Multiresolution wavelet analysis of integer scale Bessel functions

TL;DR

This paper develops a multiresolution wavelet framework for analyzing Bessel functions via Hilbert spaces, linking it to operator algebras, Markov chains, and quantum group representations.

Contribution

It introduces a novel multiresolution analysis for Bessel functions, connecting wavelet theory with operator algebras and quantum groups.

Findings

Identification of multiresolution subspaces for Bessel functions

Connection between wavelet analysis and $C^{st}$-algebra representations

Derivation of Markov traces for quantum groups

Abstract

We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived from geometric Hilbert-space considerations, the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of Hilbert spaces. We study the theory of representations of the $C^{\ast}$-algebra $% O_{\nu +1}$ arising from this multiresolution analysis. A connection with Markov chains and representations of $O_{\nu +1}$ is found. Projection valued measures arising from the multiresolution analysis give rise to a Markov trace for quantum groups $SO_q$.