Dimension functions, scaling sequences, and wavelet sets
The paper is a continuation of our study of dimension functions of orthonormal wavelets on the real line with dyadic dilations. The main result provides an explicit reconstruction of the underlying generalized multiresolution analysis for any MSF wavelet. We also 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 an explicit construction of an admissible generalized multiresolution analysis (and, a posteriori, of a wavelet) from an arbitrary dimension function. Several examples are included.