{"title":"On a sampling expansion with partial derivatives for functions of several variables","authors":"S. Norvidas","doi":"10.2298/FIL2010339N","DOIUrl":null,"url":null,"abstract":"Let $B^p_{\\sigma}$, $1\\le p 0$, denote the space of all $f\\in L^p(\\mathbb{R})$ such that the Fourier transform of $f$ (in the sense of distributions) vanishes outside $[-\\sigma,\\sigma]$. The classical sampling theorem states that each $f\\in B^p_{\\sigma}$ may be reconstructed exactly from its sample values at equispaced sampling points $\\{\\pi m/\\sigma\\}_{m\\in\\mathbb{Z}} $ spaced by $\\pi /\\sigma$. Reconstruction is also possible from sample values at sampling points $\\{\\pi \\theta m/\\sigma\\}_m $ with certain $1< \\theta\\le 2$ if we know $f(\\theta\\pi m/\\sigma) $ and $f'(\\theta\\pi m/\\sigma)$, $m\\in\\mathbb{Z}$. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2019-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Classical Analysis and ODEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2298/FIL2010339N","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
Let $B^p_{\sigma}$, $1\le p 0$, denote the space of all $f\in L^p(\mathbb{R})$ such that the Fourier transform of $f$ (in the sense of distributions) vanishes outside $[-\sigma,\sigma]$. The classical sampling theorem states that each $f\in B^p_{\sigma}$ may be reconstructed exactly from its sample values at equispaced sampling points $\{\pi m/\sigma\}_{m\in\mathbb{Z}} $ spaced by $\pi /\sigma$. Reconstruction is also possible from sample values at sampling points $\{\pi \theta m/\sigma\}_m $ with certain $1< \theta\le 2$ if we know $f(\theta\pi m/\sigma) $ and $f'(\theta\pi m/\sigma)$, $m\in\mathbb{Z}$. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives.