Structure and Stability of Weighted Alpert Wavelets

Abstract

In this thesis we present a number of results concerning Alpert wavelet bases for L2(µ), with µ a locally finite positive Borel measure on Rn. Alpert wavelets generalize Haar wavelets while retaining their orthonormality, telescoping, and moment vanishing properties. We show that the properties of such a basis are determined by the geometric structure of µ; in particular they are the result of linear dependences in L2(µ) among the functions from which the wavelets are constructed; this completes an investigation begun by Rahm, Sawyer, and Wick. These dependences can be efficiently detected using a Grobner basis algorithm, which provides enough information to determine the structure of any Alpert basis constructed on µ. We present a generalization of the usual Alpert wavelet construction, where the degree of moment vanishing is allowed to vary over the dyadic grid. We also show that Alpert bases in a doubling measure on R are stable under small translations of the underlying dyadic intervals, building on work by Wilson. We conclude with a partial result toward the converse, showing that a class of non-doubling measures cannot have this stability property.