ndensity of traces of solutions on subarcs of the boundary, and the existence of solutions in Hardy-type classes<br \/>\ndefined through the finiteness of L^p means on inner circles. We briefly discuss the analog of classical extremal<br \/>\nproblems in this context.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 17, 2007. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Casey Leonetti, Vanderbilt University.<\/p>\n<p>\t        <b>Title<\/b>: Error Analysis of Frame Reconstruction from Noisy Samples<\/p>\n<p>\t  \t  <b>Abstract<\/b>: This talk addresses the problem of reconstructing a continuous function from a countable collection of samples corrupted by noise. The additive noise is assumed to be i.i.d. with mean zero and variance sigma-squared. We<br \/>\nsample the continuous function f on the uniform lattice (1\/m)Z^d, and show for large enough m that the variance of the error between the frame reconstruction from noisy samples of f and the function f evaluated<br \/>\nat each point x behaves like sigma-squared divided by m^d times a (best) constant C_x. We also prove a similar result in the case that our<br \/>\ndata are weighted-average samples of f corrupted by additive noise. Joint work with Akram Aldroubi and Qiyu Sun.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 11, 2007. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Ju-Yi Yen, Vanderbilt University.<\/p>\n<p>\t        <b>Title<\/b>: Multivariate Jump Processes in Financial Returns.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: We apply a signal processing technique known as independent component<br \/>\nanalysis (ICA) to multivariate financial time series. The main idea of ICA is to decompose the observed time<br \/>\nseries into statistically independent components (ICs). We further assume that the ICs follow the variance gamma<br \/>\n(VG) process. The VG process is evaluated by Brownian motion with drift at a random time given by a gamma process. We build a multivariate VG portfolio model and analyze empirical results of the investment.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: April 4, 2007. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Kasso Okoudjou, University of Maryland.<\/p>\n<p>\t        <b>Title<\/b>: Uncertainty principle for fractals, graphs, and metric measure spaces.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: We formulate and prove weak uncertainty principles for functions defined on fractals, graphs and more generally on metric measure spaces. In particular, this uncertainty inequality is proved under different assumptions such as an appropriate measure growth condition with respect to a specific metric, or in the absence of such a metric, we assume the Poincare inequality and the reverse volume doubling property.\n <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: March 21, 2007. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>: Johann S. Brauchart, Vanderbilt University.<\/p>\n<p>\t        <b>Title<\/b>: Optimal logarithmic energy points on the unit sphere in $\\mathbb{R}^{d+1}$, $d\\geq2$.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: We study minimum energy point charges on the unit sphere in $\\Rset^{d+1}$, $d\\geq2$, that interact according to the logarithmic potential $\\log(1\/r)$, where $r$ is the Euclidean distance between points. Such optimal $N$-point configurations are uniformly distributed as $N\\to\\infty$. We quantify this result by estimating the spherical cap discrepancy of optimal energy configurations. The estimate<br \/>\nis of order $\\mathcal{O}(N^{-1\/(d+2)})$. Essential is an improvement of the lower bound of the optimal logarithmic energy which yields the second term $(1\/d)(\\log N)\/N$ in the asymptotical expansion of the optimal energy. Previously, the latter<br \/>\nhas been known for the unit sphere in $\\mathbb{R}^{3}$ only. From the proof of our discrepancy estimates we get an upper bound for the error of integration for polynomials of degree at most $n$ when using an equally-weighted<br \/>\nnumerical integration rule $\\numint_{N}$ with the $N$ nodes forming an optimal logarithmic energy configuration. This bound is $C_{d} ( N^{1\/d} \/ n )^{-d\/2} \\| p \\|_{\\infty}$ as $n\/N^{1\/d}\\to0$.\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: March 14, 2007. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>:  Elena Berdysheva, University of Hohenheim, Germany.<\/p>\n<p>\t        <b>Title<\/b>: On Tur\\&#8217;an&#8217;s Problem for $\\ell$-1 Radial, Positive Definite Functions.<\/p>\n<p>\t  \t  <b>Abstract<\/b>:  Tur\\&#8217;an&#8217;s problem is to determine the greatest possible value of the<br \/>\nintegral $\\int_{{\\mathbb R}^d}f(x)\\,dx \/ f(0)$ for positive definite functions $f(x)$, $x \\in {\\mathbb R}^d$,<br \/>\nsupported in a given convex centrally symmetric body $D \\subset {\\mathbb R}^d$. In this talk we consider<br \/>\nthe Tur\\&#8217;an problem for positive definite functions of the form $f(x) = \\varphi(\\|x\\|_1)$, $x \\in {\\mathbb R}^d$, with $\\varphi$ supported in $[0,\\pi]$. An essential part of the talk is devoted to the planar<br \/>\ncase ($d=2$), in this case we could settle and solve the corresponding discrete problem. Some of our results are<br \/>\nproved for an arbitrary dimension. Joint work with H. Berens (University of Erlangen-Nuremberg, Germany).\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 14, 2007. 4:10 pm, room 1310.<\/p>\n<p>\t        <b>Speaker<\/b>:  Ming-Jun Lai, University of Georgia.<\/p>\n<p>\t        <b>Title<\/b>: Bivariate Splines for Statistical Applications.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: I will use bivariate splines for functional data analysis and rank restricted<br \/>\napproximation of data.\n\t        <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t        <b>Time<\/b>: February 7, 2007. 4:10 pm, room 1312.<\/p>\n<p>\t        <b>Speaker<\/b>:  Maxim Yattselev, Vanderbilt University.<\/p>\n<p>\t        <b>Title<\/b>: On uniform convergence of AAK approximants.<\/p>\n<p>\t  \t  <b>Abstract<\/b>: In this talk we present some results on uniform convergence of AAK<br \/>\napproximants to functions of the form<br \/>\n$$F(z) = \\int_{[a,b]}\\frac{1}{z-t}\\frac{s_{\\alpha,\\beta}(t)s(t)dt}{\\sqrt{(t-a)(b-t)}}+R(z), \\;\\;\\; \\alpha,\\beta\\in[0,1\/2),$$ where $s_{\\alpha,\\beta}(t)=(t-a)^\\alpha(b-t)^\\beta$, $R$<br \/>\nis a rational function analytic at infinity having no poles on $[a,b]$, and $s$ is a complex-valued Dini<br \/>\ncontinuous nonvanishing function on $[a,b]$ with an argument of bounded variation there.\n\t        <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: January 31, 2007. 4:10 pm, room 1312.<\/p>\n<p>      <b>Speaker<\/b>:  Alexander Aptekarev, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences.<\/p>\n<p>      <b>Title<\/b>: Discrete Entropy of Orthogonal Polynomials.<\/p>\n<p>\t  <b>Abstract<\/b>: Information entropy has been introduced by Shanon as a density<br \/>\nfunctional for measuring of uncertainness of  distributions. In<br \/>\nquantum mechanics this functional is used to provide more sharp<br \/>\nbounds in  uncertainness relations (sharper than Heisenberg<br \/>\nuncertainness relation for the first moments &#8211; i.e. for the<br \/>\nmathematical expectations). Since the density of the distributions<br \/>\nof many classical quantum mechanical systems (oscillators, Coulomb<br \/>\npotential, hydrogen-like atoms) are represented by means of<br \/>\northogonal polynomials, there is a demand from quantum physicists<br \/>\nto compute entropy of orthogonal polynomials. In this talk we<br \/>\npresent some  computational and explicit results.