arXiv subjects

Hans G. Feichtinger

Publications and source records attributed to Hans G. Feichtinger.

At least 19 recordsLinked to original sources

arXiv2026

A Distributional Approach to Generalized Stochastic Processes on Locally Compact Abelian Groups

This paper is dedicated to Paul Butzer on the occasion of his 85th birthday. His work and example have strongly influenced not only the first author, but also generations of mathematicians working in approximation theory and Fourier analysis. He has shown younger colleagues the importance of remaining open to applied areas, avoiding an overly narrow scope, and exploring different ways of understanding mathematical facts. A recurring theme in his work is the logical equivalence of fundamental statements in analysis. It may be less widely known that, besides his central role in approximation theory, Paul Butzer has also made significant contributions to probability theory. We hope that he will enjoy this note, which shows that a purely functional-analytic treatment of generalized stochastic processes is possible. The approach is based on the Segal algebra S0(G) and avoids several technical difficulties associated with the customary framework of vector-valued integration and topological vector spaces.

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arXiv2026

The Concept of Wiener Amalgam Spaces

This article concerns Wiener amalgam spaces % , recalls their basic properties and provides some hints about their usefulness in various branches of Harmonic Analysis. Despite the fact that the underlying construction principles % of Wiener amalgam spaces is are quite easy to understand and basic facts follow naturally by simple rules, these spaces have not obtained the same popularity as certain other function spaces which are much more complicated to describe and often just serve a very particular purpose. \newline \indent This situation has motivated the author to provide here a summary of the foundations of the theory of Wiener amalgam spaces (and the motivation behind their construction) and a selection of relevant applications, some 45 years years after the key paper published in 1983. \newline \indent We recall first that the so-called {\it classical Wiener amalgam spaces} using local $\HFLpsp$-norms combined with a global $\HFlqsp$-behaviour are already quite useful, e.g.\ for an improvement of the Hausdorff-Young Theorem with some interesting consequences for Sobolev algebras. However, the main emphasis will be based on the idea of allowing more general local components (describing for example smoothness or membership in the Fourier algebra). This opened the door to the introduction of {\it modulation spaces}, which are now recognized as standard tools in time-frequency analysis. \newline \indent We will demonstrate in this article how Wiener amalgam spaces methods can be used to prove the Sobolev embedding theorem or determine the pointwise multipliers of Sobolev algebras. We also demonstrate that the space of multipliers from the classical Wiener algebra $\HFWCOliRd$ into its dual can be identified with $\HFSOPRd$, the space of mild distributions. }

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arXiv2026

A Sequential Approach to Mild Distributions

We describe an elementary sequential realization of the Banach Gelfand triple (S0(R^d), L2(R^d), S0'(R^d)). Here S0(R^d) is a Segal algebra of test functions, L2(R^d) is the usual Hilbert space, and S0'(R^d) is its dual space of mild distributions. This framework is fundamental for Gabor analysis and provides a natural setting for the generalized Fourier transform and the short-time Fourier transform. Inspired by Lighthill's sequential approach to tempered distributions, we construct an extended domain for the short-time Fourier transform from equivalence classes of extended mild Cauchy sequences, abbreviated as ECmiCS. Their representatives are sequences of bounded continuous functions. The construction avoids Lebesgue integration and the theory of tempered distributions. Our main result identifies the resulting sequential space canonically with S0'(R^d), thereby recovering the Banach Gelfand triple in an elementary form.

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arXiv2026

Homogeneous Banach spaces as Banach convolution modules over $M(G)$

We develop an elementary approach to convolution and Fourier analysis on a locally compact Abelian group (G), based on bounded measures and bounded uniform partitions of unity. In earlier work, the author introduced convolution and the Fourier--Stieltjes transform on the Banach space (M(G)) of bounded measures, viewed as linear functionals, in a direct Euclidean setting. The present paper constructs arbitrarily fine bounded uniform partitions of unity on general locally compact Abelian groups. The construction is designed to avoid structure theory and does not presuppose Haar measure or Lebesgue integration. It is then used to establish a natural convolution-module structure of (M(G)) on a broad class of homogeneous Banach spaces on (G). This class includes (L^p(G)), for (1\leq p<\infty), the Fourier--Stieltjes algebra, and, in particular, Segal algebras. After introducing Haar measure, we identify (L^1(G)) with the closure in (M(G)) of the measure-embedded space (C_c(G)). We prove that the homogeneous Banach spaces under consideration are essential (L^1(G))-modules. Consequently, standard approximate identities act in the expected manner and converge strongly to the identity operator. The method follows the spirit of Hans Reiter: it avoids the customary reliance on LCA-group structure theory and on vector-valued integration arguments based on duality. It is intended as a foundation for a subsequent elementary treatment of the extended Fourier transform in the Banach Gelfand triple generated by a Segal algebra.

