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

TL;DR

This paper extends classical sampling theorems to multivariable functions, providing new sampling series that incorporate samples of functions and their partial derivatives for exact reconstruction.

Contribution

It introduces multivariable sampling series that utilize both function samples and partial derivatives, advancing sampling theory beyond single-variable cases.

Findings

Sampling series for multivariable functions involving derivatives

Exact reconstruction from samples and derivatives in multiple dimensions

Generalization of classical sampling theorems to several variables

Abstract

Let $B^p_{\sigma}$, $1\le p<\infty$, $\sigma>0$, 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 $f\in B^p_{\sigma}$ 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.