From Schoenberg Coefficients to Schoenberg Functions

Christian Berg, Emilio Porcu

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58 Scopus citations


In his seminal paper, Schoenberg (Duke Math J 9:96–108, 1942) characterized the class P(Sd) of continuous functions f: [- 1 , 1] → R such that f(cos θ(ξ, η)) is positive definite on the product space Sd× Sd, with Sd being the unit sphere of Rd + 1 and θ(ξ, η) being the great circle distance between ξ, η∈ Sd. In the present paper, we consider the product space Sd× G, for G a locally compact group, and define the class P(Sd, G) of continuous functions f: [- 1 , 1] × G→ C such that f(cos θ(ξ, η) , u- 1v) is positive definite on Sd× Sd× G× G. This offers a natural extension of Schoenberg’s theorem. Schoenberg’s second theorem corresponding to the Hilbert sphere S is also extended to this context. The case G= R is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of planet Earth.

Original languageBritish English
Pages (from-to)217-241
Number of pages25
JournalConstructive Approximation
Issue number2
StatePublished - 1 Apr 2017


  • Positive definite
  • Space-time covariances
  • Spherical harmonics


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