 Solution

For $f=1_{[-\pi ,\pi )}$, its $2\pi$-periodization is one almost everywhere. The refinement mask is the periodic function equal to one on $[-\pi /2,\pi /2)$ and zero on the rest of $[-\pi ,\pi )$. Fourier inversion gives the Shannon scaling functionSince $m(t)=\frac{1}{2}{\sum }_n{a}_n{e}^{-int}$, its Fourier coefficients give