Geometric Structure in Weighted Alpert Wavelets

TL;DR

The paper develops a comprehensive framework for Alpert wavelets in $L^2(\mu)$, showing that the basis structure depends on linear dependencies among generating functions and that Gröbner bases can efficiently detect these dependencies. It introduces polynomial Alpert bases and establishes dimension formulas via Hilbert dimension, showing that $\dim P_{Q,F^n_k}(\mu)$ grows as $O(k^d)$ with Hilbert dimension $d$, and provides a mechanism to compute these dimensions from a single Gröbner basis $G$ for $I_Q$. A generalization to variable orthogonality across the dyadic grid is then presented, yielding a complete, orthonormal decomposition with telescoping identities while allowing the amount of orthogonality to vary by cube. Together, these results extend Alpert wavelet theory to irregular measures and offer tools for dimension control and adaptable basis constructions in harmonic analysis and Calderón-Zygmund operator contexts.

Abstract

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

Theorems & Definitions (23)

• definition 1: Component Space
• definition 2: Alpert Space
• definition 3
• definition 4: Gröbner Basis
• definition 5: Hilbert Dimension
• definition 6
• definition 7
• lemma 1
• proof
• theorem 1