The closure of the set of tight frame wavelets (Q1038736)

The paper deals with the properties of the closure of the set of tight frame wavelets. It is shown that the collection of such wavelets is not dense in \(L^2(\mathbb{R}^n)\). Therefore a necessary and a sufficient conditions are given for a function \(f\in L^2(\mathbb{R}^n)\) to belong to the closure of this set. These are, respectively, fulfilling one of the familiar equations characterizing Parseval wavelets, and sufficiently small support of \(f\) in the Fourier domain. It is established that for dilations admitting only minimally supported frequency (MSF) wavelets the latter is also a necessary condition, and that such a class of framelets is pathwise connected as a subset of \(L^2(\mathbb{R}^n)\).