Distance function wavelets - Part III: "Exotic" transforms and series

TL;DR

This paper explores unconventional distance function wavelets (DFW) using PDE dimension and order as scale parameters, introducing new series, connections, and applications in neural networks, with a focus on flexibility and intuition over rigor.

Contribution

It introduces novel DFW transforms and series based on PDE properties, develops anisotropic and inhomogeneous DFWs, and links these to neural network kernel functions, expanding the scope of wavelet analysis.

Findings

DFW series formulated using PDE dimension and order.

Development of anisotropic and inhomogeneous DFWs.

Potential applications in neural networks and machine learning.

Abstract

Part III of the reports consists of various unconventional distance function wavelets (DFW). The dimension and the order of partial differential equation (PDE) are first used as a substitute of the scale parameter in the DFW transforms and series, especially with the space and time-space potential problems. It is noted that the recursive multiple reciprocity formulation is the DFW series. The Green second identity is used to avoid the singularity of the zero-order fundamental solution in creating the DFW series. The fundamental solutions of various composite PDEs are found very flexible and efficient to handle a borad range of problems. We also discuss the underlying connections between the crucial concepts of dimension, scale and the order of PDE through the analysis of dissipative acoustic wave propagation. The shape parameter of the potential problems is also employed as the "scale parameter" to create the non-orthogonal DFW. This paper also briefly discusses and conjectures the DFW correspondences of a variety of coordinate variable transforms and series. Practically important, the anisotropic and inhomogeneous DFW's are developed by using the geodesic distance variable. The DFW and the related basis functions are also used in making the kernel distance sigmoidal functions, which are potentially useful in the artificial neural network and machine learning. As or even worse than the preceding two reports, this study scarifies mathematical rigor and in turn unfetter imagination. Most results are intuitively obtained without rigorous analysis. Follow-up research is still under way. The paper is intended to inspire more research into this promising area.