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

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 $V_{j+1}\ominus V_j$.