Some addenda on distance function wavelets

TL;DR

This paper expands on distance function wavelets by defining new variants, exploring their transforms, discussing error estimates, and proposing interpolation methods to improve computational efficiency.

Contribution

It introduces new distance function wavelet variants, derives their transforms, and discusses error estimation and interpolation techniques for improved application.

Findings

Defined general distance via Riesz potential

Derived Abel wavelets from fractional integrals and Laplacian

Presented a conjecture on truncation error formula

Abstract

This report will add some supplements to the recently finished report series on the distance function wavelets (DFW). First, we define the general distance in terms of the Riesz potential, and then, the distance function Abel wavelets are derived via the fractional integral and Laplacian. Second, the DFW Weyl transform is found to be a shifted Laplace potential DFW. The DFW Radon transform is also presented. Third, we present a conjecture on truncation error formula of the multiple reciprocity Laplace DFW series and discuss its error distributions in terms of node density distributions. Forth, we point out that the Hermite distance function interpolation can be used to replace overlapping in the domain decomposition in order to produce sparse matrix. Fifth, the shape parameter is explained as a virtual extra axis contribution in terms of the MQ-type Possion kernel. The report is concluded with some remarks on a range of other issues.