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: January 24, 2007. 4:10 pm, room 1312.<\/p>\n<p>      <b>Speaker<\/b>: Alex Powell, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Finding good dual frames for reconstructing quantized frame expansions.<\/p>\n<p>\t  <b>Abstract<\/b>: This talk will begin by reviewing the basics of Sigma-Delta quantization. Sigma-Delta quantization is an algorithm for digitizing\/rounding the coefficients in a redundant signal expansion. We shall work in the setting of finite frames and address the problem of finding dual frames which are better suited for signal reconstruction than the canonical dual frame.\n      <\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2006 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2006<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2006\">\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: December 5, 2006. 4:00 pm, <!-- not normal time -->room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Peter Grabner, Graz University of Technology.<\/p>\n<p>      <b>Title<\/b>: Periodicity Phenomena in the Analysis of Algorithms and Related Dirichlet Series.<\/p>\n<p>\t  <b>Abstract<\/b>: Average case analysis of algorithms studies the behaviour of an algorithm under a probabilistic model on the data. Many algorithms have a recursive structure, which gives a recursion for the average<br \/>\nperformance. In many cases, the asymptotic behaviour of the solutions of this recursion shows a periodicity in the logarithmic scale, which corresponds to complex poles of the generating Dirichlet series. We discuss a method for acceleration of convergence of such series and give several examples for its application.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: November 28, 2006. 3:00 pm, room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Nikos Stylianopoulos, University of Cyprus.<\/p>\n<p>      <b>Title<\/b>: Finite-term recurrence relations for planar orthogonal polynomials.<\/p>\n<p>\t  <b>Abstract<\/b>: We prove by elementary means that, if the Bergman orthogonal polynomials of a bounded simply-connected planar<br \/>\ndomain, satisfy a finite-term relation, then the domain is algebraic and characterized by the fact that<br \/>\nDirichlet&#8217;s problem with boundary polynomial data has a polynomial solution. This, and an additional compactness<br \/>\nassumption, is known to imply that the domain is an ellipse. In particular, we show that if the Bergman orthogonal polynomials satisfy a three-term relation then the domain is an ellipse. This completes an inquiry started forty years ago by Peter Duren. (A report of joint work with Mihai Putinar.)\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: November 14, 2006. 4:00 pm, room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Yuan Xu, University of Oregon.<\/p>\n<p>      <b>Title<\/b>: Radon transforms, orthogonal polynomials and CT.<\/p>\n<p>\t  <b>Abstract<\/b>: The central problem for computered tomography (CT) is to reconstruct a function<br \/>\n(an image) from a finite set of its Radon projections. We propose a reconstruction algorithm, called OPED, based<br \/>\non Orthogonal Polynomial Expansion on the Disk. The algorithm works naturally with the fan data and can be<br \/>\nimplemented efficiently. Furthermore, it is proved that the algorithm converges uniformly under a mild condition on the function. Numerical experiments have shown that the method is fast, stable, and has a small global error.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: Novmeber 7, 2006. 4:00 pm, room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Darrin Speegle, St. Louis University.<\/p>\n<p>      <b>Title<\/b>: The Feichtinger Conjecture for special classes of frames.<\/p>\n<p>\t  <b>Abstract<\/b>: Feichtinger conjectured that every frame for a Hilbert space can be partitioned<br \/>\ninto the finite union of sets, each of which is a Riesz basis for its closed linear span. It was quickly realized<br \/>\nthat this conjecture was closely related to the paving problem for matrices, and thus to the Kadison-Singer problem. More recently, it has been shown that settling the Feichtinger Conjecture is equivalent to solving the paving problem. In this talk I will review the partial results<br \/>\non the paving problem, primarily by Bourgain and Tzafriri, and translate them into partial results on<br \/>\nthe Feichtinger Conjecture. Then, I will describe the progress that has been made for Gabor frames, wavelet<br \/>\nframes and frames of<br \/>\nexponentials. For these restricted classes of frames, it is not clear whether settling the Feichtinger Conjecture<br \/>\nis equivalent to solving the corresponding paving problems. Despite progress, the Feichtinger Conjecture remains open even in this restricted setting.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: October 10, 2006. 4:00 pm, room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Bruce Atkinson, Samford University.<\/p>\n<p>      <b>Title<\/b>: An introduction to Markovian image models.<\/p>\n<p>\t  <b>Abstract<\/b>: A random field is a probability measure on the set of images, where an image is an<br \/>\nassignment of grey levels to vertices of a graph. We use the Gibbs sampler to realize a field, and explain how<br \/>\nthe sampler is improved if the field is Markovian. We assume a given image is a realization of a Markovian field and the observed image is a local degradation of it. The posterior distribution of the true image, given the degraded one, is also Markovian and a modification of the Gibbs sampler (an analog of simulated annealing) is<br \/>\nused to restore the true image as a maximum likelihood estimate based on the posterior distribution.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: October 3, 2006. 4:00 pm, room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Doug Hardin, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Orthogonal wavelets centered on non-uniform knot sequences.<\/p>\n<p>\t  <b>Abstract<\/b>:We develop a general notion of orthogonal non-uniform wavelets centered on a knot<br \/>\nsequence. As an application, we construct C^0 and C^1 piecewise polynomial multiwavelets for a knot sequence<br \/>\nassociated with a golden-mean refinement scheme.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: September 26, 2006. 4:00 pm, room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Larry Schumaker, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Bounds on the dimension of trivariate spline spaces.<\/p>\n<p>\n\t  <b>Abstract<\/b>:We discuss recent results with Peter Alfeld giving upper and lower bounds on the<br \/>\ndimensions of trivariate spline spaces defined on tetrahedral partitions. The results hold for general partitions<br \/>\nand for all degrees of smoothness r and polynomial degrees d.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: September 19, 2006. 4:00 pm, room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Simon Foucart, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: The Orthogonal Projector Onto Splines &#8212; Ongoing Development.<\/p>\n<p>\n\t  <b>Abstract<\/b>:A few years ago, the long-standing conjecture that the max-norm of the orthogonal<br \/>\nspline projector is bounded independently of the underlying knot sequence was settled. However, a delicate<br \/>\nquestion remains open, namely: what is the exact value [or order] of the bound? I will present some precise estimates for splines of low smoothness. I will also discuss some approaches for answering the previous question.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: September 12, 2006. 