On the Wavelet Analysis of Square Integrable Functions.

Abstract

In this study, we derive the conditions under which an arbitrary L^2(R) function can be expanded in terms of the wavelet basis. This is done by rst deriving the conditions under which an arbitrary scaling function and the subspaces Vj form a multiresolution analysis for the L^2(R) space. Speci c examples like the Haar multiresolution analysis and the Meyer multiresolution analysis have been constructed using the Haar scaling function and the Meyer scaling function respectively. Furthermore, we derive conditions under which the set of wavelet functions form an orthonormal wavelet basis for the subspaces Wj . Examples such as the Haar wavelet and the Meyer wavelets have been constructed to span the L^2(R) space. Lastly, we demonstrate the approximation and smoothening of L^2(R) functions using discrete wavelets.