On fixed point sets of wavelet induced isomorphisms

Abstract

In this paper, we show that [½, 1) is not a fixed point set of any wavelet induced isomorphism. The same holds for [0, ½). This settles Problem 1 posed in the paper "On wavelet induced isomorphisms" by Singh in Int. J. Wavelets Multiresolut. Inf. Process.8 (2010) 359–371. Further, we obtain that every other subinterval of [0, 1) of measure ½ is a fixed point set of some wavelet induced isomorphism.