Positive definite functions on spheres

Abstract

Positive definite functions on metric spaces were considered by Schoenberg (26). We write σkfor the unit hypersphere in (k+ 1)-space; then σkis a metric space under geodesic distance. The functions which are positive definite (p.d.) on σkwere characterized by Schoenberg (27), who also obtained a necessary condition for a function to be p.d. on the it sphere σ∞ in Hilbert space. We extend this result by showing that Schoenberg's necessary condition for a function to be p.d. on σ∞ is also sufficient.