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Series: Analysis Seminar

We study Hardy spaces on spaces X which are the n-fold product of homogeneous spaces. An important tool is the remarkable orthonormal wavelet basis constructed Hytonen. The main tool we develop is the Littlewood-Paley theory on X, which in turn is a consequence of a corresponding theory on each factor space. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving Littlewood-Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing function spaces and the boundedness of singular integrals on spaces of homogeneous type.
This is joint work with Yongsheng Han and Lesley Ward.

Series: Analysis Seminar

I will discuss a generalization of the KP hierarchy, which is intimately related to the cyclic quiver and the Calogero-Moser problem for the wreath-product $S_n\wr\mathbb Z/m\mathbb Z$.

Series: Analysis Seminar

A classical theorem of John Wermer asserts that the algebra of continuous
functions on the circle with holomophic extensions to the disc is a maximal
subalgebra of the algebra of all continuous functions on the circle.
Wermer's theorem has been extended in numerous directions. These will be
discussed with an emphasis on extensions to several complex variables.

Series: Analysis Seminar

In this talk we will discuss applications of a new method of proving
vector-valued inequalities discovered by M. Bateman and C. Thiele. We
give new proofs of the Fefferman-Stein inequality (without using
weighted theory) and vector-valued estimates of the Carleson operator
using this method. Also as an application to bi-parameter problems, we
give a new proof for bi-parameter multipliers without using product
theory. As an application to the bilinear setting, we talk about new
vector-valued estimates for the bilinear Hilbert transform, and
estimates for the paraproduct tensored with the bilinear Hilbert
transform. The first part of this work is joint work with Ciprian
Demeter.

Series: Analysis Seminar

The Bochner Classification Theorem (1929) characterizes the
polynomial sequences $\{p_n\}_{n=0}^\infty$, with $\deg p_n=n$ that
simultaneously form a complete set of eigenstates for a second order
differential operator and are orthogonal with respect to a positive
Borel measure having finite moments of all orders: Hermite, Laguerre,
Jacobi and Bessel polynomials. In 2009, G\'{o}mez-Ullate, Kamran, and
Milson found that for sequences $\{p_n\}_{n=1}^\infty$, with $\deg
p_n=n$ (i.e.~without the constant polynomial) the only such sequences
are the \emph{exceptional} Laguerre and Jacobi polynomials. They also
studied two Types of Laguerre polynomial sequences which omit $m$
polynomials. We show the existence of a new "Type III" family of
Laguerre polynomials and focus on its properties.

Series: Analysis Seminar

This will be a survey talk on the ongoing classification problem for
Carleson and reverse Carleson measures for the de Branges-Rovnyak
spaces. We will relate these problems to some recent work of Lacey and
Wick on the boundedness of the Cauchy transform operator.

Series: Analysis Seminar

Abstract: In the beginning, the basics about random matrix models andsome facts about normal random matrices in relation with conformal map-pings will be explained. In the main part we will show that for Gaussianrandom normal matrices the eigenvalues will fill an elliptically shaped do-main with constant density when the dimension n of the matrices tends to infinity. We will sketch a proof of universality, which is based on orthogonalpolynomials and an identity which plays a similar role as the Christoff el-Darboux formula in Hermitian random matrices.Especially we are interested in the density at the boundary where we scalethe coordinates with n^(-1/2). We will also consider the off -diagonal part of thekernel and calculate the correlation function. The result will be illustratedby some graphics.

Series: Analysis Seminar

Series: Analysis Seminar

I will introduce the cluster value problem, and its relation to the
Corona problem, in the setting of Banach algebras of analytic functions
on unit balls. Then I will present a reduction of the cluster value
problem in separable Banach spaces, for the algebras $A_u$ and
$H^{\infty}$, to those spaces that are $\ell_1$ sums of a sequence of
finite dimensional spaces. This is joint work with William B. Johnson.

Series: Analysis Seminar

This talk discusses exponential frames and Riesz sequences in L^2 over a set
of finite measure. (Roughly speaking, Frames and Riesz sequences are over
complete bases and under complete bases, respectively). Intuitively, one
would assume that the frequencies of an exponential frame can not be too
sparse, while those of an exponential Riesz sequence can not be too dense.
This intuition was confirmed in a very general theorem of Landau, which
holds for all bounded sets of positive measure.
Landau's proof involved a deep study of the eigenvalues of compositions of
certain projection operators. Over the years Landaus technique, as well as
some relaxed version of it, were used in many different setting to obtain
results of a similar nature.
Recently , joint with A. Olevskii, we found a surprisingly simple approach
to Landau's density theorems, which provides stronger versions of these results. In particular,
the theorem for Riesz sequences was extended to unbounded sets (for frames,
such an extension is trivial).
In this talk we will discuss Landau's results and our approach for studying
questions of this type.