arXiv subjects

Hartmut Führ

Publications and source records attributed to Hartmut Führ.

At least 37 records · Page 2Linked to original sources

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

On R-duals of type III in Hilbert spaces

Following work by Casazza, Kutyniok and Lammers, and extensions by Stoeva and Christensen, we provide some novel characterizations of R-dual sequences of type III in Hilbert spaces. We systematically extend the construction procedure by basing it on a choice of an antiunitary involution. For certain classes of R-duals of type III, we derive a representation of the associated frame operator in terms of spectral measures.

math.RT↗

arXiv2019

Embeddings of shearlet coorbit spaces into Sobolev spaces

We investigate the existence of embeddings of shearlet coorbit spaces associated to weighted mixed $L^p$-spaces into classical Sobolev spaces in dimension three by using the description of coorbit spaces as decomposition spaces. This different perspective on these spaces enables the application of embedding results that allow the complete characterization of embeddings for certain integrability exponents, and thus provides access to a deeper understanding of the smoothness properties of coorbit spaces, and of the influence of the choice of shearlet groups on these properties. We give a detailed analysis, identifying which features of the dilation groups have an influence on the embedding behavior, and which do not. Our results also allow to comment on the validity of the interpretation of shearlet coorbit spaces as smoothness spaces.

math.FA↗

arXiv2017

Density of Sampling and Interpolation in Reproducing Kernel Hilbert Spaces

We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is small compared to the volume of balls (weak annular decay property) and if the kernel possesses some off-diagonal decay or even some weaker form of localization, then there exists a critical density $D$ with the following property: a set of sampling has density $\geq D$, whereas a set of interpolation has density $\leq D$. The main theorem unifies many known density theorems in signal processing, complex analysis, and harmonic analysis. For the special case of bandlimited function we recover Landau's fundamental density result. In complex analysis we rederive a critical density for generalized Fock spaces. In harmonic analysis we obtain the first general result about the density of coherent frames.

math.FA↗

arXiv2017

System bandwidth and the existence of generalized shift-invariant frames

We consider the question whether, given a countable system of lattices $(Γ_j)_{j \in J}$ in a locally compact abelian group $G$, there exists a sequence of functions $(g_j)_{j \in J}$ such that the resulting generalized shift-invariant system $(g_j(\cdot - γ))_{j \in J, γ\in Γ_j}$ is a tight frame of $L^2(G)$. This paper develops a new approach to the study of almost periodic functions for generalized shift-invariant systems based on an \emph{unconditionally convergence property}, replacing previously used local integrability conditions. From this theory, we derive characterizing relations for tight and dual frame generators, we introduce the \emph{system bandwidth} as a measure of the total bandwidth a generalized shift-invariant system can carry, and we show that the so-called Calderón sum is uniformly bounded from below for generalized shift-invariant frames. We exhibit a condition on the lattice system for which the unconditionally convergence property is guaranteed to hold. Without the unconditionally convergence property, we show, counter intuitively, that even orthonormal bases can have arbitrary small system bandwidth. Our results show that the question of existence of frame generators for a general lattice system can be rather subtle, depending on analytical properties, such as the system bandwidth, as well as on algebraic properties of the lattice system.

math.FA↗

arXiv2016

A classification of continuous wavelet transforms in dimension three

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H < {\rm GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.

math.FA↗

arXiv2016

A classification of anisotropic Besov spaces

We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices $A \in {\rm GL}(d,\mathbb{R})$, with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative description as decomposition spaces. This result allows to relate properties of function spaces to combinatorial properties of the underlying coverings. This principle is applied to the question of classifying dilation matrices. It turns out the scales of homogeneous and inhomogeneous Besov spaces differ in the way they depend on the dilation matrix: Two matrices $A,B$ that induce the same scale of homogeneous Besov spaces also induce the same scale of inhomogeneous spaces, but the converse of this statement is generally false. Furthermore, the question whether $A,B$ induce the same scale of homogeneous spaces is closely related to the question whether they induce the same scale of Hardy spaces; the latter question had been previously studied by Bownik. We give a complete characterization of the different types of equivalence in terms of the Jordan normal forms of $A,B$.

math.FA↗

arXiv2016

Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings

The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: First, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is possible to treat the full class of generalized shearlet groups in a comprehensive and unified way, thus generalizing known results to an infinity of new cases. Our presentation emphasizes the interplay between the algebraic structure underlying the construction of the shearlet dilation groups, the geometric properties of the dual action, and the analytic properties of the associated shearlet transforms.

