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A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (22 November 2022) | Viewed by 7695

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Special Issue Editors

Dr. Nenad Teofanov

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Guest Editor

Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, 21101 Novi Sad, Serbia
Interests: time-frequency analysis; harmonic analysis; generalized functions; modulation spaces; pseudodifferential operators; microlocal analysis

Prof. Dr. Filip Tomić

E-Mail Website
Co-Guest Editor

Department of Fundamental Sciences, Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Serbia
Interests: functional analysis; harmonic analysis; generalized functions; microlocal analysis

Special Issue Information

Dear Colleagues,

This Special Issue aims to promote the potential arising from connections between time–frequency analysis, operators, and distributions. Theory of test function spaces and their dual spaces of distributions offers a solid theoretical background for the wide range of research topics related to diverse applications. In particular, it is useful when decay or growth conditions are considered in combination with regularity properties of the considered objects. In the last two decades, tools from time–frequency analysis have offered a new perspective on these classical issues. Apart from new insights into classical theory, the new methodology has found applications ranging from physics and engineering to harmonic analysis and partial differential equations in mathematical sciences. Emerging from explorations in signal analysis, quantum mechanics and (abstract) harmonic analysis, time–frequency analysis has evolved into a fascinating discipline with strong indicators that new methods in its theory and applications will be introduced and developed in the future.

This perspective of using time–frequency analysis in operator theory and in the background of distribution spaces has turned out to be successful in different contexts, e.g., approximate diagonalization of operators, resolution of wave-front sets, well-posedness of nonlinear Schrödinger equations, and sparse representations by frame expansions.

Researchers working in this interdisciplinary field are welcome to submit their original research results as well as expository and review papers. Potential topics include harmonic analysis, function spaces, pseudodifferential and Fourier integral operators, wave-front sets, frame theory, time–frequency representations, and their applications, provided that the aspects and connections with time–frequency analysis, operators, and distributions are emphasized.

Contributions may be submitted on a continuous basis before the deadline. After a peer-review process, submissions will be selected for publication based on their quality and relevance.

Dr. Nenad Teofanov
Prof. Dr. Filip Tomić
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

Test function spaces and spaces of distributions
Function spaces of harmonic analysis
Time–frequency analysis
Gabor and wavelet analysis
Frames
Pseudo-differential and fourier integral operators
Microlocal analysis and wave-front sets

Published Papers (5 papers)

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Research

19 pages, 394 KiB

Open AccessArticle

A Characterization of Multipliers of the Herz Algebra

by Hans G. Feichtinger

Axioms 2023, 12(5), 482; https://doi.org/10.3390/axioms12050482 - 16 May 2023

Cited by 1 | Viewed by 757

Abstract

For the characterization of multipliers of

L^{p} (R^{d})

or more generally, of

L^{p} (G)

for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra

A_{p} (G)

plays an important role. Following [...] Read more.

For the characterization of multipliers of

L^{p} (R^{d})

or more generally, of

L^{p} (G)

for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra

A_{p} (G)

plays an important role. Following Larsen’s book, we describe multipliers as bounded linear operators that commute with translations. The main result of this paper is the characterization of the multipliers of

A_{p} (G)

. In fact, we demonstrate that it coincides with the space of multipliers of

(L^{p} (G), {∥ \cdot ∥}_{p})

. Given a multiplier T of

(A_{p} (G), ∥ \cdot ∥_{A_{p} (G)})

and using the embedding

(A_{p} (G) {, ∥ \cdot ∥}_{A_{p} (G)}) ↪ (C_{0} (G), {∥ \cdot ∥}_{\infty})

, the linear functional

f \mapsto [T (f) (0)]

is bounded, and T can be written as a moving average for some element in the dual

P M_{p} (G)

(A_{p} (G), ∥ \cdot ∥_{A_{p} (G)})

. A key step for this identification is another elementary fact: showing that the multipliers from

(L^{p} (G), {∥ \cdot ∥}_{p})

(C_{0} (G), {∥ \cdot ∥}_{\infty})

are exactly the convolution operators with kernels in

(L^{q} (G), {∥ \cdot ∥}_{q})

for

1 < p < \infty

and

1 / p + 1 / q = 1

. The proofs make use of the space of mild distributions, which is the dual of the Segal algebra

(S_{0} (G), {∥ \cdot ∥}_{S_{0}})

