IFSM, wavelets and fractal-wavelets, three methods of approximation

Abstract

This thesis deals with representations and approximations of functions using iterated function systems (IFS), wavelets and fractal-wavelets.

IFS use self-similarity to approximate a function by contracted and translated copies of itself. Results covered include the Banach Contraction Mapping Principle, the completeness of IFS space and the Collage Theorem. IFS on grey-level maps (IFSM) are defined to generalize IFS to real-valued functions.

Wavelets are discussed, using multiresolution analysis. Stronger convergence results are shown to hold for wavelet expansions than for Fourier expansions. An application of the Mallat algorithm to compression is given.

Fractal-wavelets use the fact that given an orthonormal basis of L^2(R), the mapping which sends a function of L^2(R) to its sequence of basis coefficients is an isometry. An identification is made between IFSM and operators on coefficients. Local IFS on wavelet coefficients are defined and shown to induce IFSM-type operators.