arXiv subjects

Jordy Timo van Velthoven

Publications and source records attributed to Jordy Timo van Velthoven.

At least 37 records · Page 2Linked to original sources

arXiv2024

Symplectic projective orbits of unimodular exponential Lie groups

For an exponential Lie group $G$ and an irreducible unitary representation $(π,\mathcal{H}_π)$ of $G$, we consider the natural action defined by $π$ on the projective space of $\mathcal{H}_π$, and show that the stabilisers of this action coincide with the projective kernel of $π$. Using this, we prove that, if $G/\mathrm{pker}(π)$ is unimodular, then $π$ admits a symplectic projective orbit if and only if $π$ is square-integrable modulo its projective kernel $\mathrm{pker}(π)$.

math.RT↗

arXiv2024

Classification of anisotropic local Hardy spaces and inhomogeneous Triebel-Lizorkin spaces

This paper provides a characterization of when two expansive matrices yield the same anisotropic local Hardy and inhomogeneous Triebel-Lizorkin spaces. The characterization is in terms of the coarse equivalence of certain quasi-norms associated to the matrices. For nondiagonal matrices, these conditions are strictly weaker than those classifying the coincidence of the corresponding homogeneous function spaces. The obtained results complete the classification of anisotropic Besov and Triebel-Lizorkin spaces associated to general expansive matrices.

math.CA↗

arXiv2024

Coorbit spaces associated to quasi-Banach function spaces and their molecular decomposition

This paper provides a self-contained exposition of coorbit spaces associated to integrable group representations and quasi-Banach function spaces, and at the same time extends and simplifies previous work. The main results provide an extension of the theory in [Studia Math., 180(3):237-253, 2007] from groups admitting a compact, conjugation-invariant unit neighborhood to arbitrary (possibly nonunimodular) locally compact groups. In addition, the present paper establishes the existence of molecular dual frames and Riesz sequences as in [J. Funct. Anal., 280(10):56, 2021] for the full scale of quasi-Banach function spaces. The theory is developed for possibly projective and reducible unitary representations in order to be easily applicable to well-studied function spaces not satisfying the classical assumptions of coorbit theory. Compared to the existing literature on quasi-Banach coorbit spaces, all our results apply under significantly weaker integrability conditions on the analyzing vectors, which allows for obtaining sharp results in concrete settings

math.FA↗

arXiv2023

Coherent systems over approximate lattices in amenable groups

Let $G$ be a second-countable amenable group with a uniform $k$-approximate lattice $Λ$. For a projective discrete series representation $(π, \mathcal{H}_π)$ of $G$ of formal degree $d_π > 0$, we show that $D^-(Λ) \geq d_π / k$ is necessary for the coherent system $π(Λ) g$ to be complete in $\mathcal{H}_π$. In addition, we show that if $π(Λ^2) g$ is minimal, then $D^+ (Λ^2) \leq d_π k$. Both necessary conditions recover sharp density theorems for uniform lattices and are new even for Gabor systems in $L^2 (\mathbb{R})$. As an application of the approach, we also obtain necessary density conditions for coherent frames and Riesz sequences associated to general discrete sets. All results are valid for amenable unimodular groups of possibly exponential growth.

math.FA↗

arXiv2023

Overcompleteness of coherent frames for unimodular amenable groups

This paper concerns the overcompleteness of coherent frames for unimodular amenable groups. It is shown that for coherent frames associated with a localized vector a set of positive Beurling density can be removed yet still leave a frame. The obtained results extend various theorems of [J. Fourier Anal. Appl., 12(3):307-344, 2006] to frames with non-Abelian index sets.

math.FA↗

arXiv2023

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I

This paper provides maximal function characterizations of anisotropic Triebel-Lizorkin spaces associated to general expansive matrices for the full range of parameters $p \in (0,\infty)$, $q \in (0,\infty]$ and $α\in \mathbb{R}$. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. As an application, the existence of dual molecular frames and Riesz sequences is obtained; the wavelet systems are generated by translations and anisotropic dilations of a single function, where neither the translation nor dilation parameters are required to belong to a discrete subgroup. Explicit criteria for molecules are given in terms of mild decay, moment, and smoothness conditions.

math.FA↗

arXiv2023

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, II

Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces $\dot{\mathbf{F}}^α_{p,q}$ for the endpoint case of $p = \infty$ and the full scale of parameters $α\in \mathbb{R}$ and $q \in (0,\infty]$. In particular, a Peetre-type characterization of the anisotropic Besov space $\dot{\mathbf{B}}^α_{\infty,\infty} = \dot{\mathbf{F}}^α_{\infty,\infty}$ is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in $\dot{\mathbf{F}}^α_{\infty,q}$.

math.FA↗

arXiv2022

Classification of anisotropic Triebel-Lizorkin spaces

This paper provides a classification theorem for expansive matrices $A \in \mathrm{GL}(d, \mathbb{R})$ generating the same anisotropic homogeneous Triebel-Lizorkin space $\dot{\mathbf{F}}^α_{p, q}(A)$ for $α\in \mathbb{R}$ and $p,q \in (0,\infty]$. It is shown that $\dot{\mathbf{F}}^α_{p, q}(A) = \dot{\mathbf{F}}^α_{p, q}(B)$ if and only if the homogeneous quasi-norms $ρ_A, ρ_B$ associated to the matrices $A, B$ are equivalent, except for the case $\dot{\mathbf{F}}^0_{p, 2} = L^p$ with $p \in (1,\infty)$. The obtained results complement and extend the classification of anisotropic Hardy spaces $H^p(A) = \dot{\mathbf{F}}^{0}_{p,2}(A)$, $p \in (0,1]$, in [Mem. Am. Math. Soc. 781, 122 p. (2003)].

