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Series: Analysis Seminar

We
study the construction of exponential bases and exponential frames
on general $L^2$ space with the measures supported on self-affine
fractals. This problem dates back to the conjecture of Fuglede. It lies
at the interface between analysis, geometry and number theory and it
relates to translational tilings. In this talk,
we give an introduction to this topic, and report on some of the recent
advances. In particular, the possibility of constructing exponential
frames on fractal measures without exponential bases will be discussed.

Series: Analysis Seminar

The Minkowski question mark function is a singular distribution function arising from Number Theory: it maps all quadratic irrationals to rational numbers and rational numbers to dyadic numbers. It generates a singular measure on [0,1]. We are interested in the behavior of the norms and recurrence coefficients of the orthonormal polynomials for this singular measure. Is the Minkowski measure a "regular" measure (in the sense of Ullman, Totik and Stahl), i.e., is the asymptotic zero distribution the equilibrium measure on [0,1]
and do the n-th roots of the norm converge to the capacity (which is 1/4)?
Do the recurrence coefficients converge (are the orthogonal polynomials in Nevai's class). We provide some numerical results which give some indication but which are not conclusive.

Series: Analysis Seminar

[Special time and location] The content of this talk is joint work with Yumeng Ou. We describe a novel framework for the he analysis of multilinear singular integrals acting on Banach-valued functions.Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces, including, in particular, noncommutative Lp spaces. A concrete case of our result is a multilinear generalization of Weis' operator-valued Hormander-Mihlin linear multiplier theorem.Furthermore, we derive from our main result a wide range of mixed Lp-norm estimates for multi-parameter multilinear multiplier operators, as well as for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilinear Hilbert transform. These respectively extend the results of Muscalu et. al. and of Silva to the mixed norm case and provide new mixed norm fractional Leibnitz rules.

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.