Some new examples of wavelets in the Hardy space \(H^2(\mathbb{R})\) (Q2739298)

In this paper, some new examples of wavelet sets and scaling sets in the Hardy space \(H^2({\mathbb R)}\) are presented. Consider the maps \(\tau:{\mathbb R}\rightarrow [2\pi,4\pi)\) and \(d:{\mathbb R}\rightarrow [2\pi, 4\pi)\) where NEWLINE\[NEWLINE\tau (x)=x+2\pi m(x), \qquad d(x)=2^{n(x)} x,NEWLINE\]NEWLINE and \(m(x), n(x)\) are the unique integers such that \(d\) and \(\tau\) map \(x\) into \([2\pi, 4\pi)\). Then \(W\) is a wavelet set if \(d|_W\) and \(\tau|_W\) are bijections modulo null sets.NEWLINENEWLINENEWLINEIt is shown that there exists a wavelet set \(W\) such that \(\varphi\) defined by \(\widehat\varphi =\chi_W\) is a continuous, bounded \(L^2\)-function, where \(\varphi \notin L^p({\mathbb R})\) for \(p<2\) and \(\varphi\) does not belong to any Sobolev space with positive smoothness.