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Mathematics & Statistics

Classical Analysis
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Friday, March 31, 2017

Title
Speaker
Time
Place

Zero distribution for Angelesco Hermite-Padé polynomials
E. A. Rakhmanov
4:00pm-5:00pm
CMC 204

Abstract

We consider the problem of zero distribution of Hermite-Padé polynomials associated with a vector function \(\vec f=\left(f_1,\dotsc,f_s\right)\) whose each component \(f_k\) is defined as element at infinity which has unlimited analytic continuation in plane with a deleted finite set (so, each component has a finite number of branch points). We assume that branch sets \(e_k\) of component functions are well enough separated (which constitute the Angelesco case). Under this condition polynomials have limit zero distribution.

The limit measures may be defined in terms of a vector equilibrium problem determined by \(\vec f\) (or, rather, by sets \(e_k\)). There is an equivalent characterization in terms of a third kind Abelian integral on an algebraic Riemann surface determined by branch sets too. We will explain definitions and go into some details.

Friday, March 24, 2017

Title
Speaker
Time
Place

Extremal hypergeometric functions and nonlinear damping on the real line
Razvan Teodorescu
4:00pm-5:00pm
CMC 204

Abstract

I will describe an (open) extremal problem on the space of measures induced by hypergeometric functions, and associated best approximations. I will then show how this problem leads to a curious “selection” theorem and an even curiouser property of classical orthogonal polynomials.

The (open) problem is joint work with Ar. Abanov (Texas A&M). The property was last rediscovered in 2006.

Friday, March 17, 2017

Spring Break — no seminar this week.

Friday, March 3, 2017

Title
Speaker

Time
Place

Bounded Point Derivations on Certain Function Algebras
James Brennan
University of Kentucky
4:00pm-5:00pm
CMC 204

Abstract

Let \(X\) be a compact nowhere dense subset of the complex plane \(\mathbb{C}\), let \(C(X)\) be the linear space of all continuous functions on \(X\) endowed with the uniform norm, and let dA denote two-dimensional Lebesgue (or area) measure in \(\mathbb{C}\). Denote by \(R(X)\) the closure in \(C(X)\) of the set of all rational functions having no poles on \(X\). It is well-known that if \(X\) is suciently massive, then the functions in \(R(X)\) can inherit many of the properties usually associated with functions smooth in a neighborhood of \(X\) such as various degrees of differentiability, and even the uniqueness property the analytic functions itself. In this talk I shall examine the extent to which some of those properties are inherited by the larger algebra \(H^\infty(X)\), which by definition is the weak-\(*\) closure of \(R(X)\) in \(L^\infty(X)=L^\infty(X,dA)\).

Friday, February 24, 2017

Title
Speaker
Time
Place

Distribution of critical points of polynomials
Vilmos Totik
4:00pm-5:00pm
CMC 204

Abstract

The distribution of the critical points (zeros of the derivatives) of polynomials is considered. Some history related to Springer's conjecture will be reviewed, and then results will be given on the asymptotic distribution of critical points when the distribution (say \(m\)) of the zeros is given. One result is that if the support of \(m\) has connected complement, then the distribution of the critical points is again \(m\) (this need not be the case when the complement is disconnected, as the example \(z^n-1\), \(n=1,2,\dotsc\) shows).

Friday, February 17, 2017

Title

Speaker
Time
Place

Weighted planar orthogonal polynomials with logarithmic potential as multiple orthogonal polynomials of Type II, Part II
Ming Yang
4:00pm-5:00pm
CMC 204

Friday, February 10, 2017

Title

Speaker
Time
Place

Weighted planar orthogonal polynomials with logarithmic potential as multiple orthogonal polynomials of Type II
Seung-Yeop Lee
4:00pm-5:00pm
CMC 204

Abstract

When the potential is given by \(Q(z)=|z|^2+\sum\alpha_j\log\left|z-a_j\right|\), we will write down the Riemann-Hilbert problem satisfied by the planar orthogonal polynomials when there are two log-singularities.

Friday, February 3, 2017

Title

Speaker
Time
Place

Optimal polynomial approximants, connections to Jacobi matrices and orthogonal polynomials: Mysteries Revealed, III
Catherine Bénéteau
4:00pm-5:00pm
CMC 204

Friday, January 27, 2017

Title

Speaker
Time
Place

Optimal polynomial approximants, connections to Jacobi matrices and orthogonal polynomials: Mysteries Revealed, II
Catherine Bénéteau
4:00pm-5:00pm
CMC 204

Friday, January 20, 2017

Title

Speaker
Time
Place

Optimal polynomial approximants, connections to Jacobi matrices and orthogonal polynomials: Mysteries Revealed
Catherine Bénéteau
4:00pm-5:00pm
CMC 204

Abstract

In this talk, I will discuss the zeros of optimal polynomial approximants in Dirichlet-type spaces. I will explain how the smallest modulus of any zero of an optimal approximant is connected to the norm of a special Jacobi matrix constructed from a three-term recursion relation. I will discuss how the solution of this extremal problem differs for spaces such as the Bergman space compared to a spaces such as the Hardy space or the Dirichlet space. Finally, I will discuss results about the accumulation of the zeros of these approximants that are similar to Jentzsch's theorem about the accumulation of the zeros of Taylor polynomials. This talk could also be called: what Daniel did not say.

Friday, January 13, 2017

Title
Speaker

Time
Place

Derivatives of rational inner functions
Alan Sola
Stockholm University
4:00pm-5:00pm
CMC 204

Abstract

Rational inner functions are important examples of bounded analytic functions in polydisks. I will present several results concerning boundary behavior of rational inner functions, with a focus on integrability properties of their derivatives.

This reports on joint work with K. Bickel and J. Pascoe.