Frame wavelet sets in \(\mathbb{R}\) (Q2718970)

Let \(E\) be a Lebesgue measurable set with finite measure and define the function \(\psi \in L^2(R)\) by \(\widehat{\psi}= \frac{1}{\sqrt{2\pi}}\chi_E\). If the functions \(\{2^{n/2}\psi(2^n-l)\}_{n,l\in Z}\) constitute a (tight) frame for \(L^2(R)\), the set \(E\) is called a (tight) frame wavelet set. Sufficient conditions for \(E\) to be a frame wavelet set are given, and in the tight case equivalent conditions are obtained. As a consequence it is proved that the frame bound is an integer if \(E\) is a tight frame wavelet set.