Seminars and Colloquia by Series

Wednesday, March 31, 2010 - 14:00 , Location: Skiles 269 , Paul Terwilliger , University of Wisconsin - Madison , Organizer: Plamen Iliev
Wednesday, March 17, 2010 - 14:00 , Location: Skiles 269 , Brett Wick , Georgia Tech , Organizer: Plamen Iliev
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.
Wednesday, March 3, 2010 - 14:00 , Location: Skiles 269 , Doron Lubinsky , Georgia Tech , Organizer: Plamen Iliev
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.
Wednesday, February 24, 2010 - 14:00 , Location: Skiles 269 , Craig Sloane , Georgia Tech , Organizer: Plamen Iliev
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
Wednesday, February 10, 2010 - 14:00 , Location: Skiles 269 , Jeff Geronimo , Georgia Tech , Organizer: Plamen Iliev
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.
Wednesday, February 3, 2010 - 14:00 , Location: Skiles 269 , Francisco Marcellán , Universidad Carlos III de Madrid , Organizer: Plamen Iliev
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).
Thursday, January 28, 2010 - 13:00 , Location: Skiles 255 , Mishko Mitkovski , Texas A&M , Organizer: Michael Lacey
Given a set of complex exponential  e^{i \lambda_n x}  how large do you have to take r so that  the sequence is independent in  L^2[-r,r] ?  The answer is given in terms of the Beurling-Mallivan density. 
Wednesday, January 20, 2010 - 14:00 , Location: Skiles 269 , Úlfar Stefánsson , Georgia Tech , Organizer: Plamen Iliev
Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics on the interval, and the zero spacing behavior follows. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval, and the asymptotic properties of the associated Christoffel functions.
Wednesday, January 13, 2010 - 14:00 , Location: Skiles 269 , Gerald Beer , California State University, Los Angeles , Organizer: Plamen Iliev
Sandro Levi and I have investigated variational strengthenings of uniform continuity and uniform  convergence of nets or sequences of functions with respect to a family of subsets of the domain. Out of our theory comes  an answer to this basic question: what is the weakest topology  stronger than the topology of pointwise convergence in which continuity is preserved under taking limits?   We argue that the classical theory constitues a misunderstanding of what is fundamentally a variational phenomenon.
Tuesday, December 8, 2009 - 16:00 , Location: Skiles 269 , Xuan Duong , Macquarie University , Organizer: Michael Lacey
In this talk,we study  weighted L^p-norm inequalities  for general spectralmultipliersfor self-adjoint positive definite operators on L^2(X), where X is a space of homogeneous type. We show that the sharp weighted Hormander-type spectral multiplier theorems  follow  from the appropriate estimatesof the L^2 norm of the kernel of spectral multipliers and the Gaussian boundsfor the corresponding heat kernels. These results are applicable to spectral multipliersfor group invariant  Laplace operators acting on Lie groups of polynomialgrowth and   elliptic operators on compact manifolds. This is joint work with Adam Sikora and Lixin Yan.