A multidimensional Hermite-Gauss sampling formula for analytic functions of several variables

Recently, Norvidas has introduced the general multidimensional Hermite sampling series, which involves samples from a function and its mixed and non-mixed partial derivatives. The convergence of this sampling series is slow unless the sample values |f(x)| rapidly decay as |xj|→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|x_j| \rightarrow \infty $$\end{document} for all 1≤j≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le j \le n$$\end{document}. In this paper, we investigate a modified version of this sampling series that utilizes a multivariate Gaussian multiplier to approximate functions from two classes of multivariate analytic functions using a complex approach. The first class comprises entire functions of exponential type in n variables that fulfill a decay condition, while the second class includes analytic functions in n variables defined on a multidimensional horizontal strip. It has a significantly higher convergence rate compared to the general multidimensional Hermite sampling series.