4:00 pm, room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Fumiko Futamura, Vanderbilt University<\/p>\n<p>      <b>Title<\/b>: Localized Operators and the Construction of Localized Frames.<\/p>\n<p>\n\t  <b>Abstract<\/b>: A frame for a Hilbert space is a kind of generalized orthonormal basis which is useful in signal processing. A localized frame is a frame whose elements are &#8220;well-localized&#8221;, in the sense that the inner products of their elements decay as the differences of their indices increase. Grochenig in 2004 proved that localized frames for Hilbert spaces extend to frames for a family of associated Banach spaces. We generalize localized frames to the operator setting, and say an operator is<br \/>\nlocalized with respect to given frames if there is an off-diagonal decay of the matrix representation of an<br \/>\noperator with respect to the frames. We prove that operators<br \/>\nlocalized with respect to localized frames are bounded on the same family of Banach spaces, and that they can<br \/>\nbe used in the construction of new localized frames. We also consider the special case where the frames are unitary shifts of a single atom function.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: September 5, 2006. 4:00 pm, room 1310.<\/p>\n<p>      <b>Speaker<\/b>: Mike Neamtu, Vanderbilt University<\/p>\n<p>      <b>Title<\/b>: Splines on Triangulations for CAGD.<\/p>\n<p>\n\t  <b>Abstract<\/b>: In this talk I will discuss the question of whether piecewise (algebraic) polynomials<br \/>\nare the appropriate tools for defining splines in CAGD.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: April 29, 2006. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Ed Saff, Vanderbilt University<\/p>\n<p>      <b>Title<\/b>: Asymptotics for Polynomial Zeros: Beware of Predictions from Plots.<\/p>\n<p>\n\t  <b>Abstract<\/b>:\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: April 20, 2006. 4:10-5 pm, room 1308.<\/p>\n<p>      <b>Speaker<\/b>: David Benko (Western Kentucky University).<\/p>\n<p>      <b>Title<\/b>: Approximation by homogeneous polynomials.<\/p>\n<p>\n\t  <b>Abstract<\/b>: Let K be a convex origin symmetric surface in R^d. Kroo conjectures that any<br \/>\ncontinuous function on K can be uniformly approximated by a sum of two homogeneous polynomials. Using potential<br \/>\ntheory and weighted polynomials we resolve this problem on the plane. We also give a positive answer in higher dimensions under a smoothness condition on K.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: April 11, 2006. 4:10-5 pm, room 1308.<\/p>\n<p>      <b>Speaker<\/b>: Vasily Prokhorov (Univ. South Alabama and Vanderbilt).<\/p>\n<p>      <b>Title<\/b>: On Estimates for the Ratio of Errors in Best Rational Approximation of Analytic Functions.<\/p>\n<p>\n\t  <b>Abstract<\/b>:<br \/>\nLet E be an arbitrary compact subset of the extended complex plane<br \/>\nwith non-empty interior. For a function f continuous on E and<br \/>\nanalytic<br \/>\nin the interior of E denote by $\\rho_n(f; E)$ the least uniform<br \/>\ndeviation<br \/>\nof f on E  from the   class of all rational functions of order at<br \/>\nmost<br \/>\nn. We will show  that  if K is an arbitrary compact subset of the<br \/>\ninterior of E, then $ \\prod_{k=0}^n (\\rho_k(f; K)  \/\\rho_k(f; E) ),$<br \/>\nthe ratio of the errors in best rational approximation,  converges<br \/>\nto<br \/>\nzero geometrically as $n \\to \\infty$ and  the rate of convergence is<br \/>\ndetermined  by the capacity of the condenser  (\\partial E, K).\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: April 4, 2006. 4:10-5 pm, room 1308.<\/p>\n<p>      <b>Speaker<\/b>: Arthur David Snider, University of South Florida.<\/p>\n<p>      <b>Title<\/b>: High Dynamic Range Resampling for Software Radio.<\/p>\n<p>\t  <b>Abstract<\/b>:The classic problem of recovering arbitrary values of a band-limited signal from<br \/>\nits samples has an added compli- cation in software radio applications; namely, the resampling calculations<br \/>\ninevitably fold aliases of the analog signal back into the original bandwidth. The phenomenon is quantifified<br \/>\nby the spur-free dynamic range. We demonstrate how a novel application of the Remez (Parks-McClellan) algorithm<br \/>\npermits optimal signal recovery and SFDR, far surpassing state-of-the-art resamplers.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: March 28,2006. 4:10-5 pm, room 1308.<\/p>\n<p>      <b>Speaker<\/b>: Maxim Yattselev, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Strong asymptotics on a segment and its application to<br \/>\n\t\tmeromorphic and Pad\\&#8217;e approximation (joint work with Prof. L.<br \/>\n\t\tBaratchart, INRIA, Sophia Antipolis, France)<\/p>\n<p>\t  <b>Abstract<\/b>:We consider a strong (Szeg\\H{o}-type) asymptotics for<br \/>\npolynomials orthogonal with varying complex measures on a segment.<br \/>\nWe  take the approach of G. Baxter of transferring the problem to<br \/>\nthe  unit circle and dealing with the symmetric rational functions.<br \/>\nWe  apply this result to obtain the uniform convergence and the<br \/>\ndistribution of poles of meromorphic and Pad\\&#8217;e approximants of<br \/>\ncomplex Cauchy transforms.<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: March 20,2006. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Laurent Baratchart (INRIA).<\/p>\n<p>      <b>Title<\/b>: Bounded Extremal Problems in Hardy Spaces of the ball in $ {\\bf R}^n$.<\/p>\n<p>\t  <b>Abstract<\/b>:Carleman-type integral formulas for the asymptotic recovery of holomorphic functions in the disk from partial boundary data turn out to solve extremal problems where a function given on a subset of the circle is to be best-approximated in the $L2$-norm on that subset by a $H2$- function subject to certain constraints on<br \/>\nthe rest of the circle. We develop the case of a $L2$ constraint and of a pointwise constraint. The approximant can be further characterized as the solution to a spectral Toeplitz equation, and this<br \/>\nformulation carries over to Stein-Weiss divergence free Hardy spaces of the ball in ${\\bf R}^n$ where it solves a similar approximation problem on the<br \/>\nsphere (the case of a half-space is also covered this way via the Kelvin transform). The extremal problem can itself be viewed as a regularization scheme for inverse Dirichlet-Neumann problems.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: February 13, 2006. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Ozgur Yilmaz (University of British Columbia).<\/p>\n<p>      <b>Title<\/b>: The Role of Sparsity in Blind Source Separation. (Shanks Workshop).<\/p>\n<p>\t  <b>Abstract<\/b>: Certain inverse problems can be solved quite efficiently if the solution is known to have a sparse atomic decomposition with respect to some basis or frame in a Hilbert space. One particular example of such an inverse problem is the so-called cocktail party (or blind source separation) problem: Suppose we use a few microphones to record several people speaking simultaneously. How can we separate individual speech signals from these mixtures? In this talk, I will<br \/>\ndescribe an algorithm adressing the blind source separation problem<br \/>\nwhen the number of speakers is larger than the number of available mixtures. The algorithm is based on the key observation that Gabor expansions of speech signals are sparse. The<br \/>\nseparation is done in two stages: First, the &#8220;mixing matrix&#8221; A is estimated<br \/>\nvia clustering. Next, the Gabor coefficients of individual sources are computed by solving many q-norm minimization problems of<br \/>\ntype {min ||x||_q subject to Ax=b}. Several choices for the value of q will be compared.