Repeated averaging trades sharp edges for spectral rolloff

Use $$ \widehat f(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}\,dx,\qquad f(x)=\frac1{2\pi}\int_{-\infty}^{\infty}\widehat f(k)e^{ikx}\,dk. $$ Define the unit-area box $r_a(x)=1/a$ for $|x|<a/2$ and zero for $|x|>a/2$. For integer $n\ge1$, let $B_{n,a}=r_a^{\ast n}$, the $n$-fold convolution. Prove the area, support, polynomial degree, and differentiability claims for $B_{n,a}$. Derive its transform and the spectral-amplitude and intensity envelopes. Interpreting $B_{n,a}$ as a probability density, calculate its spatial variance. Work out $B_{2,a}$ explicitly and verify its squared norm directly and by Parseval. Explain why compact support does not imply finite bandwidth, even when convolution suppresses tails. For $a=0.200\,\mathrm s,n=4$, report support, differentiability, standard deviation, first positive spectral zero, and transform amplitude and intensity at $k=\pi/a$. Compare that intensity with the original box. Declare $\operatorname{sinc}u=\sin u/u$; do not silently use cyclic-frequency sinc.

First show $\widehat r_a(k)=\operatorname{sinc}(ka/2)$.