A remark concerning sinc integrals

TL;DR

This paper provides a simplified proof of a known sinc integral result and extends the analysis to identify when the integral values fall below , revealing new thresholds at larger n.

Contribution

It offers a straightforward proof of Schmid's sinc integral result and determines new n-values where the integral drops below , expanding understanding of these integrals.

Findings

Proved that K_n= for n up to 55.

Showed K_n< for n .

Identified n-values (418 and 3091) where the integral drops below .

Abstract

We give a simple proof of Hanspeter Schmid's result that $K_n:=2\int_0^\infty\cos t\,\prod_{k=0}^n\mathrm{sinc}\left(\frac{t}{2k+1}\right)\mathrm{d}t=\pi/2$ if $n\in\{0,1,\ldots,55\}$, and $K_n<\pi/2$ if $n\geq 56$. Furthermore, we present two sinc integrals where the value $\pi/2$ is undercut as soon as $n\geq 418$ and $n\geq 3091$, respectively.