We provide a sufficient condition for sets of mobile sampling in terms of the surface density of the set.
In this paper, we study forms of the uncertainty principle suggested by problems in control theory. First, we prove an analogue of the Paneah-Logvinenko-Sereda Theorem characterizing sets which satisfy the Geometric Control Condition (GCC). This resu
lt is applied to get a uniqueness result for functions with spectrum supported on sufficiently flat sets. One corollary is that a function with spectrum in an annulus of a given thickness can be bounded, in $L^2$-norm, from above by its restriction to any open GCC set, independent of the radius of the annulus. This result is applied to the energy decay rates for damped fractional wave equations.
This paper builds upon two key principles behind the Bourgain-Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theo
rem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah-Logvinenko-Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on a fractal set. As well as recovering the result of Bourgain-Dyatlov, we obtain analogous uniqueness results for denser fractals.
In this paper, we give a harmonic analysis proof of the Neumann boundary observability inequality for the wave equation in an arbitrary space dimension. Our proof is elementary in nature and gives a simple, explicit constant. We also extend the metho
d to prove the observability inequality of a visco-elastic wave equation.
We derive necessary conditions for localization of continuous frames in terms of generalized Beurling densities. As an important application we provide necessary density conditions for sampling and interpolation in a very large class of reproducing kernel Hilbert spaces.
Toeplitz operators are met in different fields of mathematics such as stochastic processes, signal theory, completeness problems, operator theory, etc. In applications, spectral and mapping properties are of particular interest. In this survey we wil
l focus on kernels of Toeplitz operators. This raises two questions. First, how can one decide whether such a kernel is non trivial? We will discuss in some details the results starting with Makarov and Poltoratski in 2005 and their succeeding authors concerning this topic. In connection with these results we will also mention some intimately related applications to completeness problems, spectral gap problems and P{o}lya sequences. Second, if the kernel is non-trivial, what can be said about the structure of the kernel, and what kind of information on the Toeplitz operator can be deduced from its kernel? In this connection we will review a certain number of results starting with work by Hayashi, Hitt and Sarason in the late 80s on the extremal function.
We introduce the concept of weak-localization for generalized frames and use this concept to define a class of weakly localized operators. This class contains many important operators, including: Short Time Fourier Transform multipliers, Calderon-Toe
plitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transform.
We study the general moment problem for measures on the real line, with polynomials replaced by more general spaces of entire functions. As a particular case, we describe measures that are uniquely determined by a restriction of their Fourier transfo
rm to a finite interval. We apply our results to prove an extension of a theorem by Eremenko and Novikov on the frequency of oscillations of measures with a spectral gap (high-pass signals) near infinity.
It was known to von Neumann in the 1950s that the integer lattice $mathbb{Z}^2$ forms a uniqueness set for the Bargmann-Fock space. It was later demonstrated by Seip and Wallsten that a sequence of points $Gamma$ that is uniformly close to the intege
r lattice is still a uniqueness set. We show in this paper that the uniqueness sets for the Fock space are preserved under much more general perturbations.
In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing ke
rnels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when $1 < p < infty$, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the $1 < p < infty$ case.
