Generalized bivariate Hermite–Gauss sampling

Abstract

Recently, Asharabi and Al-Haddad in (Turk J Math 41: 387–403, 2017) have established a generalized bivariate sampling series involving samples from the function and its mixed and non-mixed partial derivatives for an entire function of two-variables which satisfies certain growth conditions. This series converges slowly unless the sample values, $f^{(i,j)}\left( nh,mh\right)$, $i,j\in \mathbb {N}_{\circ }$, decay rapidly as $n,m\longrightarrow \mp \infty$. In this paper, we derive a modification of this sampling using a bivariate Gaussian multiplier. This modification highly improved the convergence rate of the bivariate sampling series which will be of an exponential type. Moreover, it is valid for wider classes, the class of entire functions including unbounded functions on $\mathbb {R}^{2}$ and the class of analytic functions in a bivariate strip. We show that many known results included in Asharabi and Prestin (IMA J Numer Anal 36: 851–871, 2016) are special cases of our results. Various illustrative examples are presented and they show a rightly good agreement with our theoretical analysis.