David Sermoneta: A guided tour of Wavelet theory via the constructions of Multiresolution analyses

Abstract

We give an exposition of the theory leading up to, and including, the subject of wavelet analysis. Wavelets provide an elegant and practical way to decompose functions in a fashion similar to that of decomposing vectors in Rⁿ , the key difference being that the bases for most function spaces are not finite. The main goal for this thesis is to first develop the necessary theory for what essentially amounts to ”doing linear algebra” in an infinite-dimensional context, and then apply that frame-work to the study of Wavelet analysis.

To start off, we develop a more flexible notion of integration called the Lebesgue integral, and use it to construct normed spaces of functions, such as L²(R), the inner product space of square-integrable functions. To build towards Wavelet analysis, we define the Fourier transform, a powerful tool in functional analysis that lets us describe functions in terms of complex exponentials. Finally, we introduce the theory of multiresolution analyses, a central object in the study of Wavelets, as they let us construct wavelet bases of L²(R) by decomposing the space into nested subspaces corresponding to different levels of ”detail”. This provides a frame-work for analyzing how the components of a function are distributed across these evels. Using this theory, we construct the famous Haar and Shannon wavelets and their corresponding bases.