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

Abstract

Let $B_{\sigma }^p$, $1\le p<\infty$, $\sigma >0$, 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 $f\in B_{\sigma }^p$ 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 $f^{\prime }(\theta \pi m/\sigma )$, $m\in \mathbb{Z}$. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives.

arXiv comment10 pages