Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 5

An orthonormal wavelet is $\psi \in L^2(\mathbb{R})$ such that $\{2^{j/2}\psi (2^{j}x-k){\}}_{j,k\in \mathbb{Z}}$ is an orthonormal basis. A multiresolution analysis is a nested family of closed spaces $V_j$ with trivial intersection, dense union, dyadic scaling, integer-translation invariance of $V_0$, and a generator $\varphi$ whose integer translates form an orthonormal basis of $V_0$. The Meyer-Mallat theorem says every such analysis has an orthonormal wavelet whose translates span $V_{j+1}\ominus V_j$.

Fourier transforming the refinement equation and changing variables givesBy Parseval identity, orthonormality of the translates is equivalent toThese are precisely the Fourier coefficients of the periodization $P(t)={\sum }_k|f(t+2\pi k){|}^2$. Therefore

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