Positive definite functions on spheres
- 1 January 1973
- journal article
- research article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 73 (1) , 145-156
- https://doi.org/10.1017/s0305004100047551
Abstract
Positive definite functions on metric spaces were considered by Schoenberg (26). We write σk for the unit hypersphere in (k + 1)-space; then σk is a metric space under geodesic distance. The functions which are positive definite (p.d.) on σk were 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.Keywords
This publication has 17 references indexed in Scilit:
- Random walk on spheresProbability Theory and Related Fields, 1972
- Integral representations for Jacobi polynomials and some applicationsJournal of Mathematical Analysis and Applications, 1969
- The arithmetic of certain semigroups of positive operatorsMathematical Proceedings of the Cambridge Philosophical Society, 1968
- Le mouvement Brownien fonction d'un point de la sphère de RiemannRendiconti del Circolo Matematico di Palermo Series 2, 1959
- Positive definite functions on spheresDuke Mathematical Journal, 1942
- Metric Spaces and Positive Definite FunctionsTransactions of the American Mathematical Society, 1938
- Certain integrals and infinite series involving ultra-spherical polynomials and Bessel functionsDuke Mathematical Journal, 1938
- Metric spaces and positive definite functionsTransactions of the American Mathematical Society, 1938
- Sur la détermination d’un système orthogonal complet dans un espace de riemann symétrique closRendiconti del Circolo Matematico di Palermo Series 2, 1929
- A Theorem of Sonine in Bessel Functions, with two Extensions to Spherical HarmonicsProceedings of the Edinburgh Mathematical Society, 1918