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arXiv · 2503.21046

Geometric Structure in Weighted Alpert Wavelets

Fletcher Gates·Scott Rodney
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Abstract

In this paper we present a number of results concerning Alpert wavelet bases for $L^2(\mu)$, with $\mu$ a locally finite positive Borel measure on $\mathbb{R}^n$. We show that the properties of such a basis depend on linear dependences in $L^2(\mu)$ 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 Gr\"{o}bner 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.

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Fletcher Gates, Scott Rodney. 2025-03-26. Geometric Structure in Weighted Alpert Wavelets. https://arxiv.org/abs/2503.21046

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