math.FA↗

arXiv2015

Geometric Space-Frequency Analysis on Manifolds

This paper gives a survey of methods for the construction of space-frequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace-Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections. After the discussion of the real line, our presentation starts out in the most abstract setting proving rather general sampling-type results for appropriately defined Paley-Wiener vectors in Hilbert spaces. These results allow a handy construction of Paley-Wiener frames in $L_2(\mfd{M})$, for a Riemann manifold of bounded geometry, essentially by taking a partition of unity in frequency domain. The discretization of the associated integral kernels then gives rise to frames consisting of smooth functions in $L_2(\mfd{M})$, with fast decay in space and frequency. These frames are used to introduce new norms in corresponding Besov spaces on $\mfd{M}$. For compact Riemannian manifolds the theory extends to $L_p$ and Besov spaces. Moreover, for compact homogeneous manifolds, one obtains the so-called product property for eigenfunctions of certain operators and proves a cubature formulae with positive coefficients which allow to construct Parseval frames that characterize Besov spaces in terms of coefficient decay. Throughout the paper, the general theory is exemplified with the help of various concrete and relevant examples, such as the unit sphere and the Poincaré half plane.

math.FA↗

arXiv2015

Wavelet Riesz bases associated to nonisotropic dilations

A bounded, Riemann integrable and measurable set $K\subset \mathbb{R}^d$, which fulfills \[\sum\limits_{γ\inΓ}\mathbb{1}_K(x-γ)=k\text{ almost everywhere, $x\in\mathbb{R}^d$}\] for a lattice $Γ\subset\mathbb{R}^d$ is called $k$-tiling. If $K\subset\mathbb{R}^d$ is $k$-tiling $L^2(K)$ will admit a Riesz basis of exponentials. We use this result to construct generalized Riesz wavelet bases of $L^2(\mathbb{R}^2)$, arising from the action of suitable subsets of the affine group. One example of our construction is the first known shearlet Riesz basis.

math.FA↗

arXiv2015

Integrable wavelet transforms with abelian dilation groups

We consider a class of semidirect products $G = \mathbb{R}^n \rtimes H$, with $H$ a suitably chosen abelian matrix group. The choice of $H$ ensures that there is a wavelet inversion formula, and we are looking for criteria to decide under which conditions there exists a wavelet such that the associated reproducing kernel is integrable. It is well-known that the existence of integrable wavelet coefficients is related to the question whether the unitary dual of $G$ contains open compact sets. Our main general result reduces the latter problem to that of identifying compact open sets in the quotient space of all orbits of maximal dimension under the dual action of $H$ on $\mathbb{R}^n$. This result is applied to study integrability for certain families of dilation groups; in particular, we give a characterization valid for connected abelian matrix groups acting in dimension three.

math.RT↗

arXiv2015

On biunimodular vectors for unitary matrices

A biunimodular vector of a unitary matrix $A \in U(n)$ is a vector $v \in \mathbb{T}^n\subset\bc^n$ such that $Av \in \mathbb{T}^n$ as well. Over the last 30 years, the sets of biunimodular vectors for Fourier matrices have been the object of extensive research in various areas of mathematics and applied sciences. Here, we broaden this basic harmonic analysis perspective and extend the search for biunimodular vectors to arbitrary unitary matrices. This search can be motivated in various ways. The main motivation is provided by the fact, that the existence of biunimodular vectors for an arbitrary unitary matrix allows for a natural understanding of the structure of all unitary matrices.

math.RT↗

arXiv2015

Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

We consider the coorbit theory associated to general continuous wavelet transforms arising from a square-integrable, irreducible quasi-regular representation of a semidirect product group $G = \mathbb{R}^d \rtimes H$. The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups, as well as a class called {\em generalized shearlet dilation groups}, containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups arise. As a result the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved.

