Research

Analysis

(Leader: )

Friday, April 24, 2015

Abstract

Finding the cyclic vectors of Fock spaces for functional Hilbert spaces is a problem of considerable interest, which arises under different formulations in function theory, non-commutative algebras, and group representation theories. Symmetry-based arguments developed in the operator approach provide a direct method which is sometimes much simpler than the original functional formulation. We discuss some of these cases and their explicit solutions.

Friday, April 17, 2015

Abstract

How far does the Newtonian potential of a solid bounded by an algebraic surface extend inside the solid? Why is the celebrated Schwarz reflection principle never discussed in dimensions higher than 2? How does one find singularities of an axially symmetric harmonic function in the ball from the coefficients in its expansion by spherical harmonics? If a line intersects a domain over two disjoint segments and a harmonic function in the domain vanishes on one, does it have to vanish on the other one? We shall discuss these questions in the unified light of analytic continuation of solutions to linear analytic pde. The talk will be accessible to undergraduate and graduate students majoring in math, physics and engineering.

Friday, April 10, 2015

Friday, April 3, 2015

Abstract

A formula will be discussed which gives strong asymptotics for a polynomial whose zeros are exactly uniformly distributed with respect to a given (analytic or smooth) density function.

The formula makes it possible to systematically derive a number of old and new results on asymptotics of extremal polynomials (in particular, orthogonal and Chebyshev ones). There are many other applications.

Friday, March 27, 2015

Abstract

In the space of univalent polynomials (on the unit disk), an extreme point gives the conformal image of the disk with the maximal number of double points (self-intersections). Similar consideration on the exterior unit disk gives unbounded domains with the maximal number of double points. The latter domains imply the existence of the special (critically finite) harmonic polynomials with the maximal number of roots (as shown by Khavinson and Swiatek). This is a joint work with Makarov.

Friday, March 13, 2015

Abstract

In a very influential paper from 1969, H. Widom described completely the behavior of Chebyshev and orthogonal polynomials on a system of Jordan curves. When Jordan arcs were also present, the behavior of orthogonal polynomials was the same, but for Chebyshev polynomials things changed, and the situation is still not clear. The talk will discuss this problem.

Friday, February 20, 2015

Abstract

Given a polynomial \(P\) on the unit disk, form the polynomial \(P˜\) by replacing the coefficients of \(P\) with their absolute values. There is a maximal radius \(R\), so that the supremum of \(P˜\) on the disk of radius \(R\) is bounded by the supremum of P on the whole unit disk, for every choice of P. Bohr’s famous theorem shows that \(R=1/3\), and this radius is called the Bohr radius.

Friday, February 6, 2015

Abstract

We present several results in which the interplay between the Fourier transform and the Hilbert transform is of special form and importance.

1. In the 1950s (Kahane, Izumi-Tsuchikura, Boas, etc.), the following problem in Fourier Analysis attracted much attention: Let \(\{a_k\}\), \(k=0,1,2,\dotsc\), be the sequence of the Fourier coefficients of the absolutely convergent sine (cosine) Fourier series of a function \(f:\mathbb T=[-\pi,\pi)\to \mathbb C\), that is \(\sum |a_k|<\infty\). Under which conditions on \(\{a_k\}\) the re-expansion of \(f(t)\) (\(f(t)-f(0)\), respectively) in the cosine (sine) Fourier series will also be absolutely convergent?

We solve a similar problem for functions on the whole axis and their Fourier transforms. Generally, the re-expansion of a function with integrable cosine (sine) Fourier transform in the sine (cosine) Fourier transform is integrable if and only if not only the initial Fourier transform is integrable but also the Hilbert transform of the initial Fourier transform is integrable.

2. The following result is due to Hardy and Littlewood: If a (periodic) function \(f\) and its conjugate \(\widetilde f\) are both of bounded variation, their Fourier series converge absolutely.

We generalize the Hardy-Littlewood theorem (joint work with U. Stadtmüller) to the Fourier transform of a function on the real axis and its modified Hilbert transform. The initial Hardy-Littlewood theorem is a partial case of this extension, when the function is taken to be with compact support.

3. These and other problems are integrated parts of harmonic analysis of functions of bounded variation. We have found the maximal space for the integrability of the Fourier transform of a function of bounded variation. Along with those known earlier, various interesting new spaces appear in this study. Their inter-relations lead, in particular, to improvements of Hardy's inequality.

There are multidimensional generalizations of these results.

January 23, 2015

Abstract

An extremal problem for formal power series leads to a sequence of weighted convolution inequalities. This result and its limit versions can be presented in terms of various probability functions. The related applications involve a limit binomial theorem, weighted norm inequalities for differential operators and special functions, inequalities for expectations, etc.