arXiv subjects

Hartmut Führ

Publications and source records attributed to Hartmut Führ.

At least 19 recordsLinked to original sources

arXiv2026

Characterization and Construction of Pairwise Orthogonal Parseval Frames with Applications to Sampling

In this paper, we provide a characterization of pairwise orthogonal frames with generalized translation-invariant (GTI) structures, based on the unconditional convergence property (UCP). These GTI systems are generated by translating functions over a countable family of closed, co-compact subgroups of a locally compact abelian (LCA) group $G$, where the families of subgroups associated with each system may differ. As an application of this characterization, we establish necessary and sufficient criteria for the orthogonality of various structured systems, including Gabor, wavelet, and shearlet systems on LCA groups. Furthermore, we derive a characterization of GTI Parseval (tight) frames and present explicit constructions of pairs of GTI systems using filters. Each constructed system satisfies the $\infty$-UCP and admits a Calder\'on sum equal to one. As a consequence of these results, the constructed systems form Parseval frames and are pairwise orthogonal. The proposed construction improves upon the technique in \cite{RGS} by relaxing the stationary assumption on the families of subgroups. Finally, we illustrate the results with examples using $B$-splines as generating functions and discuss applications of pairwise orthogonal frames in sampling theory.

math.FA↗

arXiv2026

Heisenberg uncertainty inequalities for locally compact abelian groups

We prove a version of Heisenberg's uncertainty principle for a rather general class of locally compact abelian groups. We compare the lower bound provided by our approach with the optimal lower bound in the Euclidean case, and formulate the Heisenberg uncertainty principle for local fields.

math.FA↗

arXiv2026

Translation complete subgroups of affine Weyl-Heisenberg groups and their generalized wavelet systems

The $n$-dimensional affine Weyl-Heisenberg group is a Lie group typically parameterized as $G_{aWH} = \mathbb{T} \times \mathbb{R}^n \times \widehat{\mathbb{R}^n} \times \mathrm{GL}(n, \mathbb{R})$, generated by all translation, dilation, and modulation operators acting on $L^2(G)$. It was introduced by Torr\'esani and his coauthors as a common framework to discuss both wavelet and time-frequency analysis, as well as possible intermediate constructions. In this paper, we focus on a particular class of subgroups of $G_{aWH}$, namely those of the form $G = \mathbb{T} \times \mathbb{R}^n \times V \times H$, where $V$ is a subspace of $\mathbb{R}^n$ and $H$ is a closed subgroup of $\mathrm{GL}(n, \mathbb{R})$. The main goal is to identify pairs $(V, H)$ that ensure the existence of an associated inversion formula, through the notion of square-integrable representations. We derive an admissibility criterion that is largely analogous to the well-known Calder\'on condition for the fully affine case, corresponding to $V = \{ 0 \}$. %The criteria for such a characterization can be formulated and proved in a way that is in many respects analogous to the affine case. We then identify $G_{aWH}$ as a subgroup of the semidirect product of the $n$-dimensional Heisenberg group and the symplectic group $Sp(n,\mathbb{R})$, which acts via the extended metaplectic representation, and compare our admissibility conditions to existing criteria based on Wigner functions. Finally, we present a list of novel examples in dimensions two and three which illustrate the potential of our approach, and present some foundational results regarding the systematic construction, classification, and conjugacy of these groups.

math.FA↗

arXiv2025

Cheeger Bounds for Stable Phase Retrieval in Reproducing Kernel Hilbert Spaces

Phase retrieval seeks to reconstruct a signal from phaseless intensity measurements and, in applications where measurements contain errors, demands stable reconstruction. We study local stability of phase retrieval in reproducing kernel Hilbert spaces. Motivated by Grohs-Rathmair's Cheeger-type estimate for Gabor phase retrieval, we introduce a kernel Cheeger constant that quantifies connectedness relative to kernel localization. This notion yields a clean stability certificate: we establish a unified lower bound over both real and complex fields, and in the real case also an upper bound, each in terms of the reciprocal kernel Cheeger constant. Our framework treats finite- and infinite-dimensional settings uniformly and covers discrete, semi-discrete, and continuous measurement domains. For generalized wavelet phase retrieval from (semi-)discrete frames, we bound the kernel Cheeger constant by the Cheeger constant of a data-dependent weighted graph. We further characterize phase retrievability for generalized wavelet transforms and derive a simple sufficient criterion for wavelet sign retrieval in arbitrary dimension for transforms associated with irreducibly admissible matrix groups.

