On the density of polyharmonic splines

This article treats the question of fundamentality of the translates of a polyharmonic spline kernel (also known as a surface spline) in the space of continuous functions on a compact set $\Omega\subset \RR^d$ when the translates are restricted to $\Omega$. Fundamentality is not hard to demonstrate when a low degree polynomial may be added or when translates are permitted to lie outside of $\Omega$; the challenge of this problem stems from the presence of the boundary, for which all successful approximation schemes require an added polynomial. When $\Omega$ is the unit ball, we demonstrate that translates of polyharmonic splines are fundamental by considering two related problems: the fundamentality in the space of functions vanishing at the boundary and fundamentality of the restricted kernel in the space of continuous function on the sphere. This gives rise to a new approximation scheme composed of two parts: one which approximates purely on $\partial \Omega$, and a second part involving a shift invariant approximant of a function vanishing outside of a neighborhood $\Omega$.