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arXiv · 2412.12835

Estimates on the decay of the Laplace-Polya integral

Gergely Ambrus·Barnabás Gárgyán
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Abstract

The Laplace--P\'olya integral, defined by $J_n(r) = \frac1\pi\int_{-\infty}^\infty \mathrm{sinc}^n t \cos(rt) \mathrm{d} \, t$, appears in several areas of mathematics. We study this quantity by combinatorial methods; accordingly, our investigation focuses on the values at integer $r$'s. Our main result establishes a lower bound for the ratio $\frac{J_n(r+2)}{J_n(r)}$ which extends and generalises the previous estimates of Lesieur and Nicolas, and provides a natural counterpart to the upper estimate established in our previous work. We derive the statement by purely combinatorial, elementary arguments. As a corollary, we deduce that no subdiagonal central sections of the unit cube are extremal, apart from the minimal, maximal, and the main diagonal sections. We also prove several consequences for Eulerian numbers.

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Gergely Ambrus, Barnabás Gárgyán. 2024-12-17. Estimates on the decay of the Laplace-Polya integral. https://doi.org/10.1112/blms.70157

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