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Karlheinz Gröchenig

Publications and source records attributed to Karlheinz Gröchenig.

At least 37 records · Page 2Linked to original sources
arXiv2021

Necessary Density Conditions for Sampling and Interpolation in Spectral Subspaces of Elliptic Differential Operators

We prove necessary density conditions for sampling in spectral subspaces of a second order uniformly elliptic differential operator on $R^d$ with slowly oscillating symbol. For constant coefficient operators, these are precisely Landaus necessary density conditions for bandlimited functions, but for more general elliptic differential operators it has been unknown whether such a critical density even exists. Our results prove the existence of a suitable critical sampling density and compute it in terms of the geometry defined by the elliptic operator. In dimension 1, functions in a spectral subspace can be interpreted as functions with variable bandwidth, and we obtain a new critical density for variable bandwidth. The methods are a combination of the spectral theory and the regularity theory of elliptic partial differential operators, some elements of limit operators, certain compactifications of $R^d $, and the theory of reproducing kernel Hilbert spaces.

math.FA↗
arXiv2021

Marcinkiewicz-Zygmund Inequalities for Polynomials in Fock Space

We study the relation between Marcinkiewicz-Zygmund families for polynomials in a weighted $L^2$-space and sampling theorems for entire functions in the Fock space and the dual relation between uniform interpolating families for polynomials and interpolating sequences. As a consequence we obtain a description of signal subspaces spanned by Hermite functions by means of Gabor frames.

math.CV↗
arXiv2021

Sampling the flow of a bandlimited function

We analyze the problem of reconstruction of a bandlimited function $f$ from the space-time samples of its states $f_t=ϕ_t\ast f$ resulting from the convolution with a kernel $ϕ_t$. It is well-known that, in natural phenomena, uniform space-time samples of $f$ are not sufficient to reconstruct $f$ in a stable way. To enable stable reconstruction, a space-time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space-time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function $\widehat f$ away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Turán type inequalities. En route, we obtain a Remez-Turán inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.

math.CA↗
arXiv2020

New Function Spaces Associated to Representations of Nilpotent Lie Groups and Generalized Time-Frequency Analysis

We study function spaces that are related to square-integrable, irreducible, unitary representations of several low-dimensional nilpotent Lie groups. These are new examples of coorbit theory and yield new families of function spaces on $\mathbb{R}^d $. The concrete realization of the representation suggests that these function spaces are useful for generalized time-frequency analysis or phase-space analysis.

math.FA↗
arXiv2020

Marcinkiewicz-Zygmund Inequalities for Polynomials in Bergmann and Hardy Spaces

We study the relationship between sampling sequences in infinite dimensional Hilbert spaces of analytic functions and Marcinkiewicz-Zygmund inequalities in subspaces of polynomials. We focus on the study of the Hardy space and the Bergman space in one variable because they provide two settings with a strikingly different behavior.

math.CV↗
arXiv2020

Balian-Low type theorems on homogeneous groups

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let $N$ be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let $(π, \mathcal{H}_π)$ be an irreducible, square-integrable representation modulo the center $Z(N)$ of $N$ on a Hilbert space $\mathcal{H}_π$ of formal dimension $d_π$. If $g \in \mathcal{H}_π$ is an integrable vector and the set $\{ π(λ)g : λ\in Λ\}$ for a discrete subset $Λ\subseteq N / Z(N)$ forms a frame for $\mathcal{H}_π$, then its density satisfies the strict inequality $D^-(Λ)> d_π$, where $D^-(Λ)$ is the lower Beurling density. An analogous density condition $D^+(Λ) < d_π$ holds for a Riesz sequence in $\mathcal{H}_π$ contained in the orbit of $(π, \mathcal{H}_π)$. The proof is based on a deformation theorem for coherent systems, a universality result for $p$-frames and $p$-Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.

math.FA↗
arXiv2019

Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator

We study the problem of recovering a function of the form $f(x) = \sum _{k\in \mathbb{Z} } c_k e^{-(x-k)^2}$ from its phaseless samples $|f(λ)|$ on some arbitrary countable set $Λ\subseteq \mathbb{R} $. For real-valued functions this is possible up to a sign for every separated set with Beurling density $D^-(Λ) >2$. This result is sharp. For complex-valued functions we find all possible solutions with the same phaseless samples.

