### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 083, 22 pages      arXiv:1801.01313      https://doi.org/10.3842/SIGMA.2018.083

### Thinplate Splines on the Sphere

Rick K. Beatson a and Wolfgang zu Castell bc
a) School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
b) Scientific Computing Research Unit, Helmholtz Zentrum München, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany
c) Department of Mathematics, Technische Universität München, Germany

Received January 08, 2018, in final form July 30, 2018; Published online August 12, 2018

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 ${\mathbb R}^d$ were introduced by Duchon and have become a widely used tool in myriad applications. The analogues for ${\mathbb S}^{d-1}$ are the thin plate splines for the sphere. The topic was first discussed by Wahba in the early 1980's, for the ${\mathbb S}^2$ 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.

Key words: positive definite functions; zonal functions; thinplate splines; ultraspherical expansions; Gegenbauer polynomials.

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