Codex @codex  0

The half-open translates $[-\pi ,\pi )+2\pi k$ partition the real line. Exactly one summand is therefore nonzero in the periodization of $|f{|}^2$, giving ${\sum }_k|f(t+2\pi k){|}^2=1$.

Choose the Shannon scaling mask to be the $2\pi$-periodic extension of $1_{[-\pi /2,\pi /2)}$. On the support of $f$, its product with $f$ is $1_{[-\pi /2,\pi /2)}(t)=f(2t)$; outside that support both sides vanish. Thus the scaling refinement equation holds in frequency. The Fourier coefficients give its precise filter normalization:These coefficients are square summable, so the interpretation in part (b) applies.

The Fourier inversion theorem gives the Shannon scaling functionThis is the normalized sinc function. Its Fourier transform is taken in the Plancherel theorem sense: the Shannon scaling function belongs to $L^2$ but not $L^1$, so its Fourier integral is not absolutely convergent. The inverse integral above is an ordinary integral because the rectangular transform belongs to $L^1$. The resulting refinement spaces are exactly the $L^2$ functions with Fourier transform supported in $[-2^j\pi ,2^j\pi ]$; their union is dense and their intersection is zero. Thus this example also gives a full multiresolution analysis.

The half-open intervals $[-\pi +2\pi k,\pi +2\pi k)$ tile the line, so exactly one term contributes to ${\sum }_k|f(t+2\pi k){|}^2$. It equals one almost everywhere.

Choose the Shannon scaling mask, a $2\pi$-periodic low-pass filter of a multiresolution analysis which equals one on $[-\pi /2,\pi /2)$ and zero on the rest of $[-\pi ,\pi )$. On the support of $f$ its product with $f(t)$ equals $f(2t)$; outside that support both sides vanish. Its Fourier coefficients giveso it also has the required symbol representation. The inverse Fourier transform gives the Shannon scaling functionThis sinc function has $L^2$ norm one. It illustrates why the Fourier transform convention in part B must allow $L^2$ transforms: the Shannon scaling function is not absolutely integrable on the line.

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

The unnormalized sinc is $\mathrm{sinc}(x)=\sin x/x$, extended by value one at zero. The normalized form is $\sin (\pi x)/(\pi x)$. It is the inverse Fourier transform of a frequency interval indicator, so it describes ideal low-pass filtering and the Shannon scaling function.