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Mathematics
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Microlocal and Time-Frequency Analysis
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Microlocal and Time-Frequency Analysis

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 October 2021) | Viewed by 14619

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Elena Cordero

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Guest Editor
Dipartimento di Matematica “G. Peano,”, Università degli Studi di Torino. Via Carlo Alberto 10, 10123 Torino, Italy
Interests: time-frequency analysis; harmonic analysis; evolution equations; Gabor matrix representations of operators; modulation and Wiener amalgam spaces; signal analysis
Dr. S. Ivan Trapasso

E-Mail Website
Guest Editor
MaLGa Center - Department of Mathematics (DIMA), University of Genoa. Via Dodecaneso 35, 16146 Genova (GE), Italy
Interests: time-frequency analysis, microlocal and semiclassical analysis; applications to mathematical physics and partial differential equations

Special Issue Information

Dear Colleagues,

The focus of this Special Issue of Mathematics lies in two fascinating areas of modern mathematics with a broad spectrum of applications ranging from theoretical physics to signal processing, namely, microlocal and time-frequency analysis. The fruitful interaction between the two disciplines is witnessed by the vast body of literature published in recent decades. It is worth mentioning the following problems among several ones which have benefited from this joint perspective, without any claim of being exhaustive: properties of quantization rules, pseudodifferential and Fourier integral operators; algebras of sparse operators in phase space; well-posedness of nonlinear dispersive PDEs and representation of their solutions; wave front sets and propagation of singularities.

In order to further explore these research trends, we solicit original, high-quality papers on microlocal and time-frequency analysis and their applications. Contributions on related topics, including for instance mathematical signal processing, harmonic analysis, and mathematical physics are welcome as well, provided that they mainly focus on aspects of or connections with microlocal and Gabor analysis. We also invite expository and review papers by senior researchers aimed at elucidating finer points or highlighting techniques of broad interest.

The contributions may be submitted on a continuous basis before the deadline and will be selected, after a peer-review process by leading experts, in view of both their quality and relevance.

Prof. Dr. Elena Cordero
Dr. S. Ivan Trapasso
Guest Editors

Manuscript Submission Information

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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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • Microlocal analysis
  • Time-frequency analysis
  • Harmonic analysis
  • Partial differential equations
  • Pseudodifferential and Fourier integral operators
  • Localization operators
  • Symplectic methods in harmonic analysis
  • Propagation of singularities
  • Semiclassical analysis
  • Function spaces of harmonic analysis
  • Frames
  • Wavelets
  • Group theory
  • Wave front sets
  • Signal analysis
  • Reproducing formulae

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Published Papers (7 papers)

