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Series: Analysis Seminar

Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems. It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of $s \log(n/s)$ - $n$ is ambient dimension and $s$ is the sparsity threshold. The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix. A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing. Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere. We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.

Series: Analysis Seminar

In this work we prove that the space of two parameter, matrix-valued BMO functions can be
characterized by considering iterated commutators with the Hilbert transform. Specifically, we
prove that the norm in the BMO space is equivalent to the norm of the
commutator of the BMO function with the Hilbert transform, as an
operator on L^2.
The upper bound estimate relies on a representation of the Hilbert transform as an average of dyadic
shifts, and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and
Lacey's wavelet proof for the scalar case.

Series: Analysis Seminar

We will describe sufficient conditions on a set $\Lambda \subset [0,2\pi) $ so that the maximal operator below is bound on $\ell^2(Z)$. $$\sup _{\lambda \in \Lambda} \Big| \sum_{n\neq 0} e^{i \lambda n^2} f(x-n)/n\Big|$$ The integral version of this result is an influential result to E.M. Stein. Of course one should be able to take $\Lambda = [0,2\pi) $, but such a proof would have to go far beyond the already complicated one we will describe. Joint work with Ben Krause.

Series: Analysis Seminar

A signal is a high dimensional vector x, and a measurement is the inner product . A one-bit measurement is the sign of . These are basic objects, as will be explained in the talk, with the help of some videos of photons. The import of this talk is that one bit measurements can be as effective as the measurements themselves, in that the same number of measurements in linear and one bit cases ensure the RIP property. This is explained by a connection with variants of classical spherical cap discrepancy. Joint work with Dimtriy Bilyk.

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