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: February 7, 2006. 4:10-5 pm, room 1308.<\/p>\n<p>      <b>Speaker<\/b>: Yuliya Babenko, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: On asymptotically optimal partitions and the error of approximation by linear and bilinear splines.<\/p>\n<p>\t  <b>Abstract<\/b>: In this talk we shall present exact asymptotics of the optimal error of linear<br \/>\nspline interpolation of an arbitrary function in various settings, in particular for the case of $L_p$-norm, $1\\leq p \\leq \\infty$, and $f \\in C^2([0,1]^2)$, and for the case of $L_{\\infty}$-norm and $f \\in C^2([0,1]^d)$. We shall present review of existing results as well as a series of new ones. Proofs of these results lead<br \/>\nto algorithms for construction of asymptotically optimal sequences of triangulations for linear interpolation.<br \/>\nSimilar results are obtained for near interpolating bilinear splines.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n      <b>Time<\/b>: January 31, 2006. 4-10-5pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Maxym Yattselev, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Meromorphic Approximants for Complex Cauchy Transforms with Polar Singularities.<\/p>\n<p>\t  <b>Abstract<\/b>: We consider a distribution of poles and convergence of meromorphic approximants to<br \/>\nfunctions of the type $$\\int\\frac{d\\mes(t)}{z-t}+R(z),$$ where $R$ is a rational function vanishing at infinity<br \/>\nand $\\mu$ is a complex measure with the regular support on $(-1,1)$ and whose argument is of bounded variation.\n      <\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2005 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2005<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2005\">\n<tr>\n<td valign=\"top\">\n<p>\t  <b>Time<\/b>: December 6, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Casey Leonetti, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Non-Uniform Sampling and Reconstruction From Sampling Sets with Unknown Jitter.<\/p>\n<p>\t  <b>Abstract<\/b>: This talk will<br \/>\naddress the problem of\ufffd non-uniform sampling and reconstruction in the presence of jitter.\ufffd In sampling applications, the countable set X on which a signal f is sampled is not precisely known.\ufffd Two main questions are considered.\ufffd First, if sampling a function f on the countable set X leads to unique and stable reconstruction of f, then when does<br \/>\nsampling on the set X&#8217;, a perturbation of X, also lead to unique and stable reconstruction?\ufffd Second, if we attempt to recover a sampled function f using the reconstruction<br \/>\noperator corresponding to the sampling set X (because the precise<br \/>\nsample points are unknown), is the recovered function a good approximation of the original f?\ufffd Based on work with Akram Aldroubi.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: November 29, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Vincent Lunot, INRIA, France.<\/p>\n<p>      <b>Title<\/b>: A Zolotarev Problem with Application to Microwave Filters.<\/p>\n<p>\t  <b>Abstract<\/b>:\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: November 15,2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Dr. Karin Hunter, University of Stellenbosch, South Africa.<\/p>\n<p>      <b>Title<\/b>: A class of symmetric interpolatory subdivision schemes.<\/p>\n<p>\t  <b>Abstract<\/b>: The well known Dubuc-Deslauriers subdivision masks are symmetric, interpolatory and<br \/>\nsatisfy a certain polynomial filling property. Here we define a class of symmetric interpolatory masks that<br \/>\ninclude the Dubuc-Deslauriers masks and then give a method to generate masks in this class. We conclude by<br \/>\nproviding a condition for convergence of a subdivision scheme for a subset of masks in this class.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: November 8, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Jorge Stolfi, Institute of Computing, State University of Campinas (Brazil).<\/p>\n<p>      <b>Title<\/b>: Splines on the Sphere (A View from the Other Hemisphere).<\/p>\n<p>\t  <b>Abstract<\/b>: Polynomial splines on the sphere with triangular topology were defined and thoroughly<br \/>\nstudied by Alfeld, Neamtu and Schumaker ca. 1996. In this talk we will review the theory of spherical<br \/>\npolynomials, their relation to spherical harmonics, and the basics of spherical polynomial spliines. We will then<br \/>\ndiscuss the use of such splines for function approximation and the integration of differential equations on the<br \/>\nsphere. (Joint work with Anamaria Gomide)\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: November 1, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Alex Powell, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Analog to digital conversion for finite frame expansions.<\/p>\n<p>\t  <b>Abstract<\/b>: We shall dicuss the mathematical aspects of analog-to-digital conversion for redundant<br \/>\nsignal expansions. We restrict ourselves to the case of finite dimensional data, and consider the naturally<br \/>\nassociated class of signal expansions given by finite frames. Our focus will be on a special class of algorithms,<br \/>\nknown as Sigma-Delta quantizers, which are related to error diffusion. We explain the basics of Sigma-Delta<br \/>\nschemes and point to ongoing directions of research such as error estimates and stability theorems.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: October 18, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Prof. Terry P. Lybrand, Vanderbilt University Center for Structural Biology.<\/p>\n<p>      <b>Title<\/b>: Computer simulation of biomacromolecules and complexes.<\/p>\n<p>\t  <b>Abstract<\/b>: Computational approaches have become indispensable for study of large biological<br \/>\nmolecules over the past twenty-plus years. It is also possible, at least in principle, to use simulations and<br \/>\nother computational techniques to predict structural and thermodynamic properties. In my group, we are interested primarily in equilibrium thermodynamic properties of biomolecules and complexes, so we use statistical mechanical calculations to estimate these properties. Direct calculation of a partition function for these complex systems is not possible, so we utilize simulation methods like molecular dynamics or (less frequently) Monte Carlo to calculate approximate partition<br \/>\nfunctions via ensemble averaging. I will present some general details of our calculations, discuss common<br \/>\nproblems and limitations we encounter, and highlight some areas where we hopefully can take advantage of recent<br \/>\nmathematical developments to improve our calculations.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: September 27, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Yuliya Babenko, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: On asymptotically optimal methods of approximation by linear and bilinear splines.<\/p>\n<p>\t  <b>Abstract<\/b>: In this talk we shall present exact asymptotics of the optimal error in different metrics of linear and bilinear spline interpolation of an arbitrary function $f \\in C^2([0,1]^2)$.<\/p>\n<p>We shall present review of existing results as well as a series of new ones. Proofs of these results lead to<br \/>\nalgorithms for construction of asymptotically optimal sequences of triangulations (in the case of interpolation<br \/>\nby linear splines) and non uniform rectangular partitions (in the case of interpolation by bilinear splines).\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: September 20, 2005. 