Georgia Institute of Technology College of Sciences

We show how to construct frames for square integrable functionsout of modulated Gaussians. Using the frame representation of the Cauchydata, we show that we can build a suitable approximation to the solutionfor low regularity, time dependent wave equations. The talk will highlightthe relationship of the construction to harmonic analysis and will explorethe differences of the new construction to the standard Gaussian beamansatz.

Must the Fourier series of an L^2 function converge pointwise almost everywhere? In the 1960's, Carleson answered this question in the affirmative, by studying a particular type of maximal singular integral operator, which has since become known as the Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In this talk we will describe new joint work with Po Lam Yung that introduces curved structure to the setting of Carleson operators.

In this talk we will connect functional analysis and analytic function theory by studying the compact linear operators on Bergman spaces. In particular, we will show how it is possible to obtain a characterization of the compact operators in terms of more geometric information associated to the function spaces. We will also point to several interesting lines of inquiry that are connected to the problems in this talk. This talk will be self-contained and accessible to any mathematics graduate student.

A brief overview of some integrable and exactly-solvable Schroedinger equations with trigonometric potentials of Calogero-Moser-Sutherland type is given.All of them are characterized bya discrete symmetry of the Hamiltonian given by the affine Weyl group,a number of polynomial eigenfunctions and eigenvalues which are usually quadratic in the quantum number, each eigenfunction is an element of finite-dimensionallinear space of polynomials characterized by the highest root vector, anda factorization property for eigenfunctions.

Self-adjoint $n$-by-$n$ matrices have a natural partial ordering, namely $ A \leq B $ if the matrix $ B - A$ is positive semi-definite. In 1934 K. Loewner characterized functions that preserve this ordering; these functions are called $n$-matrix monotone. The condition depends on the dimension $n$, but if a function is $n$-matrix monotone for all $n$, then it must extend analytically to a function that maps the upper half-plane to itself. I will describe Loewner's results, and then discuss what happens

Asymptotics for L2 Christoffel functions are a classical topic in orthogonal polynomials. We present asymptotics for Lp Christoffel functions for measures on the unit circle. The formulation involves an extremal problem in Paley-Wiener space. While there have been estimates of the Lp Christoffel functions for a long time, the asymptotics are noew for p other than 2, even for Lebesgue measure on the unict circle.

In this talk, we investigate the structures of C*-algebras generated by collections of linear-fractionally-induced composition operators and either the forward shift or the ideal of compact operators. In the setting of the classical Hardy space, we present a full characterization of the structures, modulo the ideal of compact operators, of C*-algebras generated by a single linear-fractionally-induced composition operator and the forward shift. We apply the structure results to compute spectral information for algebraic combinations of

In this talk we discuss some nonlinear transformations between moment sequences. One of these transformations is the following: if (a_n)_n is a non-vanishing Hausdorff moment sequence then the sequence defined by 1/(a_0 ... a_n) is a Stieltjes moment sequence. Our approach is constructive and use Euler's idea of developing q-infinite products in power series. Some others transformations will be considered as well as some relevant moment sequences and analytic functions related to them. We will also propose some conjectures about moment transformations

Using integral formulas based on Green's theorem and in particular a lemma of Uchiyama, we give simple proofs of comparisons of different BMO norms without using the John-Nirenberg inequality while we also give a simple proof of the strong John-Nirenberg inequality. Along the way we prove the inclusions of BMOA in the dual of H^1 and BMO in the dual of real H^1. Some difficulties of the method and possible future directions to take it will be suggested at the end.