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A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (30 June 2019) | Viewed by 17546

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Special Issue Editors

Dr. Peter Balazs

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Guest Editor

Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12–14, 1040 Vienna, Austria
Interests: frame theory; time frequency analysis; Gabor analysis; signal processing; scientific computing; acoustics and psychoacoustics

Prof. Dr. Martin Ehler

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Guest Editor

Department of Mathematics, University of Vienna, 1010 Vienna, Austria
Interests: numerical analysis; imaging; applied mathematics; Fourier analysis; wavelet analysis; wavelet

Prof. Dr. Hans Georg Feichtinger

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Guest Editor

Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
Interests: Gabor analysis; harmonic analysis; function spaces; time-frequency analysis; modulation spaces
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Strobl18 Harmonic Analysis and Applications will be held in Strobl, Austria, on June 4–8, 2018.

The idea behind this Special Issue is to create a comprehensive collection of peer-reviewed articles that address the specific themes of research, that were covered by “Strobl18 Harmonic Analysis and Applications” (www.nuhag.eu/strobl18). Authors of outstanding papers related to Harmonic Analysis presented at the conference are invited to submit extended versions of their work to the Special Issue for publication. A 15% discount of the Article Processing Charges is available for all attendees.

Researchers not attending the conference are also welcome to present their recent research, as well as review papers. Potential topics include, but are not limited to: function spaces, time-frequency analysis and Gabor analysis, sampling theory and compressed sensing, frame theory, pseudodifferential operators and Fourier integral operators, numerical harmonic analysis, abstract harmonic analysis, and applications of harmonic analysis.

Dr. Peter Balazs
Prof. Dr. Martin Ehler
Prof. Dr. Hans G. Feichtinger
Guest Editors

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

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Keywords

function spaces
time-frequency analysis
Gabor analysis
sampling theory
compressed sensing
frame theory
pseudodifferential operators
Fourier integral operators
numerical harmonic analysis
abstract harmonic analysis
applications of harmonic analysis

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Published Papers (6 papers)

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Research

Jump to: Review

18 pages, 374 KiB

Open AccessArticle

A Sequential Approach to Mild Distributions

by Hans G. Feichtinger

Axioms 2020, 9(1), 25; https://doi.org/10.3390/axioms9010025 - 24 Feb 2020

Cited by 7 | Viewed by 2187

Abstract

The Banach Gelfand Triple

(S_{0}, L^{2}, S_{0}^{'}) (R^{d})

consists of

(S_{0} (R^{d}), {∥ \cdot ∥}_{S_{0}})

, a very specific Segal algebra as algebra of test [...] Read more.

The Banach Gelfand Triple

(S_{0}, L^{2}, S_{0}^{'}) (R^{d})

consists of

(S_{0} (R^{d}), {∥ \cdot ∥}_{S_{0}})

, a very specific Segal algebra as algebra of test functions, the Hilbert space

(L^{2} (R^{d}), {∥ \cdot ∥}_{2})

and the dual space

S_{0}^{'} (R^{d})

, whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce

(S_{0} (R^{d}), {∥ \cdot ∥}_{S_{0}})

and hence

(S_{0}^{'} (R^{d}), ∥ \cdot ∥_{S_{0}^{'}})

, the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra

(S_{0} (R^{d}), {∥ \cdot ∥}_{S_{0}})

can be used to establish this natural identification. Full article

(This article belongs to the Special Issue Harmonic Analysis and Applications)

25 pages, 1921 KiB

Open AccessArticle

Gabor Frames and Deep Scattering Networks in Audio Processing

by Roswitha Bammer, Monika Dörfler and Pavol Harar

Axioms 2019, 8(4), 106; https://doi.org/10.3390/axioms8040106 - 26 Sep 2019

Cited by 2 | Viewed by 2894

Abstract

This paper introduces Gabor scattering, a feature extractor based on Gabor frames and Mallat’s scattering transform. By using a simple signal model for audio signals, specific properties of Gabor scattering are studied. It is shown that, for each layer, specific invariances to certain signal characteristics occur. Furthermore, deformation stability of the coefficient vector generated by the feature extractor is derived by using a decoupling technique which exploits the contractivity of general scattering networks. Deformations are introduced as changes in spectral shape and frequency modulation. The theoretical results are illustrated by numerical examples and experiments. Numerical evidence is given by evaluation on a synthetic and a “real” dataset, that the invariances encoded by the Gabor scattering transform lead to higher performance in comparison with just using Gabor transform, especially when few training samples are available. Full article

