Samplet limits and multiwavelets
This provides a novel construction of multiwavelets for scattered data analysis, which is incremental as it extends existing samplet frameworks to probabilistic limits and multiwavelet theory.
The paper tackles the problem of constructing samplets, which are data-adapted multiresolution analyses for scattered data, and shows that in the infinite data limit with polynomial primitives, they converge to multiwavelets with broken polynomial densities, enabling flexible prescription of vanishing moments beyond tensor product methods.
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.