Summation of Gaussian shifts as Jacobi's third Theta function

TL;DR

This paper demonstrates that summing Gaussian shifts on infinite periodic grids can be expressed using Jacobi's third Theta function, linking it to Schrödinger equation solutions and providing bounds for approximation errors.

Contribution

It establishes a novel connection between Gaussian shift summations and Jacobi's Theta functions, offering insights into parameter selection and approximation accuracy.

Findings

Summation of Gaussian shifts can be represented as Jacobi's third Theta function.

Explicit bounds for approximation errors are derived based on Gaussian shape parameters.

The work links Gaussian summations to solutions of the Schrödinger equation.

Abstract

A proper choice of parameters of the Jacobi modular identity (Jacobi Imaginary transformation) implies that the summation of Gaussian shifts on infinity periodic grids can be represented as the Jacobi's third Theta function. As such, connection between summation of Gaussian shifts and the solution to a Schr\"{o}dinger equation is explicitly shown. A concise and controllable upper bound of the saturation error for approximating constant functions with summation of Gaussian shifts can be immediately obtained in terms of the underlying shape parameter of the Gaussian. This shed light on how to choose a shape parameter and provides further understanding on using Gaussians with increasingly flatness.