Constructing $3$-Dimensional Monogenic Homogeneous Functions

TL;DR

This paper develops methods to construct 3D spherical monogenic functions using Dirac operators, orthogonal bases, and reproducing kernels, advancing the mathematical tools for multidimensional harmonic analysis.

Contribution

It introduces a novel approach to generate 3D spherical monogenics through Dirac operators and reproducing kernels, expanding the existing methods for multidimensional harmonic functions.

Findings

Constructed orthogonal spherical monogenics in 3D

Derived various types of 3D spherical harmonics and monogenics

Applied optimization techniques to enhance function construction

Abstract

This paper is dedicated to the construction of multidimensional spherical monogenics. Firstly, we investigate the construction of monogenic functions in dimension $3$ by applying the Dirac operator to the orthonormal bases of spherical harmonics, resulting in orthogonal spherical monogenics. Additionally, we employ the reproducing kernel for monogenic functions and a specialized optimization method to derive various types of $3$-dimensional spherical harmonics and spherical monogenics.