Dimension functions, scaling sequences, and wavelet sets

Abstract

The paper is a continuation of our study of dimension functions of orthonormal wavelets on the real line with dyadic dilations. The main result of Section 2 is Theorem 2.8 which provides an explicit reconstruction of the underlying generalized multiresolution analysis for any MSF wavelet. In Section 3 we reobtain a result of Bownik, Rzeszotnik and Speegle which states that for each dimension function D there exists an MSF wavelet whose dimension function coincides with D. Our method provides a completely new explicit construction of an admissible generalized multiresolution analysis (and, a posteriori, of a wavelet) from an arbitrary dimension function. Several examples are included.