MULTIRESOLUTION APPROXIMATIONS AND ORTHONORMAL WAVELET BASES

Abstract

Wavelets were introduced in France in the early 1980's by Jean Morlet and Alexander Grossman to analyse seismic signal. The mathematical theory of wavelets took off in 1985 when Yves Meyer, also in France, constructed the first orthogonal system of smooth wavelets such that their Fourier Tranform have compact support. In 1986, Meyer and Stephane Mallat developed the theory of Multiresolution Approximations which provide a natural framework for the theory of wavelet analysis and construction of orthonormal basis. An extensive overview on the subject of wavelets is given in [1], which also has an extensive list of references on the subject. Wavelets have shown great promises in the areas of mathematics such as Approximation Theory, Harmonic Analysis, Operator Theory and Numerical Partial Differential Equations. Much of the interest in the engineering areas are in their application to signal processing, where wavelets seem to hold great promises for detection of edges and singularities, provide efficient decomposition and reconstruction algorithms for signals, and data compression. Another area where wavelets are being used is in the theory of Quantum Field. The main object of this thesis is to introduce the theory of Multiresolution Approximations as an intermediary to obtain orthonormal wavelet bases for L ² (IR). As such, the construction of multiresolution approximations, the algorithm to obtain orthonormal wavelets bases constituted the bulk of this thesis.