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

Abstract

We improve on the inequality $\frac{1}{\pi }{\int }_{-\infty }^{\infty }(\frac{{\sin }^{2}t}{t^2}{)}^{p}dt\le \frac{1}{\sqrt{p}},0.2cmp\ge 1,$ showing that $\frac{1}{\pi }{\int }_{-\infty }^{\infty }(\frac{{\sin }^{2}t}{t^2}{)}^{p}dt\le C(p)\frac{\sqrt{3/\pi }}{\sqrt{p}},$ with $\underset{p⟶\infty }{\lim }C(p)=1,$ and indeed that {align*} \displaystyle{\lim_{p\longrightarrow \infty}\frac{1}π\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt/ \frac{\sqrt{3/π}}{\sqrt p}=1.} {align*}

3 pages