On Supersets of Wavelet Sets

Considering a single dyadic orthonormal wavelet ψ in L2(ℝ), it is still an open problem whether the support of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\widehat{\psi}$\end{document} always contains a wavelet set. As far as we know, the only result in this direction is that if the Fourier support of a wavelet function is “small” then it is either a wavelet set or a union of two wavelet sets. Without assuming that a set S is the Fourier support of a wavelet, we obtain some necessary conditions and some sufficient conditions for a “small” set S to contain a wavelet set. The main results, which are in terms of the relationship between two explicitly constructed subsets A and B of S and two subsets T2 and D2 of S intersecting itself exactly twice translationally and dilationally respectively, are (1) if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A\cup B\not\subseteq T_{2}\cap D_{2}$\end{document} then S does not contain a wavelet set; and (2) if A∪B⊆T2∩D2 then every wavelet subset of S must be in S∖(A∪B) and if S∖(A∪B) satisfies a “weak” condition then there exists a wavelet subset of S∖(A∪B). In particular, if the set S∖(A∪B) is of the right size then it must be a wavelet set.