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Mathematics & Statistics

# Classical Analysis (Leader: Prof. Dmitry Khavinson )

## Friday, April 17, 2009

Title
Speaker
Time
Place

Some Results on Unbounded Bergman Operators
Sherwin Kouchekian
4:00pm-5:00pm
PHY 109

Abstract

We will consider Bergman operators. These are operators of multiplication by the independent variable $$z$$ over the space of square integrable analytic functions defined on an open (not necessarily bounded) subset of the complex plane. We will present some results regarding the self-commutator of these operators and also discuss the different techniques used in the proof of the obtained results. Furthermore, we will discuss some unpublished results regarding density for the core of the Bergman operators.

## Friday, April 10, 2009

Title
Speaker
Time
Place

Asymptotics for Christoffel functions on curves
Vilmos Totik
4:00pm-5:00pm
PHY 109

Abstract

A new approach is given for almost everywhere asymptotics for Christoffel functions on families of curves. It involves fast decreasing polynomials, a sharpened form of Hilbert's lemniscate theorem and a new type of polynomial estimates.

## Friday, April 3, 2009

Title
Speaker
Time
Place

A seminorm problem, recursive inequalities, and negative binomial weights
4:00pm-5:00pm
PHY 109

Abstract

We will discuss the multiparameter recursive inequalities that are generated by a seminorm problem for formal power series. These inequalities and their applications involve negative binomial weights, arbitrary complex vectors, entire functions, weighted integrals, and integral equations.

## Friday, March 6, 2009

Title
Speaker
Time
Place

Asymptotics of Christoffel Functions for General Measures in the Plane
Marty Findley
4:00pm-5:00pm
PHY 109

Abstract

Our main concern is the precise asymptotic behavior of the sequence of Christoffel functions associated with measures having general supports. We will describe the frontier of this subject and a sharp asymptotic formula for certain weighted means of Faber polynomials. Time permitting, we will also discuss some applications to classical analysis and ill-posed problems as well as a refinement of the mean ergodic theorem for a class of unitary operators.

## Friday, February 27, 2009

Title
Speaker
Time
Place

Zeros of certain kernel functions in Fock spaces, Part II
Catherine Bénéteau
4:00pm-5:00pm
PHY 109

## Friday, February 20, 2009

Title
Speaker
Time
Place

Zeros of certain kernel functions in Fock spaces, Part I
Catherine Bénéteau
4:00pm-5:00pm
PHY 109

Abstract

This talk is devoted to examining some extremal problems in the Fock space, which is a space of entire functions with certain growth restrictions. I will discuss some generalities on order and type of Fock space functions, pose an extremal problem for a finite zero based subspace of the Fock space, and examine the zeros of the extremals.

## Friday, February 13, 2009

Title
Speaker
Time
Place

Minimal projections, Part III
Lesław Skrzypek
4:00pm-5:00pm
PHY 109

## Friday, February 6, 2009

Title
Speaker
Time
Place

Minimal projections, Part II
Lesław Skrzypek
4:00pm-5:00pm
PHY 109

## Friday, January 23, 2009

Title
Speaker
Time
Place

Minimal projections, Part I
Lesław Skrzypek
4:00pm-5:00pm
PHY 109

Abstract

I will start with general theorems about minimal projections (this will involve some knowledge from functional analysis) and then later on we will move to concrete examples and problems.

## Friday, January 16, 2009

Title
Speaker
Time
Place

Looking for singularities of the solutions to the Dirichlet Problem, Part III (3rd and final talk)
Dmitry Khavinson
4:00pm-5:00pm
PHY 109

Abstract

The classical Dirichlet Problem (DP) reduces to matching a given data function on the boundary of a domain with a function harmonic in the whole domain. When the boundary is smooth the solution exists and unique. Yet, even for the best imaginable data functions (e.g., polynomials) the solutions mysteriously develop singularities outside the initial domain where the DP was initially posed.

Questions like: (i) for which domains it never happens, (ii) how is the appearance of singularities related to the geometry of the initial domain, (iii) is there a data that produces ALL possible singularities at once turn out to be excruciatingly difficult.

In this final talk we shall discuss some recent developments in understanding this problem and its relation to an old problem on orthogonal polynomials.