Multiresolution analysis for Markov Interval Maps

TL;DR

This paper develops a multiresolution analysis framework on fractal sets from Markov Interval Maps, introducing wavelets and analyzing function spaces with respect to measures supported on these fractals.

Contribution

It establishes an abstract multiresolution analysis on fractals derived from Markov Interval Maps and constructs multiwavelets despite non-unitary scaling operators.

Findings

Constructed mother wavelets for fractal measures.

Provided a complete description of multiresolution analysis on these fractals.

Analyzed the space of square integrable functions with respect to the fractal measures.

Abstract

We set up a multiresolution analysis on fractal sets derived from limit sets of Markov Interval Maps. For this we consider the $\mathbb{Z}$-convolution of a non-atomic measure supported on the limit set of such systems and give a thorough investigation of the space of square integrable functions with respect to this measure. We define an abstract multiresolution analysis, prove the existence of mother wavelets, and then apply these abstract results to Markov Interval Maps. Even though, in our setting the corresponding scaling operators are in general not unitary we are able to give a complete description of the multiresolution analysis in terms of multiwavelets.