Reduced-quaternion inframonogenic functions on the ball

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 $f=f_0+f_1{e}_1+f_2{e}_2$ 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 ${\mathbb{R}}^3$.

References in corpus (1)

• A Cauchy-Kowalevski theorem for inframonogenic functions