On fixed point sets of wavelet induced isomorphisms and frame induced monomorphisms (Q2812454)

The paper is concerned with dyadic wavelets on the real line, i.e., with functions \(\psi\in L^2(\mathbb R)\) for which a system of the form NEWLINE\[NEWLINE \{\psi_{j,k}=2{^\frac{j}{2}}\psi(2^j\cdot -k):j,k\in \mathbb Z\} NEWLINE\]NEWLINE makes up an orthonormal basis for \( L^2(\mathbb R)\). In particular, the authors consider WSF wavelets which are wavelets whose Fourier transforms are of the form \(\chi_W\) for some measurable set (so called wavelet set) \(W\). The authors investigate properties of symmetric wavelet sets consisting of a larger number of intervals. It is proved that the Lebesgue measure of the fixed point set of the wavelet induced isomorphisms is zero for a class of symmetric \(2n\)-interval wavelet sets. The authors also provide an example of a symmetric wavelet set consisting of six intervals whose associated wavelet induced isomorphism possesses a fixed point set of positive measure. The last section of the paper is devoted to the study of frame wavelet induced isomorphisms and fixed point sets of such maps.