Estimates on the decay of the Laplace-Polya integral

TL;DR

This paper investigates the decay properties of the Laplace-Polya integral at integer points using combinatorial methods, establishing bounds and implications for geometric extremality and Eulerian numbers.

Contribution

It provides a new lower bound for the ratio of Laplace-Polya integrals at shifted points, extending previous estimates with elementary combinatorial arguments.

Findings

Established a lower bound for J_n(r+2)/J_n(r) ratio.

Showed that only certain sections of the unit cube are extremal.

Derived new consequences for Eulerian numbers.

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.