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Series: Analysis Seminar

We will discuss the problem of restricting the Fourier transform
to manifolds for which the curvature vanishes on some nonempty set. We
will give background and discuss the problem in general terms, and then
outline a proof of an essentially optimal (albeit conditional) result for a
special class of hypersurfaces.

Series: Analysis Seminar

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.

Series: Analysis Seminar

Series: Analysis Seminar

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

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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?

Series: Analysis Seminar

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).

Series: Analysis Seminar

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.