Bandlimited approximations to the truncated Gaussian and applications

TL;DR

This paper develops explicit methods for optimal bandlimited approximations of truncated and odd Gaussian functions, extending previous work and broadening the class of functions for which such approximations are known.

Contribution

It provides explicit integral representations and solves the extremal approximation problem for truncated and odd Gaussians, extending prior results to new function classes.

Findings

Recovered known examples in the literature.

Extended the class of functions with explicit extremal solutions.

Utilized properties of truncated theta functions and heat operator.

Abstract

In this paper we extend the theory of optimal approximations of functions $f: \R \to \R$ in the $L^1(\R)$-metric by entire functions of prescribed exponential type (bandlimited functions). We solve this problem for the truncated and the odd Gaussians using explicit integral representations and fine properties of truncated theta functions obtained via the maximum principle for the heat operator. As applications, we recover most of the previously known examples in the literature and further extend the class of truncated and odd functions for which this extremal problem can be solved, by integration on the free parameter and the use of tempered distribution arguments. This is the counterpart of the work \cite{CLV}, where the case of even functions is treated.