Construction of wavelet bases for L℗(R)

Classical orthonormal bases usually consist of functions whose supports are not local and which, therefore, are not very efficient for representing functions that are local both in time and frequency or in both space and wave number. Wavelet bases overcome this deficiency matching the supports of their elements to their frequencies. In addition, the elements of a wavelet basis are obtained by translation and dilation from a single function and consequently, are especially suitable for use in computationally efficient algorithms.

The purpose of this thesis is to present a readable development of orthonormal bases of wavlets. This development is approached through the concept of a multiresolution analysis. It leads to the derivation of a few key-conditions, whose solutions, as we prove, produce wavelet bases.

This thesis is roughly based on a recent presentation by R. Strichartz [8], but seeks to provide greater rigor and clarity.