math.FA↗

arXiv2014

Resolution of the wavefront set using general continuous wavelet transforms

We consider the problem of characterizing the wavefront set of a tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{d})$ in terms of its continuous wavelet transform, where the latter is defined with respect to a suitably chosen dilation group $H\subset{\rm GL}(\mathbb{R}^{d})$. In this paper we develop a comprehensive and unified approach that allows to establish characterizations of the wavefront set in terms of rapid coefficient decay, for a large variety of dilation groups. For this purpose, we introduce two technical conditions on the dual action of the group $H$, called microlocal admissibilty and (weak) cone approximation property. Essentially, microlocal admissibilty sets up a systematical relationship between the scales in a wavelet dilated by $h\in H$ on one side, and the matrix norm of $h$ on the other side. The (weak) cone approximation property describes the ability of the wavelet system to adapt its frequency-side localization to arbitrary frequency cones. Together, microlocal admissibility and the weak cone approximation property allow the characterization of points in the wavefront set using multiple wavelets. Replacing the weak cone approximation by its stronger counterpart gives access to single wavelet characterizations. We illustrate the scope of our results by discussing -- in any dimension $d\ge2$ -- the similitude, diagonal and shearlet dilation groups, for which we verify the pertinent conditions. As a result, similitude and diagonal groups can be employed for multiple wavelet characterizations, whereas for the shearlet groups a single wavelet suffices. In particular, the shearlet characterization (previously only established for $d=2$) holds in arbitrary dimensions.

math.FA↗

arXiv2014

Relevant sampling in finitely generated shift-invariant spaces

We consider random sampling in finitely generated shift-invariant spaces $V(Φ) \subset {\rm L}^2(\mathbb{R}^n)$ generated by a vector $Φ= (φ_1,\ldots,φ_r) \in {\rm L}^2(\mathbb{R}^n)^r$. Following the approach introduced by Bass and Gröchenig, we consider certain relatively compact subsets $V_{R,δ}(Φ)$ of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths $R$. Under very mild assumptions on the generators, we show that for $R$ sufficiently large, taking $O(R^n log(R^{n^2/α'}))$ many random samples (taken independently uniformly distributed within $C_R$) yields a sampling set for $V_{R,δ}(Φ)$ with high probability. Here $α' \le n$ is a suitable constant.We give explicit estimates of all involved constants in terms of the generators $φ_1, \ldots, φ_r$.

math.FA↗

arXiv2014

Wavelet Coorbit Spaces viewed as Decomposition Spaces

In this paper we show that the Fourier transform induces an isomorphism between the coorbit spaces defined by Feichtinger and Gröchenig of the mixed, weighted Lebesgue spaces $L_{v}^{p,q}$ with respect to the quasi-regular representation of a semi-direct product $\mathbb{R}^{d}\rtimes H$ with suitably chosen dilation group $H$, and certain decomposition spaces $\mathcal{D}\left(\mathcal{Q},L^{p},\ell_{u}^{q}\right)$ (essentially as introduced by Feichtinger and Gröbner), where the localized ,,parts`` of a function are measured in the $\mathcal{F}L^{p}$-norm. This equivalence is useful in several ways: It provides access to a Fourier-analytic understanding of wavelet coorbit spaces, and it allows to discuss coorbit spaces associated to different dilation groups in a common framework. As an illustration of these points, we include a short discussion of dilation invariance properties of coorbit spaces associated to different types of dilation groups.

math.FA↗

arXiv2013

Coorbit spaces and wavelet coefficient decay over general dilation groups

We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on ${\rm L}^2(\mathbb{R}^d)$. We first prove that these representations are integrable as well, with respect to a wide variety of weights, thus allowing to consistently quantify coefficient decay via coorbit space norms. We then show that these spaces always admit an atomic decomposition in terms of bandlimited Schwartz wavelets. We exhibit spaces of Schwartz functions densely contained in (most of) the coorbit spaces. We also present an example showing that for a consistent definition of coorbit spaces, the irreducibility requirement cannot be easily dispensed with. We then address the question how to predict wavelet coefficient decay from vanishing moment assumptions. To this end, we introduce a new condition on the open dual orbit associated to a dilation group: If the orbit is temperately embedded, it is possible to derive rather general weighted ${\rm L}^{p,q}$-estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function. These estimates have various applications: They provide very explicit admissibility conditions for wavelets and integrable vectors, as well as sufficient criteria for membership in coorbit spaces. As a further consequence, one obtains a transparent way of identifying elements of coorbit spaces with certain (cosets of) tempered distributions. We then show that, for every dilation group in dimension two, the associated dual orbit is temperately embedded. In particular, the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces, where they complement and generalize the known results.

math.FA↗

arXiv2013

Vanishing moment conditions for wavelet atoms in higher dimensions

We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in ${\rm L}^2(\mathbb{R}^d)$, but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of ${\rm GL}(\mathbb{R}^d)$, the so-called {\em dilation group}.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Gröchenig, and they can be informally described as follows: Given a function $ψ\in {\rm L}^2(\mathbb{R}^d)$ satisfying fairly mild decay, smoothness and vanishing moment conditions, {\em any} sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation.

math.FA↗