Seminars and Colloquia by Series

Chromatic Derivatives and Approximations Speaker

Series
Analysis Seminar
Time
Wednesday, November 9, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Aleks IgnjatovicUniversity of New South Wales
Chromatic derivatives are special, numerically robust linear differential operators which provide a unification framework for a broad class of orthogonal polynomials with a broad class of special functions. They are used to define chromatic expansions which generalize the Neumann series of Bessel functions. Such expansions are motivated by signal processing; they grew out of a design of a switch mode power amplifier. Chromatic expansions provide local signal representation complementary to the global signal representation given by the Shannon sampling expansion. Unlike the Taylor expansion which they are intended to replace, they share all the properties of the Shannon expansion which are crucial for signal processing. Besides being a promising new tool for signal processing, chromatic derivatives and expansions have intriguing mathematical properties connecting in a novel way orthogonal polynomials with some familiar concepts and theorems of harmonic analysis. For example, they introduce novel spaces of almost periodic functions which naturally correspond to a broad class of families of orthogonal polynomials containing most classical families. We also present a conjecture which generalizes the Paley Wiener Theorem and which relates the growth rate of entire functions with the asymptotic behavior of the recursion coefficients of a corresponding family of orthogonal polynomials.

Bernstein's problem on weighted polynomial approximation

Series
Analysis Seminar
Time
Wednesday, November 2, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexei PoltoratskiTexas A&M
The problem of weighted polynomial approximation of continuousfunctionson the real line was posted by S. Bernstein in 1924. It asks for adescription of theset of weights such that polynomials are dense in the space of continuousfunctions withrespect to the corresponding weighted uniform norm. Throughout the 20thcentury Bernstein's problem was studied by many prominent analysts includingAhkiezer, Carleson, Mergelyan andM. Riesz.In my talk I will discuss some of the complex analytic methods that can beapplied in Bernstein's problem along with a recently found solution.

Weierstrass Theorem for homogeneous polynomials on convex bodies and rate of approximation of convex bodies by convex algebraic level surfaces

Series
Analysis Seminar
Time
Wednesday, October 26, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Andras KrooHungarian Academy of Sciences
Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4 By the classical Weierstrass theorem, any function continuous on a compact set can be uniformly approximated by algebraic polynomials. In this talk we shall discuss possible extensions of this basic result of analysis to approximation by homogeneous algebraic polynomials on central symmetric convex bodies. We shall also consider a related question of approximating convex bodies by convex algebraic level surfaces. It has been known for some time time that any convex body can be approximated arbirarily well by convex algebraic level surfaces. We shall present in this talk some new results specifying rate of convergence.

Wavelet analysis on a metric space

Series
Analysis Seminar
Time
Wednesday, September 28, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tuomas HytonenUniversity of Helsinki
Expansion in a wavelet basis provides useful information ona function in different positions and length-scales. The simplest example of wavelets are the Haar functions, which are just linearcombinations of characteristic functions of cubes, but often moresmoothness is preferred. It is well-known that the notion of Haarfunctions carries over to rather general abstract metric spaces. Whatabout more regular wavelets? It turns out that a neat construction canbe given, starting from averages of the indicator functions over arandom selection of the underlying cubes. This is yet anotherapplication of such probabilistic averaging methods in harmonicanalysis. The talk is based on joint work in progress with P. Auscher.

Average Density of States for Hermitian Wigner Matrices

Series
Analysis Seminar
Time
Wednesday, June 15, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Dr Anna MaltsevUniversity of Bonn
We consider ensembles of $N \times N$ Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law, as $N$ tends to infinity.

On the uniqueness sets in the Bergmann-Fock space

Series
Analysis Seminar
Time
Wednesday, April 13, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mishko MitkovskiSchool of Mathematics, Georgia Tech
It is well known that, via the Bargmann transform, the completeness problems for both Gabor systems in signal processing and coherent states in quantum mechanics are equivalent to the uniqueness set problem in the Bargmann-Fock space. We introduce an analog of the Beurling-Malliavin density to try to characterize these uniqueness sets and show that all sets with such density strictly less than one cannot be uniqueness sets. This is joint work with Brett Wick.

Orthogonal Rational Functions and Rational Gauss-type Quadrature Rules

Series
Analysis Seminar
Time
Wednesday, April 6, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Karl DeckersGeorgia Tech
Consider a positive bounded Borel measure \mu with infinite supporton an interval [a,b], where -oo <= a < b <= +oo, and assume we have m distinctnodes fixed in advance anywhere on [a,b]. We then study the existence andconstruction of n-th rational Gauss-type quadrature formulas (0 <= m <= 2)that approximate int_{[a,b]} f d\mu. These are quadrature formulas with npositive weights and n distinct nodes in [a,b], so that the quadratureformula is exact in a (2n - m)-dimensional space of rational functions witharbitrary complex poles fixed in advance outside [a,b].

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