{"title":"长球面级数几乎处处收敛","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":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"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\":null,\"pages\":null},\"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}","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}
Almost everywhere convergence of prolate spheroidal series
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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