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