4:10-5pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Larry Schumaker, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Trivariate $C^r$ Polynomial Macro-Elements.<\/p>\n<p>\t  <b>Abstract<\/b>: $C^r$ macro-elements defined in terms of polynomials of degree $8r+1$ on tetrahedra<br \/>\nare analyzed. For $r=1,2$, these spaces reduce to well-known macro-element spaces used in data fitting and in the<br \/>\nfinite-element method. We determine the dimension of these spaces, and describe stable local minimal determining<br \/>\nsets and nodal minimal determining sets. We also show that the spaces approximate smooth functions to optimal<br \/>\norder.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: September 13, 2005. 4:10-5pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Kerstin Hesse, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Optimal Cubature on the Sphere.<\/p>\n<p>\t  <b>Abstract<\/b>:  In this talk I will present results from joint work with Ian H.\\,Sloan on cubature (or numerical integration) on the unit sphere $S^2$ in Sobolev spaces. We prove that the worst-case error $e(H^s;Q_m)$ of an $m$-point cubature rule $Q_m$ in the Sobolev space $H^s=H^s(S^2)$, $s&gt;1$, has the optimal order $O(m^{-s\/2})$. To achieve this we need two results: On the one hand,<br \/>\nwe show that for any $m$-point cubature rule $Q_m$ the worst-case cubature error satisfies $e(H^s;Q_m)\\geq C\\,m^{-s\/2}$, with a constant $C$ independent of the rule $Q_m$ (lower bound). On the other hand, we derive an upper bound for the optimal order of the worst-case error by identifying an infinite sequence $(Q_m)$ of $m$-point cubature<br \/>\nrules (where $m$ is from an infinite set of natural numbers) for which the worst-case cubature error has an upper bound of the order $O(m^{-s\/2})$. The results extend in a<br \/>\nnatural way to the Sobolev spaces $H^s(S^d)$, where $s&gt;d\/2$, on spheres $S^d$ of<br \/>\narbitrary dimension $d&gt;2$ (proof of the lower bound by myself and proof of the upper bound jointly with Johann S.\\,Brauchart).\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: April 19, 2005. 4:10-5 pm, room 1206.<\/p>\n<p>      <b>Speaker<\/b>: Doron Lubinsky, Georgia Tech.<\/p>\n<p>      <b>Title<\/b>: Which weights on R admit Jackson theorems?<\/p>\n<p>\t  <b>Abstract<\/b>: Let W : R ! (0;1) be continuous. Does W admit a Jackson or Jackson-Favard<br \/>\nInequality? That is, does there exist a sequence f\ufffdng1 n=1 of positive numbers with limit 0 such that for 1 \ufffd p \ufffd 1;<br \/>\ninf deg(P)\ufffdn k (f \ufffd P)W kLp(R)\ufffd \ufffdn k f0W kLp(R) for all absolutely continuous f with k f 0W kLp(R) \ufffdnite? We show<br \/>\nthat such a theorem is true i\ufffd both<br \/>\nlim x!1 W (x) Z x 0 W\ufffd1 = 0 and lim x!1\ufffdsup [0;x] W\ufffd1!Z 1 x W = 0; with analogous limits as x ! \ufffd1. In particular<br \/>\nW (x) = exp (\ufffdjxj) does not admit a<br \/>\nJackson theorem, although it is well known that W (x) = exp (\ufffdjxj\ufffd) ; \ufffd &gt; 1, does. We also construct weights that admit an L1 but not an L1 Jackson theorem (or conversely). The talk will be introductory, and might be accessible to those to whom Jackson and<br \/>\nBernstein sound like the directors of a large corporation.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: April 5, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Hong-Tae Shim, Visiting Professor, Sun Moon University, South Korea.<\/p>\n<p>      <b>Title<\/b>: On Gibbs phenomenon in wavelet expansions: its history and development.<\/p>\n<p>\t  <b>Abstract<\/b>: When a function with jump discontinuity is represented by the trigonometric series,<br \/>\none can observe that its graph exhibits overshoot or downshot near the point of discontinuity. This phenomenon<br \/>\nis called the Gibbs&#8217; phenomenon, which has been recognized for over a century. However, Gibbs phenomenon is not<br \/>\nthe special quirk of trigonometric series. It has been shown to exist for many natural approximation, e.g., those<br \/>\ninvolving Fourier series and other classical orthogonal expansions. In this talk, brief history and illustrations are given. We mainly focus on Gibbs phenomenon in wavelet expansions and provide a way to go around it.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: March 29, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Gitta Kutyniok, Univ. Giessen, Germany.<\/p>\n<p>      <b>Title<\/b>: Density of irregular wavelet systems.<\/p>\n<p>\t  <b>Abstract<\/b>: Density conditions have recently turned out to be a useful and elegant tool for<br \/>\nstudying irregular wavelet systems. In this talk we will discuss necessary and sufficient density conditions on<br \/>\nthe set of parameters for an irregular wavelet system to constitute a frame. In particular, we will derive a<br \/>\nnecessary condition on the relationship between the affine density, the frame bounds, and the admissibility<br \/>\ncondition. Several implications of this relationship will be studied. Moreover, we will prove that density<br \/>\nconditions can also be used to characterize existence of wavelet frames, thus serving in particular as sufficient conditions.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: March 9, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Fumiko Futamura, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: On Localized Frames.<\/p>\n<p>\t  <b>Abstract<\/b>: The concept of localization for frames was introduced independently by two groups for<br \/>\ntwo different purposes: one was concerned with constructing Banach frames for particular Banach spaces associated<br \/>\nto a particular Riesz basis and the other with understanding the density of frames, and how this relates to their<br \/>\nexcess. In an effort to unify their conclusions, we introduce a more generalized notion of localization. This notion, in the case of l1-self localization, comes with a natural equivalence class structure.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: March 2, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Tatyana Sorokina, The University of Georgia, Athens.<\/p>\n<p>      <b>Title<\/b>: An Octahedral $C^2$ Macro-Element.<\/p>\n<p>\t  <b>Abstract<\/b>:  (joint project with Ming-Jun Lai,The University of Georgia, Athens) A macro-element<br \/>\nof smoothness $C^2$ is constructed on the split of an octahedron into eight tetrahedra. This new element<br \/>\ncomplements those recently constructed $ Clough-Tocher and Worsey-Farin splits of a tetrahedron<br \/>\nby L.L. Schumaker, and P. Alfeld. The new element can be used to construct convenient super-spline spaces with<br \/>\nstable local bases and full approximation power that can be used for solving boundary-value problems and $\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: February 15, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Akram Aldroubi, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Robustness of sampling and reconstruction and Beurling-Landau-type theorems for shift invariant spaces.<\/p>\n<p>\t  <b>Abstract<\/b>:  Beurling-Landau-type results are known for a rather small class of functions<br \/>\nlimited to the Paley-Wiener space and certain spline spaces. Here, we show that the sampling and reconstruction<br \/>\nproblem in shift invariant spaces is robust with respect to the probing measures as well as to the underlying<br \/>\nshift invariant space. As an application we enlarge the class of functions for which a Beurling-Landau-type<br \/>\nresults hold.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: February 8, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Maxym Yattselev, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: AAK Theory and its Application to the &#8220;Crack&#8221; Problem.