Geometric Structure in Weighted Alpert Wavelets

Abstract

In this paper we present a number of results concerning Alpert wavelet bases for $L^2(\hat{I}¼)$, with $\hat{I}¼$ a locally finite positive Borel measure on ${\mathbb{R}}^n$. We show that the properties of such a basis depend on linear dependences in $L^2(\hat{I}¼)$ among the functions from which the wavelets are constructed; this result completes an investigation begun by Rahm, Sawyer, and Wick in arXiv:1808.01223. We also show that a Gröbner basis technique can be used to efficiently detect these dependences. Lastly we give a generalization of the Alpert basis construction, where the amount of orthogonality in the basis is allowed to vary over the dyadic grid.