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arXiv · 1008.4969

Gaussian Subordination for the Beurling-Selberg Extremal Problem

Emanuel Carneiro·Friedrich Littmann·Jeffrey D. Vaaler
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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.

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Emanuel Carneiro, Friedrich Littmann, Jeffrey D. Vaaler. 2010-08-29. Gaussian Subordination for the Beurling-Selberg Extremal Problem. https://doi.org/10.1090/s0002-9947-2013-05716-9

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