Seminars and Colloquia by Series

Sharp Uncertainty Principles for Shift-Invariant Spaces

Series
Analysis Seminar
Time
Wednesday, November 11, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael NorthingtonVanderbilt University
Uncertainty principles are results which restrict the localization of a function and its Fourier transform. One class of uncertainty principles studies generators of structured systems of functions, such as wavelets or Gabor systems, under the assumption that these systems form a basis or some generalization of a basis. An example is the Balian-Low Theorem for Gabor systems. In this talk, I will discuss sharp, Balian-Low type, uncertainty principles for finitely generated shift-invariant subspaces of $L^2(\R^d)$. In particular, we give conditions on the localization of the generators which prevent these spaces from being invariant under any non-integer shifts.

Reflectionless Measures for Singular Integral Operators

Series
Analysis Seminar
Time
Wednesday, October 28, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Benjamin JayeKent State University
We shall describe how the study of certain measures called reflectionless measures can be used to understand the behaviour of oscillatory singular integral operators in terms of non-oscillatory quantities. The results described are joint work with Fedor Nazarov, Maria Carmen Reguera, and Xavier Tolsa

Bochner-Riesz multipliers associated to convex planar domains with rough boundary

Series
Analysis Seminar
Time
Wednesday, September 30, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Laura CladekUniversity of Wisconsin, Madison
We consider generalized Bochner-Riesz multipliers $(1-\rho(\xi))_+^{\lambda}$ where $\rho(\xi)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sj\"{o}lin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. For integers $m\ge 2$, we find domains such that $(1-\rho(\xi))_+^{\lambda}\in M^p(\mathbb{R}^2)$ for all $\lambda>0$ in the range $\frac{m}{m-1}\le p\le 2$, but for which $\inf\{\lambda:\,(1-\rho)_+^{\lambda}\in M_p\}>0$ when $p<\frac{m}{m-1}$. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the ``additive energy" of the boundary of a convex domain. Second, we associate a set of directions to a convex domain and define a sequence of Nikodym-type maximal operators corresponding to this set of directions. We show that domains that have low higher order energy, as well as those which have asymptotically good $L^p$ bounds for the corresponding sequence of Nikodym-type maximal operators, have improved $L^p$ bounds for the associated Bochner-Riesz operators over those proved by Seeger and Ziesler.

Cyclic polynomials in two variables

Series
Analysis Seminar
Time
Wednesday, September 23, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alan Sola University of South Florida
In my talk, I will discuss coordinate shifts acting on Dirichlet spaces on the bidisk and the problem of finding cyclic vectors for these operators. For polynomials in two complex variables, I will describe a complete characterization given in terms of size and nature of zero sets in the distinguished boundary.

Sobolev orthogonal polynomials in several variables

Series
Analysis Seminar
Time
Wednesday, August 26, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lidia FernandezApplied Math Dept, University of Granada
The purpose of this talk is to introduce some recent works on the field of Sobolev orthogonal polynomials. I will mainly focus on our two last works on this topic. The first has to do with orthogonal polynomials on product domains. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions. The second one analyzes a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which involves the outward normal derivatives on the sphere. Using the representation of these polynomials in terms of spherical harmonics, algebraic and analytic properties will be deduced. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of Sobolev orthogonal polynomials. Then explicit expressions for the norms will be obtained, among other properties.

Analytic Continuation of Analytic Fractals

Series
Analysis Seminar
Time
Wednesday, June 24, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael BarnsleyMathematical Sciences Institute, Australian National University
Examples of analytic fractals are Julia sets, Koch Curves, and Sierpinski triangles, and graphs of analytic functions. Given a piece of such a set, how does one "continue" it, in a manner consistent with the classical construction of an analytic Riemannian manifold, starting from a locally convergent series expansion?

On the convergence of Hermite-Pade approximants for rational perturbations of a Nikishin system

Series
Analysis Seminar
Time
Wednesday, May 6, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guillermo LopezUniversity of Madrid Carlos III
In the recent past multiple orthogonal polynomials have attracted great attention. They appear in simultaneous rational approximation, simultaneous quadrature rules, number theory, and more recently in the study of certain random matrix models. These are sequences of polynomials which share orthogonality conditions with respect to a system of measures. A central role in the development of this theory is played by the so called Nikishin systems of measures for which many results of the standard theory of orthogonal polynomials has been extended. In this regard, we present some results on the convergence of type I and type II Hermite-Pade approximation for a class of meromorphic functions obtained by adding vector rational functions with real coefficients to a Nikishin system of functions (the Cauchy transforms of a Nikishin system of measures).

Matrix weighted function spaces and the Carleson Embedding Theorem

Series
Analysis Seminar
Time
Wednesday, April 22, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Amalia CuliucBrown University
We will prove a recent version of the weighted Carleson Embedding Theorem for vector-valued function spaces with matrix weights. Time permitting, we will discuss the applications of this theorem to estimates on well-localized operators. This result relies heavily on the work of Kelly Bickel and Brett Wick and is joint with Sergei Treil.

A pointwise estimate for positive dyadic shifts and some applications

Series
Analysis Seminar
Time
Wednesday, April 15, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guillermo ReyMichigan State
We will prove a pointwise estimate for positive dyadic shifts of complexitym which is linear in the complexity. This can be used to give a pointwiseestimate for Calderon-Zygmund operators and to answer a question posed byA. Lerner. Several applications will be discussed.- This is joint work with Jose M. Conde-Alonso.

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