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Positive definite functions on spheres

Published online by Cambridge University Press:  24 October 2008

N. H. Bingham
N. H. Bingham
Affiliation:
Westfield College, London
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 73 , Issue 1 , January 1973 , pp. 145 - 156
Copyright
Copyright © Cambridge Philosophical Society 1973

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