One-sided Band-limited Approximations of Some Radial Functions

TL;DR

This paper develops extremal majorants and minorants for Gaussian and radial functions with Fourier support constraints, extending to multidimensional and periodic cases, with applications to inequalities.

Contribution

It introduces new extremal functions for Gaussian and radial functions in multiple dimensions, using advanced methods adapted from prior techniques.

Findings

Constructed extremal majorants for Gaussian functions with Fourier support in a box.

Established asymptotic extremality of minorants as the support size increases.

Applied the results to derive inequalities of Hilbert-type.

Abstract

We construct majorants and minorants of a Gaussian function in Euclidean space that have Fourier transforms supported in a box. The majorants that we construct are shown to be extremal and our minorants are shown to be asymptotically extremal as the sides of the box become uniformly large. We then adapt the Distribution and Gaussian Subordination methods of Carneiro-Littmann-Vaaler to the multidimensional setting to obtain majorants and minorants for a class of radial functions. Periodic analogues of the main results are proven and applications to Hilbert-type inequalities are given.