<\/p>\n<p>\t  <b>Abstract<\/b>:\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: February 1, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Andras Kroo, Hungarian Academy of Sciences.<\/p>\n<p>      <b>Title<\/b>: On Density of Multivariate Homogeneous Polynomials.<\/p>\n<p>\t  <b>Abstract<\/b>: The classical Weierstrass Theorem states that every function continuous on an interval<br \/>\ncan be uniformly approximated by algebraic polynomials. This was the first significant density result in Analysis<br \/>\nwhich inspired numerous generalizations applicable to other families of functions. The famous Stone-Weierstrass<br \/>\nTheorem gave an extension to subalgebras of C(K), yielding, in particular, the density of multivariate algebraic<br \/>\npolynomials. In this talk we shall discuss the density of a special important class of polynomials: the<br \/>\nmultivariate homogeneous polynomials. Homogeneous polynomials appear in many areas of Analysis.<br \/>\nThis family is nonlinear, so its density cannot be handled by the Stone-Weierstrass Theorem. In this talk we<br \/>\nshall present some recent developments in solving the density problem for homogeneous polynomials.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: January 25, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: David Benko, Western Kentucky University.<\/p>\n<p>      <b>Title<\/b>: Weighted polynomials on the real line.<\/p>\n<p>\t  <b>Abstract<\/b>: We will consider weighted polynomials of the form $w(x)^n P_n(x)$ where $w(x)$ is a<br \/>\nnon-negative fixed weight. Professor Saff introduced the problem of finding the uniform closure of these weighted<br \/>\npolynomials. In particular the Saff conjecture also arose from him. It was a long standing conjecture for a<br \/>\nspecial class of weights which was finally proved by Professor Totik. In the talk we will give a possible<br \/>\nextension of the problem.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: January 18, 2005. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Akram Aldroubi, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Convolution, average sampling, and Calderon resolution of the identity.<\/p>\n<p>\t  <b>Abstract<\/b>:\n      <\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2004 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2004<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2004\">\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: November 17, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Paul Leopardi, University of New South Wales, Australia.<\/p>\n<p>      <b>Title<\/b>: An equal-measure partition of S^d.<\/p>\n<p>\t  <b>Abstract<\/b>: A construction is given for an equal-measure partition of the unit sphere<br \/>\n$S^d \\subset R^{d+1}$ called the Recursive-Zhou-Saff-Sloan partition. For $d &lt; = 8$ it can be proven that there<br \/>\nis a constant $K_d$ such that, for the RZ partition of $S^d$ into N regions, each region has Euclidean diameter<br \/>\nat most $K_d N^{-1\/d}$.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: November 10, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Yuliya Babenko, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: On existence of a function with prescribed norms of its derivatives.<\/p>\n<p>\t  <b>Abstract<\/b>:  In this talk we shall discuss the following problem which was posed by Kolmogorov:<br \/>\nFor given integer $d$, given numbers $M_{\\nu_i}$, %$1\\leq p_i\\leq \\infty$ and<br \/>\n$1\\leq \\nu_i \\leq r$, $1 \\leq i \\leq d$ and function space $X$ find necessary and sufficient conditions for<br \/>\nexistence $x\\in X$ such that $$ \\left\\| x ^ {\\left( \\nu_i\\right) }\\right\\| _{\\infty}= M_{\\nu_i}. $$ We shall give<br \/>\na short review of known results and present new ones. In particular, we will give a complete characterization of sets of four numbers such that there exists $l$-monotone function with prescribed smoothness that has these numbers as values of sup-norms of<br \/>\nits corresponding derivatives. Along with mentioned classical Kolmogorov problem we shall consider the following related question: if we fix any three out of four given derivatives of order $0&lt;k_1 &lt;k_2&lt;r$, what can be said about the remaining one?\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: November 3, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>:  Maxim Yattselev, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: A Remez-Type Theorem for Homogeneous Polynomials. (Joint work with A. Kroo and E.B. Saff).<\/p>\n<p>\t  <b>Abstract<\/b>:  In this presentation we are going to consider a problem of estimating of the supremum<br \/>\nnorm of a polynomials on some set when its norm on a smaller subset is known. This problem was solved by Remez<br \/>\nfor the interval case. Later A. Kroo and D. Schmidt generalized it for the multivariate polynomials on domains<br \/>\nwith different smoothness of the boundary. We have considered this problem for class of homogeneous polynomials.<br \/>\nIn this case a better estimate can be achieved due to their special structure.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: October 27, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Sergiy Borodachov, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: On minimization of the Riesz s-energy on rectifiable sets.<\/p>\n<p>\t  <b>Abstract<\/b>:  In this presentation we are going to consider a problem of estimating of the supremum<br \/>\nnorm of a polynomials on some set when its norm on a smaller subset is known. This problem was solved by Remez<br \/>\nfor the interval case. Later A. Kroo and D. Schmidt generalized it for the multivariate polynomials on domains<br \/>\nwith different smoothness of the boundary. We have considered this problem for class of homogeneous polynomials.<br \/>\nIn this case a better estimate can be achieved due to their special structure.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: October 6, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Mike Neamtu, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Bivariate B-splines Used as Basis Functions for Data Fitting.<\/p>\n<p>\t  <b>Abstract<\/b>:  We present results summarizing the utility of bivariate B-splines for solving data<br \/>\nfitting problems on bounded domains. These basis functions are defined by certain collections of points in the<br \/>\nplane, called knots. The linear span of these functions gives rise to a spline space with good approximation<br \/>\nproperties. Our numerical results show that the B-splines basis also entertains excellent spectral properties,<br \/>\nrendering the B-splines useful for, among other things, iterative solution of data fitting and collocation<br \/>\nproblems in computational electromagnetics.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: September 29, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: G. Lopez Lagomasino, Universidad Carlos III de Madrid, Spain.<\/p>\n<p>      <b>Title<\/b>: Ratio asymptotics of Hermite-Pade orthogonal poltnomials for Nikishin systems.<\/p>\n<p>\t  <b>Abstract<\/b>:  Multiple orthogonal polynomials share orthogonality relations with a system of<br \/>\nmeasures. They arise naturally when considering simultaneous interpolating rational approximations to a system<br \/>\nof analytic functions, and the interpolation conditions are distributed between the different functions. We<br \/>\nconsider so-called Nikishin systems of functions which are made up of certain types of Cauchy transforms of Borel<br \/>\nmeasures supported on a same finite interval $\\Delta$ of the real line, and multiple orthogonal polynomials with<br \/>\nrespect to the measures generating the Nikishin system with orthogonality &#8220;nearly&#8221; equally distributed between<br \/>\nthe different measures. We prove that the ratio of &#8220;consecutive&#8221; multiorthogonal polynomials converge to an<br \/>\nanalytic function uniformly on the compact subsets of $C \\setminus \\Delta$ if the Radon-Nikodym derivative of the<br \/>\nmeasures is $&gt; 0$ a.e. on $\\Delta$. This result<br \/>\nextends a well known Theorem due to E. A. Rakhmanov.