math.CV↗
arXiv2019

Linear perturbations of the Wigner transform and the Weyl quantization

We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen's class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.

math.FA↗
arXiv2019

Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions

Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density. The techniques combine methods from several complex variables (estimates for $\bar \partial$) and the theory of localized frames in general reproducing kernel Hilbert spaces (with no analyticity assumed). The abstract results on Fekete points and deformation of frames may be of independent interest.

math.CA↗
arXiv2019

Sampling of Entire Functions of Several Complex Variables on a Lattice and Multivariate Gabor Frames

We give a general construction of entire functions in $d$ complex variables that vanish on a lattice of the form $L = A (Z + i Z )^d$ for an invertible complex-valued matrix. As an application we exhibit a class of lattices of density >1 that fail to be a sampling set for the Bargmann-Fock space in $C ^2$. By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.

math.CV↗
arXiv2019

Kernel Theorems in Coorbit Theory

We prove general kernel theorems for operators acting between coorbit spaces. These are Banach spaces associated to an integrable representation of a locally compact group and contain most of the usual function spaces (Besov spaces, modulation spaces, etc.). A kernel theorem describes the form of every bounded operator between a coorbit space of test functions and distributions by means of a kernel in a coorbit space associated to the tensor product representation. As special cases we recover Feichtinger's kernel theorem for modulation spaces and the recent generalizations by Cordero and Nicola. We also obtain a kernel theorem for operators between the Besov spaces $\dot{B}^0_{1,1}$ and $\dot{B}^{0}_{\infty, \infty }$.

math.FA↗
arXiv2019

Harmonic analysis in phase space and finite Weyl-Heisenberg ensembles

Weyl-Heisenberg ensembles are translation-invariant determinantal point processes on $\mathbb{R}^{2d}$ associated with the Schrödinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl-Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with $N$ points is asymptotically close to the restriction of the infinite Ginibre ensemble to the disk of area $N$, we define finite WH ensembles as adequate finite approximations of the restriction of infinite WH ensembles to a given domain $Ω$. We provide a precise rate for the convergence of the corresponding one-point intensities to the indicator function of $Ω$, as $Ω$ is dilated and the process is rescaled proportionally (thermodynamic regime). The construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials, but rather on phase-space methods. Second, we apply our construction to study the pure finite Ginibre-type polyanalytic ensembles, which model finite particle systems in a single Landau level, and are defined in terms of complex Hermite polynomials. On a technical level, we show that finite WH ensembles provide an approximate model for finite polyanalytic Ginibre ensembles, and we quantify the corresponding deviation. By means of this asymptotic description, we derive estimates for the rate of convergence of the one-point intensity of polyanalytic Ginibre ensembles in the thermodynamic limit.

math-ph↗
arXiv2018

Zeros of the Wigner Distribution and the Short-Time Fourier Transform

We study the question under which conditions the zero set of a (cross-) Wigner distribution W (f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson's theorem for the positivity of the Wigner distribution and to Hardy's uncertainty principle. We then construct a class of step functions S so that the Wigner distribution W (f, 1 (0,1)) always possesses a zero f $\in$ S $\cap$ L p for p < $\infty$, but may be zero-free for f $\in$ S $\cap$ L $\infty$. The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

math.CA↗
arXiv2018

Gabor Frames: Characterizations and Coarse Structure

This survey offers a systematic and streamlined exposition of the most important characterizations of Gabor frames over a lattice. The goal is to collect the most important characterizations of Gabor frames and offer a systematic exposition of these structures. In the center of these characterizations is the duality theorem for Gabor frames. Most characterizations within the $L^2$-theory follow directly from this fundamental duality. In particular, the celebrated characterizations of Janssen and Ron-Shen are consequences of the duality theorem, and the characterization of Zeevi and Zibulski for rational lattices also becomes a corollary. The novelty is the streamlined sequence of proofs, so that most of the structure theory of Gabor frames fits into a single, short article. The only prerequisite is the thorough mastery of the Poisson summation formula and some basic facts about frames and Riesz sequences.

math.FA↗