On the Gaussian functions of two discrete variables

TL;DR

This paper extends the concept of Gaussian functions to two discrete variables using Jacobi theta functions, exploring their Fourier and Wigner transforms to deepen understanding of their properties.

Contribution

It introduces a novel discrete two-variable Gaussian function framework based on Jacobi theta functions and analyzes its Fourier and Wigner transforms.

Findings

Defined a new class of discrete two-variable Gaussian functions

Analyzed Fourier transform properties of these functions

Investigated Wigner functions for the discrete Gaussian

Abstract

A remarkable discrete counterpart of the Gaussian function of one continuous variable can be defined by using a Jacobi theta function, that is, as the sum of a convergent series. We extend this approach to Gaussian functions of two variables, and investigate the Fourier transform and Wigner function of the functions of discrete variable defined in this way.