On the density of polyharmonic splines

Details On the density of polyharmonic splines On the density of polyharmonic splines

2011-11-28

arxiv.org/abs/1111.6578v1

Mathematics

Classical Analysis and ODEs

12 pages

Scientific paper

This brief article treats 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 R^d when the translates are restricted to \Omega. Fundamentality is not hard to demonstrate when \Omega= R^d or when \Omega is compact but a low degree polynomial may be added; 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.

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