On a sampling expansion with partial derivatives for functions of several variables

Abstract

Let $B^p_{\sigma}$, $1\le p<\infty$, $\sigma>0$,  denote the space of all $f\in L^p(\R)$ 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\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  $f'(\theta\pi m/\sigma)$, $m\in\Z$. In this paper we present  sampling series for functions of several variables. These series involves  samples of functions  and    their partial  derivatives.