Seminars and Colloquia by Series

Hardy Inequalities for Fractional Integrals on Convex Domains

Series
Analysis Seminar
Time
Wednesday, February 24, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Craig SloaneGeorgia Tech
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

Gasper's identity and the Markov sequence problem

Series
Analysis Seminar
Time
Wednesday, February 10, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Jeff GeronimoGeorgia Tech
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.

Orthogonal Polynomials on the Unit Circle. Spectral transformations and their applications to integrable systems

Series
Analysis Seminar
Time
Wednesday, February 3, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Francisco MarcellánUniversidad Carlos III de Madrid
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).

On the independence of complex exponentials

Series
Analysis Seminar
Time
Thursday, January 28, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Mishko MitkovskiTexas A&M
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.

Asymptotic behavior of Müntz orthogonal polynomials

Series
Analysis Seminar
Time
Wednesday, January 20, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Úlfar StefánssonGeorgia Tech
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.

Uniform continuity and uniform convergence revisited

Series
Analysis Seminar
Time
Wednesday, January 13, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Gerald BeerCalifornia State University, Los Angeles
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.

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Series
Analysis Seminar
Time
Tuesday, December 8, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Xuan DuongMacquarie University
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.

An Extension of the Cordoba-Fefferman Theorem on the Equivalence Between the Boundedness Maximal and Multiplier Operators

Series
Analysis Seminar
Time
Wednesday, December 2, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Alexander StokolosGeorgia Southern University
I will speak about an extension of Cordoba-Fefferman Theorem on the equivalence between boundedness properties of certain classes of maximal and multiplier operators. This extension utilizes the recent work of Mike Bateman on directional maximal operators as well as my work with Paul Hagelstein on geometric maximal operators associated to homothecy invariant bases of convex sets satisfying Tauberian conditions.

How likely is Buffon's needle to land near a 1-dimensional Sierspinski gasket? A power estimate via Fourier analysis.

Series
Analysis Seminar
Time
Wednesday, November 18, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Matt BondMichigan State University
It is well known that a needle thrown at random has zero probability of intersecting any given irregular planar set of finite 1-dimensional Hausdorff measure. Sharp quantitative estimates for fine open coverings of such sets are still not known, even for such sets as the Sierpinski gasket and the 4-corner Cantor set (with self-similarities 1/4 and 1/3). In 2008, Nazarov, Peres, and Volberg provided the sharpest known upper bound for the 4-corner Cantor set. Volberg and I have recently used the same ideas to get a similar estimate for the Sierpinski gasket. Namely, the probability that Buffon's needle will land in a 3^{-n}-neighborhood of the Sierpinski gasket is no more than C_p/n^p, where p is any small enough positive number.

A topological separation condition for attractors of contraction mapping systems

Series
Analysis Seminar
Time
Wednesday, November 11, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sergiy BorodachovTowson University
We consider finite systems of contractive homeomorphisms of a complete metric space, which are non-redundanton every level. In general, this condition is weaker than the strong open set condition and is not equivalent to the weak separation property. We show that the set of N-tuples of contractive homeomorphisms, which satisfy this separation condition is a G_delta set in the topology of pointwise convergence of every component mapping with an additional requirement that the supremum of contraction coefficients of mappings in the sequence be strictly less than one.We also give several sufficient conditions for this separation property. For every fixed N-tuple of dXd invertible contraction matrices from a certain class, we obtain density results for vectors of fixed points, which defineN-tuples of affine contraction mappings having this separation property. Joint work with Tim Bedford (University of Strathclyde) and Jeff Geronimo (Georgia Tech).

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