{"title":"Almost everywhere convergence of prolate spheroidal series","authors":"Philippe Jaming, Michael Speckbacher","doi":"10.1215/00192082-8622664","DOIUrl":null,"url":null,"abstract":"In this paper, we show that the expansions of functions from $L^p$-Paley-Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1<p<\\infty$, even in the cases when they might not converge in $L^p$-norm. We thereby consider the classical Paley-Wiener spaces $PW_c^p\\subset L^p(\\mathcal{R})$ of functions whose Fourier transform is supported in $[-c,c]$ and Paley-Wiener like spaces $B_{\\alpha,c}^p\\subset L^p(0,\\infty)$ of functions whose Hankel transform $\\mathcal{H}^\\alpha$ is supported in $[0,c]$.As a side product, we show the continuity of the projection operator $P_c^\\alpha f:=\\mathcal{H}^\\alpha(\\chi_{[0,c]}\\cdot \\mathcal{H}^\\alpha f)$ from $L^p(0,\\infty)$ to $L^q(0,\\infty)$, $1<p\\leq q<\\infty$.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Classical Analysis and ODEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1215/00192082-8622664","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper, we show that the expansions of functions from $L^p$-Paley-Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1
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