Construction of MRA and non-MRA wavelet sets on Cantor dyadic group (Q1997737)

Let \(G\) be a Cantor dyadic group, which is a set of sequences \[ x=(\cdots x_n x_{n-1}\ldots x_0.x_{-1}x_{-2}\dots) \] with \(x_j\in \{0,1\}\), \(j\in \mathbb{Z}\) and \(x_j=0\) for every \(j\ge n\) for some \(n\in \mathbb{Z}\). The group operation on \(G\) is coordinatewise addition modulo \(2\). Under the Haar measure \(\mu\) on Borel setsubsets of \(G\), the authors consider wavelet sets in the Lebesgue space \(L^2(G)\). More precisely, a function \(\psi\in L^2(G)\) is said to be an orthonormal wavelet for \(L^2(G)\) if \(\{ 2^{j/2}\psi(\rho^j x-n) : j\in \mathbb{Z}, n\in \Lambda\}\) forms an orthonormal basis for \(L^2(G)\). In Section 2 on preliminaries, the authors study several propertied related to multiresolution analysis of \(L^2(G)\) by generalizing corresponding results on wavelets in the Euclidean space \(\mathbb{R}^n\). The main result is Theorem 4 showing that a measurable subset \(\Omega\) of \(G^\ast\) is a wavelet set (i.e., \(\psi\) is an orthonormal wavelet in \(L^2(G)\) with \(\hat{\psi}=\chi_{\Omega}\)) if and only if \(\{\sigma^k (\Omega): k\in \mathbb{Z}\}\) tiles \(G^\ast\) up to sets of measure zero and \(\Omega\) is \(\Lambda\)-translation congruent to \(D\) up to sets of measure zero. Then the authors study wavelet sets on Cantor dyadic group associated with MRA in Theorem 6 of Section 4 saying that \(\Omega\) is a wavelet set associated with an MRA if and only if \(\mu^\ast(\Omega^s\cap (\Omega^s+n))=\delta_{n,0}\) for all \(n\in \Lambda\), where \(\Omega^s=\cup_{j=1}^\infty (\sigma^j (\Omega))\). Several examples are provided to illustrate the results in this paper.