A Characterization of Dimension Functions of Wavelets

Abstract

This paper is devoted to the study of the dimension functions of (multi)wavelets, which was introduced and investigated by P. Auscher in 1995 (J. Geom. Anal.5, 181-236). Our main result provides a characterization of functions which are dimension functions of a (multi)wavelet. As a corollary, we obtain that for every function D that is the dimension function of a (multi)wavelet, there is a minimally supported frequency (multi)wavelet whose dimension function is D. In addition, we show that if a dimension function of a wavelet not associated with a multiresolution analysis (MRA) attains the value K, then it attains all integer values from 0 to K. Moreover, we prove that every expansive matrix which preserves ZN admits an MRA structure with an analytic (multi)wavelet. © 2001 Academic Press.