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Research

22 pages, 537 KiB  
Article
Homogeneous Banach Spaces as Banach Convolution Modules over M(G)
by Hans Georg Feichtinger
Mathematics 2022, 10(3), 364; https://doi.org/10.3390/math10030364 - 25 Jan 2022
Cited by 9 | Viewed by 2459
Abstract
This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform [...] Read more.
This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform on (M(G),∥·∥M), the space of bounded measures (viewed as a space of linear functionals) in an elementary fashion over Rd. Bounded uniform partitions of unity (BUPUs) are easily constructed in the Euclidean setting (by dilation). Moving on to general LCA groups, it becomes an interesting challenge to find ways to construct arbitrary fine BUPUs, ideally without the use of structure theory, the existence of a Haar measure and even Lebesgue integration. This article provides such a construction and demonstrates how it can be used in order to show that any so-called homogeneous Banach space(B,∥·∥B) on G, such as (Lp(G),∥·∥p), for 1≤p<∞, or the Fourier–Stieltjes algebra FM(G), and in particular any Segal algebra is a Banach convolution module over (M(G),∥·∥M) in a natural way. Via the Haar measure we can then identify L1(G),∥·∥1 with the closure (of the embedded version) of Cc(G), the space of continuous functions with compact support, in (M(G),∥·∥M), and show that these homogeneous Banach spaces are essentialL1(G)-modules. Thus, in particular, the approximate units act properly as one might expect and converge strongly to the identity operator. The approach is in the spirit of Hans Reiter, avoiding the use of structure theory for LCA groups and the usual techniques of vector-valued integration via duality. The ultimate (still distant) goal of this approach is to provide a new and elementary approach towards the (extended) Fourier transform in the setting of the so-called Banach–Gelfand triple(S0,L2,S0′)(G), based on the Segal algebra S0(G). This direction will be pursued in subsequent papers. Full article
(This article belongs to the Special Issue Microlocal and Time-Frequency Analysis)
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13 pages, 322 KiB  
Article
Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity
by Divyang G. Bhimani and Saikatul Haque
Mathematics 2021, 9(23), 3145; https://doi.org/10.3390/math9233145 - 6 Dec 2021
Cited by 5 | Viewed by 1963
Abstract
We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We [...] Read more.
We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0. Full article
(This article belongs to the Special Issue Microlocal and Time-Frequency Analysis)
9 pages, 255 KiB  
Article
The Pauli Problem for Gaussian Quantum States: Geometric Interpretation
by Maurice A. de Gosson
Mathematics 2021, 9(20), 2578; https://doi.org/10.3390/math9202578 - 14 Oct 2021
Cited by 5 | Viewed by 1467
Abstract
We solve the Pauli tomography problem for Gaussian signals using the notion of Schur complement. We relate our results and method to a notion from convex geometry, polar duality. In our context polar duality can be seen as a sort of geometric Fourier [...] Read more.
We solve the Pauli tomography problem for Gaussian signals using the notion of Schur complement. We relate our results and method to a notion from convex geometry, polar duality. In our context polar duality can be seen as a sort of geometric Fourier transform and allows a geometric interpretation of the uncertainty principle and allows to apprehend the Pauli problem in a rather simple way. Full article
(This article belongs to the Special Issue Microlocal and Time-Frequency Analysis)
15 pages, 818 KiB  
Article
Local Convergence of the Continuous and Semi-Discrete Wavelet Transform in Lp(R)
by Jaime Navarro-Fuentes, Salvador Arellano-Balderas and Oscar Herrera-Alcántara
Mathematics 2021, 9(5), 522; https://doi.org/10.3390/math9050522 - 3 Mar 2021
Cited by 1 | Viewed by 1503
Abstract
The smoothness of functions f in the space Lp(R) with 1<p<∞ is studied through the local convergence of the continuous wavelet transform of f. Additionally, we study the smoothness of functions in [...] Read more.
The smoothness of functions f in the space Lp(R) with 1<p<∞ is studied through the local convergence of the continuous wavelet transform of f. Additionally, we study the smoothness of functions in Lp(R) by means of the local convergence of the semi-discrete wavelet transform. Full article
(This article belongs to the Special Issue Microlocal and Time-Frequency Analysis)
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30 pages, 27582 KiB  
Article
Wavelet Energy Accumulation Method Applied on the Rio Papaloapan Bridge for Damage Identification
by Jose M. Machorro-Lopez, Juan P. Amezquita-Sanchez, Martin Valtierra-Rodriguez, Francisco J. Carrion-Viramontes, Juan A. Quintana-Rodriguez and Jesus I. Valenzuela-Delgado
Mathematics 2021, 9(4), 422; https://doi.org/10.3390/math9040422 - 21 Feb 2021
Cited by 11 | Viewed by 2365
Abstract
Large civil structures such as bridges must be permanently monitored to ensure integrity and avoid collapses due to damage resulting in devastating human fatalities and economic losses. In this article, a wavelet-based method called the Wavelet Energy Accumulation Method (WEAM) is developed in [...] Read more.
Large civil structures such as bridges must be permanently monitored to ensure integrity and avoid collapses due to damage resulting in devastating human fatalities and economic losses. In this article, a wavelet-based method called the Wavelet Energy Accumulation Method (WEAM) is developed in order to detect, locate and quantify damage in vehicular bridges. The WEAM consists of measuring the vibration signals on different points along the bridge while a vehicle crosses it, then those signals and the corresponding ones of the healthy bridge are subtracted and the Continuous Wavelet Transform (CWT) is applied on both, the healthy and the subtracted signals, to obtain the corresponding diagrams, which provide a clue about where the damage is located; then, the border effects must be eliminated. Finally, the Wavelet Energy (WE) is obtained by calculating the area under the curve along the selected range of scale for each point of the bridge deck. The energy of a healthy bridge is low and flat, whereas for a damaged bridge there is a WE accumulation at the damage location. The Rio Papaloapan Bridge (RPB) is considered for this research and the results obtained numerically and experimentally are very promissory to apply this method and avoid accidents. Full article
(This article belongs to the Special Issue Microlocal and Time-Frequency Analysis)
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14 pages, 317 KiB  
Article
Boundary Values in Ultradistribution Spaces Related to Extended Gevrey Regularity
by Stevan Pilipović, Nenad Teofanov and Filip Tomić
Mathematics 2021, 9(1), 7; https://doi.org/10.3390/math9010007 - 22 Dec 2020
Cited by 4 | Viewed by 1721
Abstract
Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In the given framework we prove that boundary values of analytic functions [...] Read more.
Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In the given framework we prove that boundary values of analytic functions with the corresponding logarithmic growth rate towards the real domain are ultradistributions. The essential condition for that purpose, known as stability under ultradifferential operators in the classical ultradistribution theory, is replaced by a weaker condition, in which the growth properties are controlled by an additional parameter. For that reason, new techniques were used in the proofs. As an application, we discuss the corresponding wave front sets. Full article
(This article belongs to the Special Issue Microlocal and Time-Frequency Analysis)
23 pages, 335 KiB  
Article
On the Sharp Gårding Inequality for Operators with Polynomially Bounded and Gevrey Regular Symbols
by Alexandre Arias Junior and Marco Cappiello
Mathematics 2020, 8(11), 1938; https://doi.org/10.3390/math8111938 - 3 Nov 2020
Cited by 1 | Viewed by 1583
Abstract
In this paper, we analyze the Friedrichs part of an operator with polynomially bounded symbol. Namely, we derive a precise expression of its asymptotic expansion. In the case of symbols satisfying Gevrey estimates, we also estimate precisely the regularity of the terms in [...] Read more.
In this paper, we analyze the Friedrichs part of an operator with polynomially bounded symbol. Namely, we derive a precise expression of its asymptotic expansion. In the case of symbols satisfying Gevrey estimates, we also estimate precisely the regularity of the terms in the asymptotic expansion. These results allow new and refined applications of the sharp Gårding inequality in the study of the Cauchy problem for p-evolution equations. Full article
(This article belongs to the Special Issue Microlocal and Time-Frequency Analysis)
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