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 Ω⊂Rd when the translates are restricted to ΩΩ. Fundamentality is not hard to demonstrate when a low degree polynomial may be added or when translates are permitted to lie outside of ΩΩ; the challenge of this problem stems from the presence of the boundary, for which all successful approximation schemes require an added polynomial.When ΩΩ 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 functions on the sphere. This gives rise to a new approximation scheme composed of two parts: one which approximates purely on ∂Ω∂Ω, and a second part involving a shift invariant approximant of a function vanishing outside of a neighborhood ΩΩ.