math.FA↗

arXiv2025

The Restricted Isometry Property for Measurements from Group Orbits

It is known that sparse recovery by measurements from random circulant matrices provides good recovery bounds. We generalize this to measurements that arise as a random orbit of a group representation for some finite group G. We derive estimates for the number of measurements required to guarantee the restricted isometry property with high probability. Following this, we present several examples highlighting the role of appropriate representation-theoretic assumptions.

cs.IT↗

arXiv2025

Wavelet Coorbit Spaces over Local Fields

This paper studies wavelet coorbit spaces on disconnected local fields $K$, associated to the quasi-regular representation of $G = K \rtimes K^*$ acting on $L^2(K)$. We show that coorbit space theory applies in this context, and identify the homogeneous Besov spaces $\dot{B}_{α,s,t}(K)$ as coorbit spaces. We identify a particularly convenient space $\mathcal{S}_0(K)$ of wavelets that give rise to tight wavelet frames via the action of suitable, easily determined discrete subsets $R \subset G$, and show that the resulting wavelet expansions converge simultaneously in the whole range of coorbit spaces. For orthonormal wavelet bases constructed from elements of $\mathcal{S}_0(K)$, the associated wavelet bases turn out to be unconditional bases for all coorbit spaces. We give explicit constructions of tight wavelet frames and wavelet orthonormal bases to which our results apply.

math.FA↗

arXiv2025

Classifying Wavelet Coorbit Spaces in Dimension 2

Coorbit spaces provide a rigorous framework for the assessment of the approximation theoretic properties of generalized wavelet systems. It is therefore useful to understand when two different wavelet systems give rise to the same scales of coorbit spaces. This paper provides an exhaustive answer to this question for the case of continuous wavelet transforms associated with matrix groups in dimension two.

math.FA↗

arXiv2025

Sparse Recovery from Group Orbits

While most existing sparse recovery results allow only minimal structure within the measurement scheme, many practical problems possess significant structure. To address this gap, we present a framework for structured measurements that are generated by random orbits of a group representation associated with a finite group. We differentiate between two scenarios: one in which the sampling set is fixed and another in which the sampling set is randomized. For each case, we derive an estimate for the number of measurements required to ensure that the restricted isometry property holds with high probability. These estimates are contingent upon the specific representation employed. For this reason, we analyze and characterize various representations that yield favorable recovery outcomes, including the left regular representation. Our work not only establishes a comprehensive framework for sparse recovery of group-structured measurements but also generalizes established measurement schemes, such as those derived from partial random circulant matrices.

cs.IT↗

arXiv2025

Consistent sampling of Paley-Wiener functions on graphons

We study sampling methods for Paley-Wiener functions on graphons, thereby adapting and generalizing methods initially developed for graphs to the graphon setting. We then derive conditions under which such a sampling estimate is consistent with graphon convergence.

eess.SP↗

arXiv2024

On wavelet coorbit spaces associated to different dilation groups

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.

math.FA↗

arXiv2024

Energy Propagation in Scattering Convolution Networks Can Be Arbitrarily Slow

We analyze energy decay for deep convolutional neural networks employed as feature extractors, including Mallat's wavelet scattering transform. For time-frequency scattering transforms based on Gabor filters, previous work has established that energy decay is exponential for arbitrary square-integrable input signals. In contrast, our main results allow proving that this is false for wavelet scattering in arbitrary dimensions. Specifically, we show that the energy decay of wavelet and wavelet-like scattering transforms acting on generic square-integrable signals can be arbitrarily slow. Importantly, this slow decay behavior holds for dense subsets of $L^2(\mathbb{R}^d)$, indicating that rapid energy decay is generally an unstable property of signals. We complement these findings with positive results that allow us to infer fast (up to exponential) energy decay for generalized Sobolev spaces tailored to the frequency localization of the underlying filter bank. Both negative and positive results highlight that energy decay in scattering networks critically depends on the interplay between the respective frequency localizations of both the signal and the filters used.

math.FA↗

arXiv2024

Wavelet characterizations of the Sobolev wavefront set: bandlimited wavelets and compactly supported wavelets

We consider the problem of characterizing the Sobolev wavefront set of a tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{d})$ in terms of its continuous wavelet transform, with the latter being defined with respect to a suitably chosen dilation group $H\subset{\rm GL}(\mathbb{R}^{d})$. We derive necessary and sufficient criteria for elements of the Sobolev wavefront set, formulated in terms of the decay behaviour of a given generalized continuous wavelet transform. It turns out that the characterization of directed smoothness of finite order can be performed in the two important cases: (1) bandlimited wavelets, and (2) wavelets with finitely many vanishing moments (e.g.~compactly supported wavelets). The main results of this paper are based on a number of fairly technical conditions on the dilation group. In order to demonstrate their wide applicability, we exhibit a large class of generalized shearlet groups in arbitrary dimensions fulfilling all required conditions, and give estimates of the involved constants.

