A New Proof Of The Asymptotic Limit Of The $Lp$ Norm Of The Sinc Function

TL;DR

This paper refines the asymptotic behavior of the Lp norm of the sinc function, providing a sharper inequality and confirming the limit of the normalized integral as p approaches infinity.

Contribution

It establishes a new bound for the integral of the sinc function's power and proves the limit of the normalized integral as p tends to infinity.

Findings

The inequality is improved with a constant C(p) approaching 1.

The limit of the normalized integral equals 1 as p approaches infinity.

The asymptotic behavior of the Lp norm of sinc is precisely characterized.

Abstract

We improve on the inequality $\displaystyle{\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt\leq \frac{1}{\sqrt p}, {0.2 cm}p\geq 1,}$ showing that $\displaystyle{\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt\leq C(p) \frac{\sqrt{3/\pi}}{\sqrt p},}$ with $\displaystyle{\lim_{p\longrightarrow \infty} C(p)=1,}$ and indeed that {align*} \displaystyle{\lim_{p\longrightarrow \infty}\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt/ \frac{\sqrt{3/\pi}}{\sqrt p}=1.} {align*}