Stability results for scattered-data interpolation on Euclidean spheres

Let S-m denote the unit sphere in Rm+1 and d(m) the geodesic distance in S-m. A spherical-basis function approximant is a function of the form s(x) = Sigma(j=1)(M) a(j)phi(d(m)(x, x(j))), x is an element of S-m, where (a(j))(1)(M) are real constants, phi : [0, pi] --> R is a fixed function, and (x(j))(1)(M) is a set of distinct points in S-m. It is known that if phi is a strictly positive definite function in S-m, then the interpolation matrix (phi(d(m)(x(j), x(k))))(j,k=1)(M) is positive definite, hence invertible, for every choice of distinct points (x(j))(1)(M) and every positive integer M. The paper studies a salient subclass of such functions phi, and provides stability estimates for the associated interpolation matrices.