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Series: Analysis Seminar

We present results on the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. Our motivation comes from the fact that the zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral defined on the real axis and having a high order stationary point. The limit distribution of these zeros is also analyzed, and we show that they accumulate along a contour in the complex plane that has the S-property in the presence of an external field. Additionally, the strong asymptotics of the orthogonal polynomials is obtained by applying the nonlinear Deift--Zhou steepest descent method to the corresponding Riemann--Hilbert problem. This is joint work with D. Huybrechs and A. Kuijlaars, Katholieke Universiteit Leuven (Belgium).

Series: Analysis Seminar

Some discrete dynamical systems defined by a Lax pair are considered. The method of investigation is based on the analysis of the matrical moments for the main operator of the pair. The solutions of these systems are studied in terms of properties of this operator, giving, under some conditions, explicit expressions for the resolvent function.

Series: Analysis Seminar

The tangent cone of a set X in R^n at a point p of X is the limit of all rays which emanate from p and pass through sequences of points p_i of X as p_i converges to p. In this talk we discuss how C^1 regular hypersurfaces of R^n may be characterized in terms of their tangent cones. Further using the real nullstellensatz we prove that convex real analytic hypersurfaces are C^1, and will also discuss some applications to real algebraic geometry.

Series: Analysis Seminar

We discuss joint work with J.-M. Martell, in which werevisit the ``extrapolation method" for Carleson measures, originallyintroduced by John Lewis to proveA_\infty estimates for certain caloric measures, and we present a purely real variable version of the method. Our main result is a general criterion fordeducing that a weight satisfies a ReverseHolder estimate, given appropriate control by a Carleson measure.To illustrate the useof this technique,we reprove a well known theorem of R. Fefferman, Kenig and Pipherconcerning the solvability of the Dirichlet problem with data in some L^p space.

Series: Analysis Seminar

Series: Analysis Seminar

The Drury-Arveson space of functions on the unit ball in C^n has recently been intensively studied from the point of view function theory and operator theory. While much is known about this space of functions, a characterization of the interpolating sequences for the space has still remained elusive. In this talk, we will discuss the relevant background of the problem, and then I will discuss some work in progress and discuss a variant of the question for which we know the answer completely.

Series: Analysis Seminar

Let mu be a measure with compact support, with orthonormal polynomials {p_{n}} and associated reproducing kernels {K_{n}}. We show that bulk universality holds in measure in {x:mu'(x)>0}. The novelty is that there are no local or global conditions on the measure. Previous results have required regularity as a global condition, and a Szego condition as a local condition.As a consequence, for a subsequence of integers, universality holds for a.e. x. Under additional conditions on the measure, we show universality holds in an L_{p} sense for all finite p>0.

Series: Analysis Seminar

We prove a sharp Hardy inequality for fractional integrals for functions that are supported in a convex domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda. Further, the Hardy term in this inequality is stronger than the one in the classical case. The result can be extended as well to more general domains

Series: Analysis Seminar

Gasper in his 1971 Annals of Math paper proved that the Jacobi polynomials
satisfy a product formula which generalized the product formula of
Gegenbauer for ultraspherical polynomials. Gasper proved this by showing that
certains sums of triple products of Jacobi polynomials are positive
generalizing results of Bochner who earlier proved a similar results for
ultraspherical polynomials. These results allow a convolution structure for
Jacobi polynomials. We will give a simple proof of Gasper's and Bochner's
results using a Markov operator found by Carlen, Carvahlo, and Loss in their study of the
Kac model in kinetic theory. This is joint work with Eric Carlen and Michael Loss.

Series: Analysis Seminar

In this talk we will present some recent results about the matrix representation of the multiplication operator in terms of a basis of either orthogonal polynomials (OPUC) or orthogonal Laurent polynomials (OLPUC) with respect to a nontrivial probability measure supported on the unit circle. These are the so called GGT and CMV matrices.When spectral linear transformations of the measure are introduced, we will find the GGT and CMV matrices associated with the new sequences of OPUC and OLPUC, respectively. A connection with the QR factorization of such matrices will be stated. A conjecture about the generator system of such spectral transformations will be discussed.Finally, the Lax pair for the GGT and CMV matrices associated with some special time-depending deformations of the measure will be analyzed. In particular, we will study the Schur flow, which is characterized by a complex semidiscrete modified KdV equation and where a discrete analogue of the Miura transformation appears. Some open problems for time-depending deformations related to spectral linear transformations will be stated.This is a joint work with K. Castillo (Universidad Carlos III de Madrid) and L. Garza (Universidad Autonoma de Tamaulipas, Mexico).