math.FA↗

arXiv2023

The metaplectic action on modulation spaces

We study the mapping properties of metaplectic operators $\widehat{S}\in \mathrm{Mp}(2d,\mathbb{R})$ on modulation spaces of the type $\mathrm{M}^{p,q}_m(\mathbb{R}^d)$. Our main result is a full characterisation of the pairs $(\widehat{S},\mathrm{M}^{p,q}(\mathbb{R}^d))$ for which the operator $\widehat{S}:\mathrm{M}^{p,q}(\mathbb{R}^d) \to \mathrm{M}^{p,q}(\mathbb{R}^d)$ is (i) well-defined, (ii) bounded. It turns out that these two properties are equivalent, and they entail that $\widehat{S}$ is a Banach space automorphism. For polynomially bounded weight functions, we provide a simple sufficient criterion to determine whether the well-definedness (boundedness) of ${\widehat{S}:\mathrm{M}^{p,q}{}(\mathbb{R}^d)\to \mathrm{M}^{p,q}(\mathbb{R}^d)}$ transfers to $\widehat{S}:\mathrm{M}^{p,q}_m(\mathbb{R}^d)\to \mathrm{M}^{p,q}_m(\mathbb{R}^d)$.

math.FA↗

arXiv2023

Dilational Symmetries of Decomposition and Coorbit Spaces

We investigate the invariance properties of general wavelet coorbit spaces and Besov-type decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasi-isometry properties with respect to a certain metric in frequency domain. We formulate versions of this phenomenon both for the decomposition and coorbit space settings. We then apply the general results to a particular class of dilation groups, the so-called shearlet dilation groups. We present a general, algebraic characterization of matrices the are coorbit compatible with a given shearlet dilation group. We determine the groups of compatible dilations for a variety of concrete examples.

math.FA↗

arXiv2023

Phase retrieval for affine groups over prime fields

We study phase retrieval for group frames arising from permutation representations, focusing on the action of the affine group of a finite field. We investigate various versions of the phase retrieval problem, including conjugate phase retrieval, sign retrieval, and matrix recovery. Our main result establishes that the canonical irreducible representation of the affine group $\mathbb{Z}_p \rtimes \mathbb{Z}_p^\ast$ (with $p$ prime), acting on the vectors in $\mathbb{C}^{p}$ with zero-sum, has the strongest retrieval property, allowing to reconstruct matrices from scalar products with a group orbit consisting of rank-one projections. We explicitly characterize the generating vectors that ensure this property, provide a linear matrix recovery algorithm and explicit examples of vectors that allow matrix recovery. We also comment on more general permutation representations.

math.RT↗

arXiv2023

Phase retrieval for nilpotent groups

We study the phase retrieval property for orbits of general irreducible representations of nilpotent groups, for the classes of simply connected connected Lie groups, and for finite groups. We prove by induction that in the Lie group case, all irreducible representations do phase retrieval. For the finite group case, we mostly focus on $p$-groups. Here our main result states that every irreducible representation of an arbitrary $p$-group with exponent $p$ and size $\le p^{2+p/2}$ does phase retrieval. Despite the fundamental differences between the two settings, our inductive proof methods are remarkably similar.

math.RT↗

arXiv2022

Classifying decomposition and wavelet coorbit spaces using coarse geometry

This paper is concerned with the study of Besov-type decomposition spaces, which are scales of spaces associated to suitably defined coverings of the euclidean space $\mathbb{R}^d$, or suitable open subsets thereof. A fundamental problem in this domain, that is currently not well understood, is deciding when two different coverings give rise to the same scale of decomposition spaces. In this paper, we establish a coarse geometric approach to this problem, and show how it specializes for the case of wavelet coorbit spaces associated to a particular class of matrix groups $H < GL(\mathbb{R}^d)$ acting via dilations. This class can be understood as a special case of decomposition spaces, and it turns out that the question whether two different dilation groups $H_1,H_2$ have the same coorbit spaces can be decided by investigating whether a suitably defined map $ϕ: H_1 \to H_2$ is a quasi-isometry with respect to suitably defined word metrics. We then proceed to apply this criterion to a large class of dilation groups called {\em shearlet dilation groups}, where this quasi-isometry condition can be characterized algebraically. We close with the discussion of selected examples.

math.FA↗

arXiv2021

Embeddings of anisotropic Besov spaces into Sobolev spaces

We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix $A$ into Sobolev spaces, with focus on the influence of $A$ on the embedding behaviour. For a large range of parameters, we derive sharp characterizations of embeddings.

math.FA↗