Gaussian Subordination for the Beurling-Selberg Extremal Problem

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 $-1<\alpha$; \,$\log ((x^2+{\alpha }^2)/(x^2+{\beta }^2))$, for $0\le \alpha <\beta$;\, $\log (x^2+{\alpha }^2)$; and $x^{2n}\log x^2$\,, for $n\in \mathbb{N}$. 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.

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