A kernel is a function where is the unit sphere in . If the kernel is rotational invariant on then there is a function such that where denotes the usual inner product in The function is called a zonal function and a zonal kernel. It is positively definite if and only if is positively definite. In the following is supposed strictly positively definite. Denote by the linear space of all homogeneous harmonic polynomials of degree and let If is an orthonormal basis of then a zonal function admits the expansion
Denote by the linear space of all finite linear combinations of zonal shifts of equipped with the inner product for and where are finite subsets of The completion of is called the native space and it is a reproducing kernel Hilbert space with kernel Considering two zonal functions, as above, and with the expansion coefficients then provided but this embedding is not isometric in general. The authors define a multiplier operator and prove that, for a finite subset of every has a unique best approximation element in and that the coefficients are determined by the interpolation condition They give also an estimate of the error, namely
where satisfies being the mesh of the finite set and the constant is independent of .