{"title":"A Note on the Operator Window of Modulation Spaces","authors":"Weichao Guo, Guoping Zhao","doi":"10.1007/s00041-023-10055-x","DOIUrl":null,"url":null,"abstract":"<p>Inspired by the recent article Skrettingland (J. Fourier Anal. Appl. <b>28</b>(2), 1–34 (2022)), this paper is devoted to the study of a suitable class of windows in the framework of bounded linear operators on <span>\\(L^2({{\\mathbb {R}}}^{d})\\)</span>. We establish a natural and complete characterization for the window class such that the corresponding STFT leads to equivalent norms on modulation spaces. The positive bounded linear operators are also characterized by its Cohen’s class distributions such that the corresponding quantities form equivalent norms on modulation spaces. As a generalization, we introduce a family of operator classes corresponding to the operator-valued modulation spaces. Some applications of our main theorems to the localization operators are also concerned.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":"15 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Fourier Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00041-023-10055-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
Inspired by the recent article Skrettingland (J. Fourier Anal. Appl. 28(2), 1–34 (2022)), this paper is devoted to the study of a suitable class of windows in the framework of bounded linear operators on \(L^2({{\mathbb {R}}}^{d})\). We establish a natural and complete characterization for the window class such that the corresponding STFT leads to equivalent norms on modulation spaces. The positive bounded linear operators are also characterized by its Cohen’s class distributions such that the corresponding quantities form equivalent norms on modulation spaces. As a generalization, we introduce a family of operator classes corresponding to the operator-valued modulation spaces. Some applications of our main theorems to the localization operators are also concerned.
期刊介绍：
The Journal of Fourier Analysis and Applications will publish results in Fourier analysis, as well as applicable mathematics having a significant Fourier analytic component. Appropriate manuscripts at the highest research level will be accepted for publication. Because of the extensive, intricate, and fundamental relationship between Fourier analysis and so many other subjects, selected and readable surveys will also be published. These surveys will include historical articles, research tutorials, and expositions of specific topics.
TheJournal of Fourier Analysis and Applications will provide a perspective and means for centralizing and disseminating new information from the vantage point of Fourier analysis. The breadth of Fourier analysis and diversity of its applicability require that each paper should contain a clear and motivated introduction, which is accessible to all of our readers.
Areas of applications include the following:
antenna theory * crystallography * fast algorithms * Gabor theory and applications * image processing * number theory * optics * partial differential equations * prediction theory * radar applications * sampling theory * spectral estimation * speech processing * stochastic processes * time-frequency analysis * time series * tomography * turbulence * uncertainty principles * wavelet theory and applications
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