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arXiv2024

Twice Fourier transformable measures and diffraction theory

Mathematical diffraction theory has been developed since about 1995. Hof's initial approach relied on tempered distributions in euclidean space. Nowadays often the Fourier theory by Argabright and Gil de Lamadrid is used, which applies to appropriate measures on locally compact abelian groups. We review diffraction theory using Wiener amalgams as test function spaces. For translation bounded measures, this unifies and simplifies the former two approaches. We treat weighted versions of Meyer's model sets as examples.

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arXiv2024

Measure-operator convolutions and applications to mixed-state Gabor multipliers

For the Weyl-Heisenberg group, convolutions between functions and operators were defined by Werner as a part of a framework called quantum harmonic analysis. We show how recent results by Feichtinger can be used to extend this definition to include convolutions between measures and operators. Many properties of function-operator convolutions carry over to this setting and allow us to prove novel results on the distribution of eigenvalues of mixed-state Gabor multipliers and derive a version of the Berezin-Lieb inequality for lattices. New results on the continuity of Gabor multipliers with respect to lattice parameters, masks and windows as well as their ability to approximate localization operators are also derived using this framework.

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arXiv2024

Modelling Signals as Mild Distributions

This note gives a summary of ideas concerning Applied Fourier Analysis, mostly formulated for those who have to give such courses to engineers or mathematicians interested in real life applications. It tries to answer recurrent questions arising regularly after my talks on the subject. It outlines alternative ways of presenting the core material of Fourier Analysis in a way which is supposed to help students to grasp the relevance of this transform in the context of applications. Essentially we consider functions in $S_0(R^d)$ (known as Feichtinger's algebra) as possible measurements, and the elements of the dual space (which can be also described by various completion procedures) is thus the collection of all "things" (in the spirit of signals) which can be measured in a reasonable way. We call them mild distributions. In other words, we want to base signal analysis on the mathematical theory of mild distributions. They do not need to be defined in a pointwise sense. Dirac's Delta or Dirac combs are natural examples. The are "as real as point masses in physics". Various equivalent approaches (including a sequential approach) help to verify the relevant results. The material is based on the experiences gained by the author in the last 30 years by teaching the subject in an abstract or application oriented manner, at different universities, and to audiences with quite different background. For related course material see www.nuhag.eu/ETH20.

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arXiv2024

On some properties of modulation spaces as Banach algebras

In this paper, we give some properties of the modulation spaces $M_s^{p,1}({\mathbf R}^n)$ as commutative Banach algebras. In particular, we show the Wiener-Lévy theorem for $M^{p,1}_s({\mathbf R}^n)$, and clarify the sets of spectral synthesis for $M^{p,1}_s ({\mathbf R}^n)$ by using the ``ideal theory for Segal algebras'' developed in Reiter [30].The inclusion relationship between the modulation space $M^{p,1}_0 ({\mathbf R})$ and the Fourier Segal algebra ${\mathcal F}\hspace{-0.08cm}A_p({\mathbf R})$ is also determined.

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arXiv2024

Further study of modulation spaces as Banach algebras

This paper discusses spectral synthesis for those modulation spaces $M^{p,q}_s({\mathbf R}^n)$ which form Banach algebras under pointwise multiplication. An important argument will be the ``ideal theory for Segal algebras'' by H. Reiter [15]. This paper is a continuation of our paper [5] where the case $q=1$ is treated. As a by-product we obtain a variant of Wiener-Lévy theorem for $M^{p,q}_s({\mathbf R}^n)$ and Fourier-Wermer algebras ${\mathcal F}L^q_s({\mathbf R}^n)$.

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arXiv2022

The inner kernel theorem for a certain Segal algebra

The Segal algebra $\mathbf{S}_{0}(G)$ is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups $G$. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from $\mathbf{S}_{0}(G_{1})$ to $\mathbf{S}'_{0}(G_{2})$ by generalized functions in $\mathbf{S}'_{0}(G_{1} \times G_{2})$ the "outer kernel theorem". The main subject of this paper is to formulate what we call the "inner kernel theorem". This is the characterization of those linear operators that have kernels in $\mathbf{S}_{0}(G_{1} \times G_{2})$. Such operators are regularizing -- in the sense that they map $\mathbf{S}'_{0}(G_{1})$ into $\mathbf{S}_{0}(G_{2})$ in a $w^{*}$ to norm continuous manner. A detailed functional analytic treatment of these operators is given and applied to the case of general LCA groups. This is done without the use of Wilson bases, which have previously been employed for the case of elementary LCA groups. We apply our approach to describe natural laws of composition for operators that imitate those of linear mappings via matrix multiplications. Furthermore, we detail how these operators approximate general operators (in a weak form). As a concrete example, we derive the widespread statement of engineers and physicists that pure frequencies "integrate" to a Dirac delta distribution in a mathematically justifiable way.

