This paper shows that weighted alpert wavelets exhibit linear dependences in L2(μ), suggesting Gr"obner basis techniques can enhance construction efficiency.
In this paper we present a number of results concerning Alpert wavelet bases for L²(μ), with μ a locally finite positive Borel measure on Rⁿ. We show that the properties of such a basis depend on linear dependences in L²(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 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.
No takes yet. Share an insight, caveat, or question.
Gates et al. (2025) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: