Gaussian Subordination for the Beurling-Selberg Extremal Problem

TL;DR

This paper develops a method using Gaussian subordination to find extremal entire functions that approximate or bound Gaussian functions and other related functions, with applications to number theory and inequalities.

Contribution

It introduces a Gaussian subordination framework to solve extremal problems for a broad class of even functions, extending previous results and providing new applications.

Findings

Solved extremal problems for Gaussian functions and their generalizations

Extended extremal function techniques to new classes of functions like |x|^α and log functions

Applied results to number theory, including theta function approximation and inequalities

Abstract

We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function $e^{-\pi\lambda x^2}$ by entire functions of exponential type. This leads to the solution of analogous extremal problems for a wide class of even functions that includes most of the previously known examples (for instance \cite{CV2}, \cite{CV3}, \cite{GV} and \cite{Lit}), plus a variety of new interesting functions such as $|x|^{\alpha}$ for $-1 < \alpha$; \,$\log \,\bigl((x^2 + \alpha^2)/(x^2 + \beta^2)\bigr)$, for $0 \leq \alpha < \beta$;\, $\log\bigl(x^2 + \alpha^2\bigr)$; and $x^{2n} \log x^2$\,, for $n \in \N$. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.