Thinplate Splines on the Sphere

Abstract

In this paper, we give explicit closed forms for the semi-reproducing kernels associated with thinplate spline interpolation on the sphere. Polyharmonic or thinplate splines for Rᵈ were introduced by Duchon and have become a widely used tool in myriad applications. The analogues for Sᵈ⁻¹ are the thin plate splines for the sphere. The topic was first discussed by Wahba in the early 1980s, for the S² case. Wahba presented the associated semi-reproducing kernels as infinite series. These semi-reproducing kernels play a central role in expressions for the solution of the associated spline interpolation and smoothing problems. The main aims of the current paper are to give a recurrence for the semi-reproducing kernels and also to use the recurrence to obtain explicit closed-form expressions for many of these kernels. The closed-form expressions will, in many cases, be significantly faster to evaluate than the series expansions. This will enhance the practicality of using these thinplate splines for the sphere in computations.