Wavelet and Fourier bases on Fractals

Abstract

In this thesis we first develop a geometric framework for spectral pairs and for orthonormal families of complex exponential functions in L2-spaces with respect to a given Borel probability measure that is compactly supported. Secondly, we develop wavelet bases on L2-spaces based on limit sets of different iteration systems. In the framework of spectral pairs we consider families of exponential functions with a countable index set G which difference set is equal to all integers, and we determine the L2-spaces in which these functions are orthonormal or constitute a basis. We also consider invariant measures on Cantor sets and study for which measures we have a family of exponential functions that is an orthonormal basis for the L2-space with respect to this measure. For the case of Cantor sets the families of exponential functions are obtained via Hadamard matrices. For the study of wavelet bases, we set up a multiresolution analysis on fractal sets derived from limit sets of Markov Interval Maps. For this we consider the translation by integers of a non-atomic measure supported on the limit set of such a system 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. We also set up a multiresolution analysis for enlarged fractals in one and two dimensions, which are sets arising from fractals that are generated by iterated function systems, so that the enlarged fractals are dense in the lin or plane, respectively. The measure supported on the fractal is also extended to a measure on the enlarged fractal. We then construct a wavelet basis via multiresolution analysis on this L2-space with respect to the measure having the enlarged fractal as the support, with the characteristic function of the original fractal as the father wavelet which gives us via the multiresolution analysis the wavelet basis for the L2-space. In this construction we have two unitary operators. Finally, we also apply the wavelet bases on enlarged fractals in two dimensions to image compression.