Quaternionic metamonogenic functions in the unit disk

TL;DR

This paper constructs a set of quaternionic metamonogenic functions in the unit disk, demonstrating their approximation capabilities and applying them to solve the imaginary-time wave equation in polar coordinates.

Contribution

It introduces a new orthogonal set of quaternionic functions in the unit disk and shows their use in approximating solutions to the imaginary-time wave equation.

Findings

Set of quaternionic metamonogenic functions forms a dense subset in $L^2$

Functions are orthogonal except for a small subspace

Application to time-dependent solutions of the imaginary-time wave equation

Abstract

We construct a set of quaternionic metamonogenic functions (that is, in $\mbox{Ker}(D+\lambda)$ for diverse $\lambda$) in the unit disk, such that every metamonogenic function is approximable in the quaternionic Hilbert module $L^2$ of the disk. The set is orthogonal except for the small subspace of elements of orders zero and one. These functions are used to express time-dependent solutions of the imaginary-time wave equation in the polar coordinate system.