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, 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 $n \in \mathbb {N}$. 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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• Transactions of the American Mathematical Society - View - PDF