A remark concerning sinc integrals

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 $n\in \{0,1,\dots ,55\}$, and $K_n<\pi /2$ if $n\ge 56$. Furthermore, we present two sinc integrals where the value $\pi /2$ is undercut as soon as $n\ge 418$ and $n\ge 3091$, respectively.

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