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arXiv2022

Completeness of Sets of Shifts in Invariant Banach Spaces of Tempered Distributions via Tauberian conditions

The main result of this paper is a far reaching generalization of the completeness result given by V.~Katsnelson in a recent paper [35]. Instead of just using a collection of dilated Gaussians it is shown that the key steps of an earlier paper [27] by the authors, combined with the use of Tauberian conditions (i.e. the non-vanishing of the Fourier transform) allow us to show that the linear span of the translates of a single function $g \in {\boldsymbol{\mathcal {S}({\mathbb{R}}^d)}}$ is a dense subspace of any Banach space satisfying certain double invariance properties. In fact, a much stronger statement is presented: for a given compact subset $M$ in such a Banach space $({\boldsymbol B}, \, \|\,\cdot\,\|_{\boldsymbol B})$ one can construct a finite rank operator, whose range is contained in the linear span of finitely many translates of $g$, and which approximates the identity operator over $M$ up to a given level of precision. The setting of tempered distributions allows to reduce the technical arguments to methods which are widely used in Fourier Analysis. The extension to non-quasi-analytic weights respectively locally compact Abelian groups is left to a forthcoming paper, which will be technically much more involved and uses different ingredients.

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arXiv2022

Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform

We study the connection between STFT multipliers $A^{g_1,g_2}_{1\otimes m}$ having windows $g_1,g_2$, symbols $a(x,ω)=(1\otimes m)(x,ω)=m(ω)$, $(x,ω)\in\mathbb{R}^{2d}$, and the Fourier multipliers $T_{m_2}$ with symbol $m_2$ on $\mathbb{R}^d$. We find sufficient and necessary conditions on symbols $m,m_2$ and windows $g_1,g_2$ for the equality $T_{m_2}= A^{g_1,g_2}_{1\otimes m}$. For $m=m_2$ the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier $A^{g_1,g_2}_{1\otimes m}$, also called localization operator, presents a smoothing effect due to the so-called two-window short-time Fourier transform which enters in the definition of $A^{g_1,g_2}_{1\otimes m}$. As a by-product we prove necessary conditions for the continuity of anti-Wick operators $A^{g,g}_{1\otimes m}: L^p\to L^q$ having multiplier $m$ in weak $L^r$ spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multiplier are better known as linear time invariant (LTI) filters.

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arXiv2021

Approximately invertible elements in non-unital normed algebras

We introduce a concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian C*-algebras.

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arXiv2020

Modulation spaces associated to tensor products of amalgam spaces

We identify the modulation spaces associated to tensor products of amalgam spaces having a large class of Banach spaces as their local component. As consequences of the main results, we describe the modulation spaces associated to tensor products of various $L^p$ spaces.

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arXiv2020

Completeness of shifted dilates in invariant Banach spaces of tempered distributions

We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V.~Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into $(L^2(\mathbb{R}), ||\cdot{}||_2)$, nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space $\mathcal{S}^{\prime}(\mathbb{R}^d)$ ($d \geq 1$) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes $({Q}_{s}(\mathbb{R}^d), ||\cdot{}||_{Q_s})$, showing that they are special cases of Katsnelson's setting (only) for $s \geq 0$.

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arXiv2020

On exceptional times for pointwise convergence of integral kernels in Feynman-Trotter path integrals

In the first part of the paper we provide a survey of recent results concerning the problem of pointwise convergence of integral kernels in Feynman path integral, obtained by means of time-frequency analysis techniques. We then focus on exceptional times, where the previous results do not hold, and we show that weaker forms of convergence still occur. In conclusion we offer some clues about possible physical interpretation of exceptional times.

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arXiv2018

Distribution Theory by Riemann Integrals

It is the purpose of this article to outline a course that can be given to engineers looking for an understandable mathematical description of the foundations of distribution theory and the necessary functional analytic methods. Arguably, these are needed for a deeper understanding of basic questions in signal analysis. Objects such as the Dirac delta and Dirac comb require a proper definition, and it should be possible to explain how one can reconstruct a band-limited function from its samples by means of simple series expansions. It should also be useful for graduate students who want to see how functional analysis can help to understand fairly practical problems, or teachers who want to offer a course related to the "Mathematical Foundations of Signal Processing". The course requires only an understanding of the basic terms from linear functional analysis, namely Banach spaces and their duals, bounded linear operators and a simple version of weak$^{*}$-convergence. As a matter of fact we use a set of function spaces which is quite different from the collection of Lebesgue spaces used normally. We thus avoid the use of Lebesgue integration theory. Furthermore we avoid topological vector spaces in the form of the Schwartz space. Although all tools developed and presented can be realized on LCA groups, we restrict our attention in the current presentation to the Euclidean setting, where we have (generalized) functions over $R^d$. This allows us to make use of simple bounded, uniform partitions of unity, to apply dilation operators and to make use of special functions such as the Gaussian. The problems of the overall current situation, with the separation of theoretical Fourier Analysis as carried out by (pure) mathematicians and Applied Fourier Analysis (as used in engineering applications) are getting bigger and therefore courses filling the gap are in strong need.

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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.

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