Multiresolution expansions of distributions: Pointwise convergence and quasiasymptotic behavior

TL;DR

This paper proves pointwise convergence of multiresolution expansions for distributions in multiple variables, extending previous one-dimensional results, and characterizes their quasiasymptotic behavior at finite points.

Contribution

It extends earlier one-variable results to multiple variables and provides new characterizations of quasiasymptotic behavior of distributions.

Findings

Proves pointwise convergence of multiresolution expansions in several variables.

Extends and improves previous one-variable results.

Provides characterizations of quasiasymptotic behavior at finite points.

Abstract

In several variables, we prove the pointwise convergence of multiresolution expansions to the distributional point values of tempered distributions and distributions of superexponential growth. The article extends and improves earlier results by G. G. Walter and B. K. Sohn and D. H. Pahk that were shown in one variable. We also provide characterizations of the quasiasymptotic behavior of distributions at finite points and discuss connections with $\alpha$-density points of measures.