Reduced-quaternion inframonogenic functions on the ball

TL;DR

This paper develops an explicit orthogonal basis for square-integrable inframonogenic functions in the ball, using reduced-quaternion homogeneous polynomials, advancing the mathematical understanding of these functions.

Contribution

It introduces a new basis for inframonogenic functions in the reduced quaternion setting, explicitly constructed from homogeneous polynomials of degree n.

Findings

Homogeneous polynomials of degree n form a 6n+3 dimensional subspace.

Constructed an explicit orthogonal basis for inframonogenic functions.

Provided a computable basis for the Hilbert space of these functions.

Abstract

A function $f$ from a domain in $\mathbb{R}^3$ to the quaternions is said to be inframonogenic if $\overline{\partial}\, f\overline{\partial} =0$, where $\overline{\partial} = \partial/\partial x_0+ (\partial/\partial x_1)e_1+(\partial/\partial x_2) e_2$. All inframonogenic functions are biharmonic. In the context of functions $f=f_0+f_1e_1+f_2e_2$ taking values in the reduced quaternions, we show that the homogeneous polynomials of degree $n$ form a subspace of dimension $6n+3$. We use them to construct an explicit, computable orthogonal basis for the Hilbert space of square-integrable inframonogenic functions defined in the ball in $\mathbb{R}^3$.