So far w have been concerned with interpolation on an arbitrary domain Ω ⊂ ℝd. However, we have not used any topological information about Ω. Instead, we have employed only the fact that it is a subset of ℝd. As a matter of fact, without having more information on Ω this is the only way. But many applications provide us with additional information on the underlying domain. For example, problems coming from geology often relate to the entire earth, so that the unit sphere would be an appropriate model and the additional information should lead to a better approximant.

Hence, in this chapter, we want to give an introduction to the theory of scattered data interpolation on spheres and other compact manifolds by radial or zonal functions.

Spherical harmonics

Generally, functions on the sphere are expressed as Fourier series with respect to an orthonormal family called spherical harmonics. In this section we will review the results on these functions. Since this material is only necessary for the present chapter we did not incorporate it into Chapter 5. Moreover, we have to skip the proofs once again. The interested reader is referred to Müller's book [140].

The domain of interest is the d-variate unit sphere Sd−1 := {x ∈ ℝd : ∥x∥2 = 1} ⊆ ℝd. It has surface area

On Sd−1, we will employ the usual inner product

where dS(x) is given by the standard measure on the sphere.

The distance between two points is the geodesic distance, which is the length of the shorter part of the great circle joining x and y or, in other words, dist(x, y) = arccos(xT y).