Radial Multiresolution, Cuntz Algebras Representations and an Application to Fractals

We study Bernoulli type convolution measures on attractor sets arising from iterated function systems on R. In particular we examine orthogonality for Hankel frequencies in the Hilbert space of square integrable functions on the attractor coming from a radial multiresolution analysis on R³. A class of fractals emerges from a finite system of contractive affine mappings on the zeros of Bessel functions. We have then fractal measures on one hand and the geometry of radial wavelets on the other hand. More generally, multiresolutions serve as an operator theoretic framework for the study of such selfsimilar structures as wavelets, fractals, and recursive basis algorithms. The purpose of the present paper is to show that this can be done for a certain Bessel–Hankel transform. Submitted: February 20, 2008., Accepted: March 6, 2008.