  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 $\mathbb{R}^3$.
