arXiv subjects
Publications and source records attributed to Jordy Timo van Velthoven.
We prove strict necessary density conditions for smooth coherent frames and Riesz sequences in the orbit of a connected, simply connected, nilpotent Lie group. This extends previous results for homogeneous groups to general nilpotent Lie groups. The main novelty in our approach is that we prove a stability result for coherent frames and Riesz sequences based on deforming their representation rather than their index set.
We construct Riesz sequences whose Beurling density is arbitrary close to the critical density in the setting of reproducing kernel Hilbert spaces on metric measure spaces. In particular, our results apply to exponential systems on unbounded sets, nonlocalized Gabor systems and general coherent systems arising from nilpotent Lie groups. The methods used in this paper are based on our previous work on the redundancy of frames and a selector form of Weaver's conjecture.
We construct wavelet orthonormal bases for nonexpansive dilations and integer translations that do not admit a wavelet set. In addition, these orthonormal bases do not satisfy the Calderón sum formula. In particular, we show the existence of a wavelet basis for a dilation matrix with determinant one.
We show that a finite number of time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the so-called HRT conjecture of Heil, Ramanathan, and Topiwala. In particular, we provide an example consisting of 12 time-frequency shifts.
We survey recent classification theorems for expansive matrices that generate the same anisotropic homogeneous Triebel-Lizorkin function space or sequence space. The function spaces are classified precisely by those matrices for which their associated homogeneous quasi-norms on Euclidean space are equivalent, whereas the sequence spaces are classified by a strictly stronger condition. We unravel this discrepancy between function spaces and sequence spaces by showing that two sequence spaces are retracts of each other whenever the corresponding function spaces are the same.
We show the existence of a coherent frame in the orbit of a connected, simply connected unimodular solvable Lie group of exponential growth for which the lower Beurling density of its index set is zero.
We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices $A$ and $B$, it is shown that $\dot{\mathbf{f}}^α_{p,q}(A) = \dot{\mathbf{f}}^α_{p,q}(B)$ for all $α\in \mathbb{R}$ and $p, q \in (0, \infty]$ if and only if the set $\{A^j B^{-j} : j \in \mathbb{Z}\}$ is finite, or in the trivial case when $p = q$ and $|\det(A)|^{α+ 1/2 - 1/p} = |\det(B)|^{α+ 1/2 - 1/p}$. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.
We show that the Calderón sum formula for orthonormal wavelet bases holds for arbitrary dilation and translation matrices under a mild condition on the wavelet function. This partially solves a conjecture by Bownik and Lemvig.
We show that the frame measure function of a frame in certain reproducing kernel Hilbert spaces on metric measure spaces is given by the reciprocal of the Beurling density of its index set. In addition, we show that each such frame with Beurling density greater than one contains a subframe with Beurling density arbitrary close to one. This confirms that the concept of frame measure function as introduced by Balan and Landau is a meaningful quantitative definition for the redundancy of a large class of infinite frames. In addition, it shows that the necessary density conditions for sampling in reproducing kernel Hilbert spaces obtained by Führ, Gröchenig, Haimi, Klotz and Romero are optimal. As an application, we also settle the open questions of the existence of frames near the critical density for exponential frames on unbounded sets and for nonlocalized Gabor frames. The techniques used in this paper combine a selector form of Weaver's conjecture and various methods for quantifying the overcompleteness of frames.
We characterize exponential systems on sets of finite measure that form a frame or a Riesz sequence at the critical density. Namely, they are precisely those systems for which the underlying point set admits a weak limit that yields a Riesz basis. In combination with a recent result by Kozma, Nitzan and Olevskii, this shows that there exist sets that fail to possess a frame or Riesz sequence at the critical density, solving an open problem posed by Olevskii. As another consequence, we show that exponential frames and Riesz sequences over repetitive point sets (such as cut-and-project sets) at the critical density are already Riesz bases.
This note provides elementary proofs for necessary density conditions for frames and Riesz sequences in the lattice orbit of a discrete series representation that involve the projective stabiliser of the vector. The presented approach extends and simplifies known density conditions for Bergman kernels and lattices in semisimple Lie groups.
Given a relatively compact set $Ω\subseteq \mathbb{R}$ of Lebesgue measure $|Ω|$ and $\varepsilon > 0$, we show the existence of a set $Λ\subseteq \mathbb{R}$ of uniform density $D (Λ) \leq (1+\varepsilon) |Ω|$ such that the exponential system $\{ \exp(2πi λ\cdot) \mathbf{1}_Ω: λ\in Λ\}$ is a frame for $L^2 (Ω)$ with frame bounds $A |Ω|, B |Ω|$ for constants $A,B$ only depending on $\varepsilon$. This solves a problem on the frame bounds of an exponential frame near the critical density posed by Nitzan, Olevskii and Ulanovskii. We also prove an extension to locally compact abelian groups, which improves a result by Agora, Antezana and Cabrelli by providing frame bounds involving the spectrum.
This paper develops a theory of Besov spaces $\dot{\mathbf{B}}^σ_{p,q} (N)$ and Triebel-Lizorkin spaces $\dot{\mathbf{F}}^σ_{p,q} (N)$ on an arbitrary homogeneous group $N$ for the full range of parameters $p, q \in (0, \infty]$ and $σ\in \mathbb{R}$. Among others, it is shown that these spaces are independent of the choice of the Littlewood-Paley decomposition and that they admit characterizations in terms of continuous maximal functions and molecular frame decompositions. The defined spaces include as special cases various classical function spaces, such as Hardy spaces on homogeneous groups and homogeneous Sobolev spaces and Lipschitz spaces associated to sub-Laplacians on stratified groups.
We study the existence of certain characteristically nilpotent Lie algebras with flat coadjoint orbits. Their connected, simply connected Lie groups admit square-integrable representations modulo the center. There are many examples of nilpotent Lie groups admitting families of dilations and square-integrable representations. Much less is known about examples admitting square-integrable representations for which the quotient by the center does not admit a family of dilations. In this paper we construct a two-parameter family of characteristically nilpotent Lie groups $G(α,β)$ in dimension $11$, admitting square-integrable representations modulo the center $Z$, such that $G(α,β)/Z$ does not admit a family of dilations.
We prove various estimates for the asymptotics of counting functions associated to point sets of coherent frames and Riesz sequences. The obtained results recover the necessary density conditions for coherent frames and Riesz sequences for general unimodular amenable groups, while providing more precise estimates under additional localization conditions on the coherent system for groups of polynomial growth.
This note considers the finite linear independence of coherent systems associated to discrete subgroups. We show by simple arguments that such coherent systems of amenable groups are linearly independent whenever the associated twisted group ring does not contain any nontrivial zero divisors. We verify the latter for discrete subgroups in nilpotent Lie groups. For the particular case of time-frequency translates of Euclidean space, our approach provides a simple and self-contained proof of the Heil--Ramanathan--Topiwala (HRT) conjecture for subsets of arbitrary discrete subgroups.
This paper develops methods based on coarse geometry for the comparison of wavelet coorbit spaces defined by different dilation groups, with emphasis on establishing a unified approach to both irreducible and reducible quasi-regular representations. We show that the use of reducible representations is essential to include a variety of examples, such as anisotropic Besov spaces defined by general expansive matrices, in a common framework. The obtained criteria yield, among others, a simple characterization of subgroups of a dilation group yielding the same coorbit spaces. They also allow to clarify which anisotropic Besov spaces have an alternative description as coorbit spaces associated to irreducible quasi-regular representations.
On a homogeneous group, we characterize the one-parameter groups of dilations whose associated Hardy spaces in the sense of Folland and Stein are the same.