On wavelet induced isomorphisms for joint (d, -d)-dilation wavelet and multiwavelet sets

Abstract

For a dyadic wavelet set W, Ionascu [A new construction of wavelet sets, Real Anal. Exchange28 (2002) 593–610] obtained a measurable self-bijection on the interval [0, 1), called the wavelet induced isomorphism of [0, 1), denoted by [Formula: see text]. Extending the result for a d-dilation wavelet set, we characterize a joint (d, -d)-dilation wavelet set, where |d| is an integer greater than 1, in terms of wavelet induced isomorphisms. Its analogue for a joint (d, -d)-dilation multiwavelet set has also been provided. In addition, denoting by [Formula: see text], the wavelet induced isomorphism associated with a d-dilation wavelet set W, we show that for a joint (d, -d)-dilation wavelet set W, the measures of the fixed point sets of [Formula: see text] and [Formula: see text] are equal almost everywhere.

Minimally Supported Frequency (MSF) d-Dilation Wavelets

“…A d-dilation wavelet set was characterized by Bownik et al [7] as a measurable set of R that partitions R to be its integral translation and by its d-dilates. Further, Singh et al [13] characterized a joint (d, −d)-dilation wavelet set and multiwavelet set in terms of wavelet-induced isomorphisms, where |d| > 1.…”