{"title":"论区间 $$[-1,1]$$ 上广义 Lipschitz 类的傅立叶-邓克尔系数","authors":"Othman Tyr","doi":"10.1007/s00009-024-02710-4","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we consider <span>\\(\\mathcal {E}\\)</span> the set of all infinitely differentiable functions with compact support included on the interval <span>\\(I=[-1,1]\\)</span>. We use the distributions in <span>\\(\\mathcal {E}\\)</span>, as a tool to prove the continuity of the Dunkl operator and the Dunkl translation. Some properties of the modulus of smoothness related to the Dunkl operator are verified. By means of generalized Dunkl–Lipschitz conditions on Dunkl–Sobolev spaces, a result of Younis on the torus, which is an analog of Titchmarsh’s theorem, is deduced as a special case. In addition, certain conditions and a characterization of the Dini–Lipschitz classes on <i>I</i> in terms of the behavior of their Fourier–Dunkl coefficients are derived.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Fourier–Dunkl Coefficients of Generalized Lipschitz Classes on the Interval $$[-1,1]$$\",\"authors\":\"Othman Tyr\",\"doi\":\"10.1007/s00009-024-02710-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we consider <span>\\\\(\\\\mathcal {E}\\\\)</span> the set of all infinitely differentiable functions with compact support included on the interval <span>\\\\(I=[-1,1]\\\\)</span>. We use the distributions in <span>\\\\(\\\\mathcal {E}\\\\)</span>, as a tool to prove the continuity of the Dunkl operator and the Dunkl translation. Some properties of the modulus of smoothness related to the Dunkl operator are verified. By means of generalized Dunkl–Lipschitz conditions on Dunkl–Sobolev spaces, a result of Younis on the torus, which is an analog of Titchmarsh’s theorem, is deduced as a special case. In addition, certain conditions and a characterization of the Dini–Lipschitz classes on <i>I</i> in terms of the behavior of their Fourier–Dunkl coefficients are derived.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-08-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00009-024-02710-4\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00009-024-02710-4","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
On the Fourier–Dunkl Coefficients of Generalized Lipschitz Classes on the Interval $$[-1,1]$$
In this paper, we consider \(\mathcal {E}\) the set of all infinitely differentiable functions with compact support included on the interval \(I=[-1,1]\). We use the distributions in \(\mathcal {E}\), as a tool to prove the continuity of the Dunkl operator and the Dunkl translation. Some properties of the modulus of smoothness related to the Dunkl operator are verified. By means of generalized Dunkl–Lipschitz conditions on Dunkl–Sobolev spaces, a result of Younis on the torus, which is an analog of Titchmarsh’s theorem, is deduced as a special case. In addition, certain conditions and a characterization of the Dini–Lipschitz classes on I in terms of the behavior of their Fourier–Dunkl coefficients are derived.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.