(This article belongs to the Special Issue Harmonic Analysis and Applications)

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20 pages, 445 KiB

Open AccessArticle

A New Generalized Projection and Its Application to Acceleration of Audio Declipping

by Pavel Rajmic, Pavel Záviška, Vítězslav Veselý and Ondřej Mokrý

Axioms 2019, 8(3), 105; https://doi.org/10.3390/axioms8030105 - 19 Sep 2019

Cited by 8 | Viewed by 3204

Abstract

In convex optimization, it is often inevitable to work with projectors onto convex sets composed with a linear operator. Such a need arises from both the theory and applications, with signal processing being a prominent and broad field where convex optimization has been used recently. In this article, a novel projector is presented, which generalizes previous results in that it admits to work with a broader family of linear transforms when compared with the state of the art but, on the other hand, it is limited to box-type convex sets in the transformed domain. The new projector is described by an explicit formula, which makes it simple to implement and requires a low computational cost. The projector is interpreted within the framework of the so-called proximal splitting theory. The convenience of the new projector is demonstrated on an example from signal processing, where it was possible to speed up the convergence of a signal declipping algorithm by a factor of more than two. Full article

(This article belongs to the Special Issue Harmonic Analysis and Applications)

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22 pages, 430 KiB

Open AccessArticle

PIP-Space Valued Reproducing Pairs of Measurable Functions

by Jean-Pierre Antoine and Camillo Trapani

Axioms 2019, 8(2), 52; https://doi.org/10.3390/axioms8020052 - 30 Apr 2019

Cited by 4 | Viewed by 2472

Abstract

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space [...] Read more.

Z : = Z (X, μ)

, where

(X, μ)

is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case. Full article

(This article belongs to the Special Issue Harmonic Analysis and Applications)

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9 pages, 303 KiB

Open AccessReview

Sampling Theorems for Stochastic Signals. Appraisal of Paul L. Butzer’s Work

by Tibor K. Pogány

Axioms 2019, 8(3), 91; https://doi.org/10.3390/axioms8030091 - 1 Aug 2019

Cited by 1 | Viewed by 2497

Abstract

The sampling reconstruction theory is one of the great areas of the analysis in which Paul Leo Butzer earned longstanding and excellent theoretical results. Thus, we are forced either by earlier exhaustive presentations of his research activity and/or the highly voluminous material to restrict ourselves to a more narrow and precise sub-area in consideration; we discuss here, giving deeper insight, Paul Butzer’s sampling theoretical work with special attention concerning sampling stochastic signals. Full article

(This article belongs to the Special Issue Harmonic Analysis and Applications)

40 pages, 418 KiB

Open AccessReview

Groups, Special Functions and Rigged Hilbert Spaces

by Enrico Celeghini, Manuel Gadella and Mariano A. del Olmo

Axioms 2019, 8(3), 89; https://doi.org/10.3390/axioms8030089 - 27 Jul 2019

Cited by 9 | Viewed by 3474

Abstract

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space

H

and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space; instead, they are functionals on the dual space,

Φ^{\times}

, of a rigged Hilbert space,

Φ \subset H \subset Φ^{\times}

. In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors

Φ

and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on

Φ

with its own topology, so that they admit continuous extensions to the dual

Φ^{\times}

and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider

S O (2)

and functions on the unit circle,

S U (2)

and associated Laguerre functions, Weyl–Heisenberg group and Hermite functions,

S O (3, 2)

and spherical harmonics,

s u (1, 1)

and Laguerre functions,

s u (2, 2)

and algebraic Jacobi functions and, finally,

s u (1, 1) \oplus s u (1, 1)

and Zernike functions on a circle. Full article

(This article belongs to the Special Issue Harmonic Analysis and Applications)

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