math.FA↗

arXiv2022

Smooth lattice orbits of nilpotent groups and strict comparison of projections

This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian-Low type theorems for the non-existence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert $C^*$-module. An important ingredient in the approach is that twisted group $C^*$-algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple $C^*$-algebra has strict comparison of projections.

math.FA↗

arXiv2022

Integrability properties of quasi-regular representations of $NA$ groups

Let $G = N \rtimes A$, where $N$ is a graded Lie group and $A = \mathbb{R}^+$ acts on $N$ via homogeneous dilations. The quasi-regular representation $π= \mathrm{ind}_A^G (1)$ of $G$ can be realised to act on $L^2 (N)$. It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from $L^2 (N)$ into $L^2 (G)$ and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for proving decomposition theorems for function spaces on nilpotent Lie groups.

math.RT↗

arXiv2022

Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains

Let $π_α$ be a holomorphic discrete series representation of a connected semi-simple Lie group $G$ with finite center, acting on a weighted Bergman space $A^2_α (Ω)$ on a bounded symmetric domain $Ω$, of formal dimension $d_{π_α} > 0$. It is shown that if the Bergman kernel $k^{(α)}_z$ is a cyclic vector for the restriction $π_α |_Γ$ to a lattice $Γ\leq G$ (resp. $(π_α (γ) k^{(α)}_z)_{γ\in Γ}$ is a frame for $A^2_α(Ω)$), then $\mathrm{vol}(G/Γ) d_{π_α} \leq |Γ_z|^{-1}$. The estimate $\mathrm{vol}(G/Γ) d_{π_α} \geq |Γ_z|^{-1}$ holds for $k^{(α)}_z$ being a $p_z$-separating vector (resp. $(π_α (γ) k^{(α)}_z)_{γ\in Γ/ Γ_z}$ being a Riesz sequence in $A^2_α (Ω)$). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for $G = \mathrm{PSU}(1, 1)$.

math.FA↗

arXiv2022

Completeness of coherent state subsystems for nilpotent Lie groups

Let $G$ be a nilpotent Lie group and let $π$ be a coherent state representation of $G$. The interplay between the cyclicity of the restriction $π|_Γ$ to a lattice $Γ\leq G$ and the completeness of subsystems of coherent states based on a homogeneous $G$-space is considered. In particular, it is shown that necessary density conditions for Perelomov's completeness problem can be obtained via density conditions for the cyclicity of $π|_Γ$.

math.FA↗

arXiv2022

Invertibility of frame operators on Besov-type decomposition spaces

We derive an extension of the Walnut-Daubechies criterion for the invertibility of frame operators. The criterion concerns general reproducing systems and Besov-type spaces. As an application, we conclude that $L^2$ frame expansions associated with smooth and fast-decaying reproducing systems on sufficiently fine lattices extend to Besov-type spaces. This simplifies and improves recent results on the existence of atomic decompositions, which only provide a particular dual reproducing system with suitable properties. In contrast, we conclude that the $L^2$ canonical frame expansions extend to many other function spaces, and, therefore, operations such as analyzing using the frame, thresholding the resulting coefficients, and then synthesizing using the canonical dual frame are bounded on these spaces.

math.FA↗

arXiv2021

On sufficient density conditions for lattice orbits of relative discrete series

This note provides new criteria on a unimodular group $G$ and a discrete series representation $(π, \mathcal{H}_π)$ of formal degree $d_π > 0$ under which any lattice $Γ\leq G$ with $\text{vol}(G/Γ) d_π \leq 1$ (resp. $\text{vol}(G/Γ) d_π \geq 1$) admits $g \in \mathcal{H}_π$ such that $π(Γ) g$ is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.

math.FA↗

arXiv2021

The density theorem for discrete series representations restricted to lattices

This article considers the relation between the spanning properties of lattice orbits of discrete series representations and the associated lattice co-volume. The focus is on the density theorem, which provides a trichotomy characterizing the existence of cyclic vectors and separating vectors, and frames and Riesz sequences. We provide an elementary exposition of the density theorem, that is based solely on basic tools from harmonic analysis, representation theory, and frame theory, and put the results into context by means of examples.

math.FA↗

arXiv2021

On dual molecules and convolution-dominated operators

We show that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the size profile of a function is fully reflected by the size profile of its sampled values. The main tool is a local holomorphic calculus for convolution-dominated operators, valid for groups with possibly non-polynomial growth. Applied to the matrix coefficients of a group representation, our methods improve on classical results on atomic decompositions and bridge a gap between abstract and concrete methods.

math.FA↗

arXiv2020

Coorbit spaces associated to integrably admissible dilation groups

This paper considers coorbit spaces parametrized by mixed, weighted Lebesgue spaces with respect to the quasi-regular representation of the semi-direct product of Euclidean space and a suitable matrix dilation group. The class of dilation groups that we allow, the so-called integrably admissible dilation groups, contains the matrix groups yielding an irreducible, square-integrable quasi-regular representation as a proper subclass. The obtained scale of coorbit spaces extends therefore the well-studied wavelet coorbit spaces associated to discrete series representations. We show that for any integrably admissible dilation group there exists a convienent space of smooth, admissible analyzing vectors that can be used to define a consistent coorbit space possessing all the essential properties that are known to hold in the setting of discrete series representations. In particular, the obtained coorbit spaces can be realized as Besov-type decomposition spaces by means of a Littlewood-Paley type characterization. The classes of anisotropic Besov spaces associated to expansive matrices are shown to coincide precisely with the coorbit spaces induced by the integrably admissible one-parameter groups.

math.FA↗

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↗