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: September 22, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Larry L. Schumaker, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Smooth Macro-Elements on Powell-Sabin-12 Splits.<\/p>\n<p>\t  <b>Abstract<\/b>:  For all r &gt;= 0, a family of macro-element spaces of smoothness Cr is constructed<br \/>\nbased on the Powell-Sabin-12 refinement of a triangulation. These new spaces complement the macro-element spaces<br \/>\nbased on Powell-Sabin-6 splits which have recently been developed. These new superspline spaces have stable local<br \/>\nbases and full approximation power, and can be used to solve boundary-value problems and interpolate Hermite data.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: September 8, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Doug Hardin, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Properties of minimum Riesz energy point sets on rectifiable manifolds.<\/p>\n<p>\t  <b>Abstract<\/b>:  For a compact set $A\\subset {\\bf R}^{d&#8217;}$, we consider minimal $s$-energy<br \/>\narrangements of $N$ points that interact through a power law (Riesz) potential $V=1\/r^{s}$, where $s&gt;0$ and $r$<br \/>\nis Euclidean distance in ${\\bf R}^{d&#8217;}$. For example, this is the classical Thomson problem of distributing<br \/>\nelectrons on a sphere in the case $A$ is the unit sphere in ${\\bf R}^3$, and $s=1$. In applications one is often<br \/>\ninterested in determining when such point sets are &#8220;uniformly&#8221; distributed on $A$ for large $N$. Physicists are<br \/>\nalso interested in &#8220;universal&#8221; (i.e. independent of $s$) properties of such configurations. In this talk I will<br \/>\npresent recent results characterizing asymptotic (as $N\\to \\infty$) properties of $s$-energy optimal $N$-point<br \/>\nconfigurations for a class of rectifiable $d$-dimensional manifolds and $s\\ge d$. This is joint work<br \/>\nwith E. B. Saff.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: April 7, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Bernd Mulansky, Technical Univ. of Clausthal, Germany.<\/p>\n<p>      <b>Title<\/b>: Delaunay configurations.<\/p>\n<p>\t  <b>Abstract<\/b>:  Delaunay configurations can be used to select collections of knot-sets in the<br \/>\nconstruction of multivariate spline spaces from simplex spline. We consider geometric and combinatorial<br \/>\nproperties of Delaunay configurations of a finite point set in the plane, including their efficient computation.<br \/>\nDecisive is an interpretation of Delaunay configurations in terms of a convex hull.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: March 31, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Johan de Villiers, University of Stellenbosh, South Africa.<\/p>\n<p>      <b>Title<\/b>: On refinable functions and subdivisions with positive masks.<\/p>\n<p>\t  <b>Abstract<\/b>:  We present some extensions of the existing theory of refinement equations with<br \/>\npositive masks. In particular, attention is given to the geometric converegnce rate of both the cascade algorithm<br \/>\nand the subdivision scheme, as well as the sequence space on which the subdivision converges. Finally, we<br \/>\nconsider the regularity (or degree of smoothness) of the underlying refinable function.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: March 24, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Frank Zeilfelder, University of Mannheim.<\/p>\n<p>      <b>Title<\/b>: Approximation and Visualization of Huge Volume Data Sets by Trivariate Splines.<\/p>\n<p>\t  <b>Abstract<\/b>: In recent years, the reconstruction of volume data became a very active area of<br \/>\nresearch since it is important for many general applications such as for instance in scientific visualization and<br \/>\nmedical imaging. It is known to be a difficult problem to keep all the practical requirements simultaneously into<br \/>\naccount: high quality visual appearance of the reconstructed objects, quick computation which aims towards the general goal of interactive frame rates, optimal approximation properties of the model and its gradients, insensitiveness for noisy data, efficiency in representation and evaluation of the models. We develop new models for the reconstruction problem of volume data. These models are<br \/>\ntrivariate splines, i.e. piecewise polynomial functions defined w.r.t. appropriate tetrahedral partitions of the<br \/>\nvolumetric domain. The talk is subdivided into two parts. In the first part we give some theoretical background<br \/>\non the complex structure of the trivariate splines, while in the second part we show how to turn these results<br \/>\ninto practical methods for volume data approximation and visualization. Numerical tests show the efficiency of<br \/>\nthe methods.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: March 17, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Ursula Molter, University of Buenos Aires.<\/p>\n<p>      <b>Title<\/b>: Thin and thick Cantor sets.<\/p>\n<p>\t  <b>Abstract<\/b>:  In this talk we will discuss the construction of Cantor sets (on the line) associated<br \/>\nto summable sequences of positive terms. We will show that to each such Cantor set we can associate an<br \/>\nappropriate function h, such that the Hausdorff-h measure of the set is positive.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: March 3, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Doug Hardin, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>:Discrete minimum energy problems on rectifiable manifolds.<\/p>\n<p>\t  <b>Abstract<\/b>:\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: February 5, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Andras Kroo, Alfred Renyi Mathematical Institute, Hungarian Academy of Sciences.<\/p>\n<p>      <b>Title<\/b>: Uniform norm estimation for factors of multivariate polynomials II.<\/p>\n<p>\t  <b>Abstract<\/b>: We shall consider the following problem of norm estimation of factors of polynomials:<br \/>\ngiven a polynomial p which factors into the product of 2 polynomials p=rq give an upper bound for the norms of<br \/>\nfactors r and q if the norm of p is known. This problem has been considered in various norms by many authors,<br \/>\nit has applications in Banach space theory, number theory, constructive function theory, etc. In this talk we<br \/>\nshall discuss this question for spaces of multivariate polynomials endowed with uniform norm on some compact set<br \/>\nK, and show how the geometry of K effects the corresponding estimates.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: January 21, 2004. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Andras Kroo, Alfred Renyi Mathematical Institute, Hungarian Academy of Sciences.<\/p>\n<p>      <b>Title<\/b>:Uniform norm estimation for factors of multivariate polynomials.