Georgia Institute of Technology College of Sciences

I will review recent and classical results concerning the asymptotic properties (as N --> \infty) of 'ground state' configurations of N particles restricted to a d-dimensional compact set A\subset {\bf R}^p that minimize the Riesz s-energy functional \sum_{i\neq j}\frac{1}{|x_{i}-x_{j}|^{s}} for s>0. Specifically, we will discuss the following (1) For s < d, the ground state configurations have limit distribution as N --> \infty given by the equilibrium measure \mu_s, while the first order asymptotic growth of the energy of these configurations is given by the 'transfinite diameter' of A. (2) We study the behavior of \mu_s as s approaches the critical value d (for s\ge d, there is no equilibrium measure). In the case that A is a fractal, the notion of 'order two density' introduced by Bedford and Fisher naturally arises. This is joint work with M. Calef. (3) As s --> \infty, ground state configurations approach best-packing configurations on A. In work with S. Borodachov and E. Saff we show that such configurations are asymptotically uniformly distributed on A.

In this talk, I will discuss some results obtained in my Ph.D. thesis. First, the point mass formula will be introduced. Using the formula, we shall see how the asymptotics of orthogonal polynomials relate to the perturbed Verblunsky coefficients. Then I will discuss two classes of measures on the unit circle -- one with Verblunsky coefficients \alpha_n --> 0 and the other one with \alpha_n --> L (non-zero) -- and explain the methods I used to tackle the point mass problem involving these measures. Finally, I will discuss the point mass problem on the real line. For a long time it was believed that point mass perturbation will generate exponentially small perturbation on the recursion coefficients. I will demonstrate that indeed there is a large class of measures such that that proposition is false.

Modulation spaces are a class of Banach spaces which provide a quantitative time-frequency analysis of functions via the Short-Time Fourier Transform. The modulation spaces are the "right" spaces for time-frequency analysis andthey occur in many problems in the same way that Besov Spaces are attached to wavelet theory and issues of smoothness. In this seminar, I will talk about embeddings of modulation Spaces into BMO or VMO (the space of functions of bounded or vanishing mean oscillation, respectively ). Membership in VMO is central to the Balian-Low Theorem, which is a cornerstone of time-frequency analysis.

We consider multipoint Padé approximation to Cauchy transforms of complex measures. First, we recap that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-continuous non-vanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. Second, we show that this convergence holds also for measures whose Radon–Nikodym derivative is a Jacobi weight modified by a Hölder continuous function. The asymptotics behavior of Padé approximants is deduced from the analysis of underlying non–Hermitian orthogonal polynomials, for which the Riemann–Hilbert–∂ method is used.