{"title":"Hardy’s uncertainty principle for Gabor transform on compact extensions of $$\\mathbb {R}^n$$","authors":"Kais Smaoui","doi":"10.1007/s00605-024-01960-4","DOIUrl":null,"url":null,"abstract":"<p>We prove in this paper a generalization of Hardy’s theorem for Gabor transform in the setup of the semidirect product <span>\\(\\mathbb {R}^n\\rtimes K\\)</span>, where <i>K</i> is a compact subgroup of automorphisms of <span>\\(\\mathbb {R}^n\\)</span>. We also solve the sharpness problem and thus obtain a complete analogue of Hardy’s theorem for Gabor transform. The representation theory and Plancherel formula are fundamental tools in the proof of our results.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"53 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01960-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove in this paper a generalization of Hardy’s theorem for Gabor transform in the setup of the semidirect product \(\mathbb {R}^n\rtimes K\), where K is a compact subgroup of automorphisms of \(\mathbb {R}^n\). We also solve the sharpness problem and thus obtain a complete analogue of Hardy’s theorem for Gabor transform. The representation theory and Plancherel formula are fundamental tools in the proof of our results.