<\/p>\n<p>\t  <b>Abstract<\/b>: We shall consider the following problem of norm estimation of factors of polynomials:<br \/>\ngiven a polynomial p which factors into the product of 2 polynomials p=rq give an upper bound for the norms of<br \/>\nfactors r and q if the norm of p is known. This problem has been considered in various norms by many authors, it<br \/>\nhas applications in Banach space theory, number theory, constructive function theory, etc. In this talk we shall<br \/>\ndiscuss this question for spaces of multivariate polynomials endowed with uniform norm on some compact set K, and<br \/>\nshow how the geometry of K effects the corresponding estimates.\n      <\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n<p><!-- 2003 --><\/p>\n<link rel='stylesheet' type='text\/css' href='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/css\/accordion.css' media='screen' \/><script type='text\/javascript' src='https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-content\/themes\/vanderbilt-brand\/js\/accordion.js'><\/script><ul class='accordion collapsible'>\n<li><a href='#'>2003<\/a>\n<div class='acitem'>\n<table cellpadding=\"10\" id=\"2003\">\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: December 10, 2003. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Wolfgang Dahmen, Institut f?r Geometrie und Praktische Mathematik.<\/p>\n<p>      <b>Title<\/b>: Adaptive application of operators in wavelet coordinates.<\/p>\n<p>\t  <b>Abstract<\/b>:\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: November 19, 2003. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Allan Pinkus, Technion.<\/p>\n<p>      <b>Title<\/b>: Herman Muntz, 1884-1956.<\/p>\n<p>\t  <b>Abstract<\/b>: The Muntz Theorem is a central theorem in approximation theory. But who was Muntz? How<br \/>\ndid he come to prove this theorem? In this talk we consider this forgotten mathematician and the odyssey of his<br \/>\nlife.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: November 5, 2003. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Allan Pinkus, Technion.<\/p>\n<p>      <b>Title<\/b>: Negative Theorems in Approximation Theory.<\/p>\n<p>\t  <b>Abstract<\/b>: Approximation theory is concerned with the ability to approximate functions and<br \/>\nprocesses by simpler and more easily calculated objects. However there are very definite and intrinsic<br \/>\nlimitations on approximation processes. In this talk I will survey some of these limitations. Little to no<br \/>\napproximation theory background is needed.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: October 29, 2003. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Pencho Petrushev, U. South Carolina.<\/p>\n<p>      <b>Title<\/b>: Nonlinear n-term approximation from hierarchical spline bases.<\/p>\n<p>\t  <b>Abstract<\/b>: Nonlinear n-term approximation from sequences of hierarchical spline bases generated<br \/>\nby multilevel nested triangulations in R2 will be discussed. The emphasis will be placed on the smoothness spaces<br \/>\n(B-spaces) governing the rates of nonlinear n-term approximation. The properties of the corresponding Franklin systems will be given as well. It will be explained how the general<br \/>\nJackson-Bernstein machinery can be utilized for characterization of the rates of nonlinear n-term<br \/>\napproximation. Also, it will be shown that the B-spaces can be used in the design of algorithms which capture the<br \/>\nrate of the best n-term spline approximation. Some related topics and open problems will be discussed as well.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: October 15, 2003. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Akram Aldroubi, Vanderbilt University.<\/p>\n<p>      <b>Title<\/b>: Wavelet frames on irregular grids, with arbitrary dilation matrices, and in multi-dimension.<\/p>\n<p>\t  <b>Abstract<\/b>: This talk will be introductory and should be understandable by all. We will first<br \/>\nintroduce the concepts of wavelet bases and wavelet frames. Then, using a one dimensional simple example, we will<br \/>\npresent the main ideas on how to construct wavelet frames on irregular lattices, and<br \/>\nwith arbitrary dilation matrices.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: October 8, 2003. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Peter Dragnev, Indiana University-Purdue University, Fort Wayne.<\/p>\n<p>      <b>Title<\/b>: On a discrete Zolotarev problem with applications to the Alternating Direction Implicit (ADI) method.<\/p>\n<p>\t  <b>Abstract<\/b>: In this talk I will consider a discrete version of the Third Zolotarev Problem. This<br \/>\nproblem arises in the investigation of optimal parameters of the ADI method for solving partial differential<br \/>\nequations. The asymptotics of these parameters are governed by a constrained<br \/>\nenergy problem for signed measures.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: September 24, 2003. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Oleg Davydov, Univ. of Giessen, Germany.<\/p>\n<p>      <b>Title<\/b>: Multilevel Bivariate Splines.<\/p>\n<p>\t  <b>Abstract<\/b>: We discuss various possibilities to construct multilevel spline bases in two variables<br \/>\nas well as some applications, including recent hierarchical Riesz basis for Sobolev spaces H2(O) on arbitrary<br \/>\npolygonal domains.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: September 18, 2003. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Peter Alfeld, University of Utah.<\/p>\n<p>      <b>Title<\/b>: Trivariate Spline Spaces on Tetrahedral Partitions.<\/p>\n<p>\t  <b>Abstract<\/b>: We consider spaces of smooth piecewise polynomial functions defined on a tetrahedral<br \/>\npartition of a three dimensional domain. These spaces can be described in terms of minimal determining sets, i.e.,<br \/>\nsets of points in the domain that correspond to a set of coefficients which can be chosen arbitrarily and which<br \/>\nuniquely determine a spline. The talk will focus on a software package that enables the computation of dimensions<br \/>\nand the design of finite elements. The code grew out of a similar package for bivariate splines that has proved<br \/>\ninstrumental in deriving a number of results in two dimensions.\n      <\/td>\n<\/tr>\n<tr>\n<td valign=\"top\">\n\t  <b>Time<\/b>: September 10, 2003. 4:10-5 pm, room 1431.<\/p>\n<p>      <b>Speaker<\/b>: Andrei Martinez Finkelshtein.<\/p>\n<p>      <b>Title<\/b>: Strong asymptotics of Jacobi polynomials with varying nonstandard parameters.<\/p>\n<p>\t  <b>Abstract<\/b>:\n      <\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The weekly Computational Analysis Seminar is attended by faculty, students, and visiting researchers working in one or more of the following areas of mathematics: constructive approximation theory, splines, wavelets, signal processing, image compression, shift-invariant spaces, constrained approximation and interpolation, computer-aided geometric design, and a few other related areas. If you need more information and\/or want&#8230;<\/p>\n","protected":false},"author":637,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-18","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-json\/wp\/v2\/pages\/18","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-json\/wp\/v2\/users\/637"}],"replies":[{"embeddable":true,"href":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-json\/wp\/v2\/comments?post=18"}],"version-history":[{"count":57,"href":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-json\/wp\/v2\/pages\/18\/revisions"}],"predecessor-version":[{"id":504,"href":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-json\/wp\/v2\/pages\/18\/revisions\/504"}],"wp:attachment":[{"href":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-json\/wp\/v2\/media?parent=18"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/my.dev.vanderbilt.edu\/constructiveapproximation\/wp-json\/wp\/v2\/tags?post=18"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}