{"title":"Symplectic analysis of time-frequency spaces","authors":"Elena Cordero , Gianluca Giacchi","doi":"10.1016/j.matpur.2023.06.011","DOIUrl":null,"url":null,"abstract":"<div><p>We present a different symplectic point of view in the definition of weighted modulation spaces <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> and weighted Wiener amalgam spaces <span><math><mi>W</mi><mo>(</mo><mi>F</mi><msubsup><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>p</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mi>q</mi></mrow></msubsup><mo>)</mo><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. All the classical time-frequency representations, such as the short-time Fourier transform (STFT), the <em>τ</em>-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions <span><math><mi>μ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>(</mo><mi>f</mi><mo>⊗</mo><mover><mrow><mi>g</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math></span>, where <span><math><mi>μ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is the metaplectic operator and <span><math><mi>A</mi></math></span> is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In <span>[13]</span>, the authors suggest that any metaplectic Wigner distribution that satisfies the so-called <em>shift-invertibility condition</em> can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes into play for Wiener amalgam spaces. The shift-invertibility property is necessary: Rihaczek and conjugate Rihaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-triangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"177 ","pages":"Pages 154-177"},"PeriodicalIF":2.1000,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782423000855","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We present a different symplectic point of view in the definition of weighted modulation spaces and weighted Wiener amalgam spaces . All the classical time-frequency representations, such as the short-time Fourier transform (STFT), the τ-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions , where is the metaplectic operator and is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [13], the authors suggest that any metaplectic Wigner distribution that satisfies the so-called shift-invertibility condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes into play for Wiener amalgam spaces. The shift-invertibility property is necessary: Rihaczek and conjugate Rihaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-triangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications.
期刊介绍:
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.