, and the fact that multipliers T from

S_{0} (G)

S_{0}^{'} (G)

are convolution operators of the form

T : f \mapsto σ * f

for some uniquely determined

σ \in S_{0}^{'}

. This setting also allows us to switch from the description of these multipliers as convolution operators (by suitable pseudomeasures) to their description as Fourier multipliers, using the extended Fourier transform in the setting of

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

. The approach presented here extends to other function spaces, but a more detailed discussion is left to future publications. Full article

(This article belongs to the Special Issue Time-Frequency Analysis, Distributions, and Operators)

11 pages, 312 KiB

Open AccessArticle

Extended Gevrey Regularity via Weight Matrices

by Nenad Teofanov and Filip Tomić

Axioms 2022, 11(10), 576; https://doi.org/10.3390/axioms11100576 - 21 Oct 2022

Cited by 1 | Viewed by 1185

Abstract

The main aim of this paper is to compare two recent approaches for investigating the interspace between the union of Gevrey spaces

G_{t} (U)

and the space of smooth functions

C^{\infty} (U)

. The first approach in [...] Read more.

The main aim of this paper is to compare two recent approaches for investigating the interspace between the union of Gevrey spaces

G_{t} (U)

and the space of smooth functions

C^{\infty} (U)

. The first approach in the style of Komatsu is based on the properties of two parameter sequences

M_{p} = p^{τ p^{σ}}

τ > 0

σ > 1

. The other one uses weight matrices defined by certain weight functions. We prove the equivalence of the corresponding spaces in the Beurling case by taking projective limits with respect to matrix parameters, while in the Roumieu case we need to consider a larger space than the one obtained as the inductive limit of extended Gevrey classes. Full article

(This article belongs to the Special Issue Time-Frequency Analysis, Distributions, and Operators)

20 pages, 386 KiB

Open AccessArticle

by Peter Balazs, Giorgia Bellomonte and Hessam Hosseinnezhad

Axioms 2022, 11(4), 180; https://doi.org/10.3390/axioms11040180 - 16 Apr 2022

Viewed by 1689

Abstract

Frames for Hilbert spaces are interesting for mathematicians but also important for applications in, e.g., signal analysis and physics. In both mathematics and physics, it is natural to consider a full scale of spaces, and not only a single one. In this paper, we study how certain frame-related properties of a certain sequence in one of the spaces, such as completeness or the property of being a (semi-) frame, propagate to the other ones in a scale of Hilbert spaces. We link that to the properties of the respective frame-related operators, such as analysis or synthesis. We start with a detailed survey of the theory of Hilbert chains. Using a canonical isomorphism, the properties of frame sequences are naturally preserved between different spaces. We also show that some results can be transferred if the original sequence is considered—in particular, that the upper semi-frame property is kept in larger spaces, while the lower one is kept in smaller ones. This leads to a negative result: a sequence can never be a frame for two Hilbert spaces of the scale if the scale is non-trivial, i.e., if the spaces are not equal. Full article

(This article belongs to the Special Issue Time-Frequency Analysis, Distributions, and Operators)

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16 pages, 331 KiB

Open AccessArticle

Wilson Bases and Ultradistributions

by Nenad Teofanov

Axioms 2021, 10(4), 241; https://doi.org/10.3390/axioms10040241 - 28 Sep 2021

Viewed by 1074

Abstract

We provide a characterization of the Gelfand–Shilov-type spaces of test functions and their dual spaces of tempered ultradistributions by means of Wilson bases of exponential decay. We offer two different proofs and extend known results to the Roumieu case. Full article

(This article belongs to the Special Issue Time-Frequency Analysis, Distributions, and Operators)

11 pages, 276 KiB

Open AccessArticle

Characterization of Wave Fronts of Ultradistributions Using Directional Short-Time Fourier Transform

by Sanja Atanasova, Snježana Maksimović and Stevan Pilipović

Axioms 2021, 10(4), 240; https://doi.org/10.3390/axioms10040240 - 28 Sep 2021

Cited by 1 | Viewed by 1589

Abstract

In this paper we give a characterization of Sobolev k-directional wave front of order

p \in [1, \infty)

of tempered ultradistributions via the directional short-time Fourier transform. Full article

(This article belongs to the Special Issue Time-Frequency Analysis, Distributions, and Operators)

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