Sinc fourth-power integral (${\int }_0^{\infty }(\sin x/x{)}^{4}dx=\pi /3$)

The triangular function $f(x)=(1-|x|)$ on $|x|<1$, zero elsewhere, has Fourier transform $\hat{f}(k)=[\sin (k/2)/(k/2){]}^2$ in the angular-frequency convention. Its squared L2 norm is $2/3$. Parseval's identity therefore givesThis proves the displayed value. The removable value at zero is one, and the fourth-power decay bounds the tail. The sinc-squared integral follows similarly from a rectangular function rather than a triangle.