Weighted inequalities for singular integral operators are central in the study of non-homogeneous harmonic analysis. Two weight inequalities for singular integral operators, in-particular attracted attention as they can be essential in the perturbation theory of unitary matrices, spectral theory of Jacobi matrices and PDE's.
Mobile sampling concerns finding surfaces upon which any function with Fourier transform supported in a symmetric convex set must have some large values. We shall describe a sharp sufficient for mobile sampling in terms of the surface density introduced by Unnikrishnan and Vetterli. Joint work with Mishko Mitkovski and Manasa Vempati.
Given a discrete set $\Lambda\subseteq\mathbb{R}$ and an interval $I$, define the sequence of complex exponentials in $L^2(I)$, $\mathcal{E}(\Lambda)$, by $\{e^{2\pi i\lambda t}\colon \lambda\in\Lambda\}$. A fundamental result in harmonic analysis says that if $\mathcal{E}(\frac{1}{b}\mathbb{Z})$ is an orthogonal basis for $L^2(I)$ for any interval $I$ of length $b$. It is also well-known that there exist sets $\Lambda$, which may be irregular, such that sets $\mathcal{E}(\Lambda)$ form nonorthogonal bases (known as Riesz bases) for $L^2(S)$, for $S\subseteq\math
We estimate the Riesz basis (RB) bounds obtained in Hruschev, Nikolskii and Pavlov' s classical characterization of exponential RB. As an application, we improve previously known estimates of the RB bounds in some classical cases, such as RB obtained by an Avdonin type perturbation, or RB which are the zero-set of sine-type functions. This talk is based on joint work with S. Nitzan
Let $H$ be a separable Hilbert space and let $\{x_n\}$ be a Bessel sequence or a frame for $H$ which does not contain any zero elements. We say that $\{x_n\}$ is a normalizable Bessel sequence or normalizable frame if the normalized sequence $\{x_n/||x_n||\}$ remains a Bessel sequence or frame. In this talk, we will present characterizations of normalizable and non-normalizable frames . In particular, we prove that normalizable frames can only have two formulations. Perturbation theorems tailored for normalizable frames will be also presented.
In 1996, C.~Heil, J.~Ramanatha, and P.~Topiwala conjectured that the (finite) set $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N$ is linearly independent for any non-zero square integrable function $g$ and subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ This problem is now known as the HRT Conjecture, and is still largely unresolved.
This talk will detail two recent papers concerning Rogers-Shephard inequalities and Zhang inequalities for various classes of measures, the first of which is a reverse form of the Brunn-Minkowsk inequality, and the second of which can be seen to be a reverse affine isoperimetric inequality; the feature of both inequalities is that they each provide a classification of the n-dimensional simplex in the volume case. The covariogram of a measure plays an essential role in the proofs of each of these inequalities.
Let f be a real-valued Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution.
What is the probability that f remains above a certain fixed line for a long period of time?
We give simple spectral(and almost tight) conditions under which this probability is asymptotically exponential, that is, that the limit of log P(f>a on [0,T])/ T, as T approaches infinity, exists.
In this talk, we show an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we present the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of Polya-Szego: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space.
