Samplet limits and multiwavelets

2026-04-02

AI Summary

This work addresses the construction of multiwavelet bases with prescribed vanishing moments on scattered data in arbitrary dimensions and analyzes their asymptotic behavior as the data size tends to infinity. Within a probabilistic framework, the authors propose a flexible method for constructing samplets that overcomes the limitations of traditional tensor-product approaches and accommodates arbitrary vanishing moments; for regular partitions, the method recovers classical multiwavelets along with their scale-invariant filters. Theoretically, it is shown that when polynomial basis functions are employed, samplets converge in the infinite-data limit to a signed measure induced by hierarchical domain partitions, whose density is piecewise polynomial. Numerical experiments on both random and low-discrepancy point sets confirm this convergence behavior.

Abstract

Samplets are data adapted multiresolution analyses of localized discrete signed measures. They can be constructed on scattered data sites in arbitrary dimension and such that they exhibit vanishing moments with respect to any prescribed set of primitives. We consider the samplet construction in a probabilistic framework and show that, when choosing polynomials as primitives, the resulting samplet basis converges in the infinite data limit to signed measures with broken polynomial densities. These densities amount to multiwavelets with respect to a hierarchical partition of the region containing the data sites. As a byproduct, we therefore obtain a construction of general multiwavelets that allows for a flexible prescription of vanishing moments going beyond tensor product constructions. For congruent partitions we particularly recover classical multiwavelets with scale- and partition- independent filter coefficients. The theoretical findings are complemented by numerical experiments that illustrate the convergence results in case of random as well as low-discrepancy data sites.

Problem

• samplets
• multiwavelets
• vanishing moments
• multiresolution analysis
• signed measures

Innovation

• samplets
• multiwavelets
• vanishing moments
• probabilistic framework
• hierarchical partition