Frame wavelets with frame set support in the frequency domain (Q1765977)

A function \(\psi\in L^2(\mathbb{R})\) is a frame wavelet if the functions \(\{2^{n/2}\psi(2^nx-k)\}_{k,n\in Z}\) constitute a frame for \(L^2(\mathbb{R})\). A Lebesgue measurable set \(E\subset \mathbb{R}\) is called a frame wavelet set if the function \(\psi\) defined via \(\frac1{2\pi}\hat{\psi}=\chi_E\) is a frame wavelet. The paper deals with the construction of frame wavelets for which the support of \(\hat{\psi}\) is a frame wavelet set. In particular it is proved that for any given positive number \(\alpha\), there exists a frame wavelet whose Fourier transform is supported by a set of measure \(\alpha\).