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 $\mathbb{ R}^3$.
Categories: math.CV
