Constructive Approximation

, Volume 45, Issue 2, pp 217–241 | Cite as

From Schoenberg Coefficients to Schoenberg Functions

  • Christian BergEmail author
  • Emilio Porcu


In his seminal paper, Schoenberg (Duke Math J 9:96–108, 1942) characterized the class \(\mathcal P(\mathbb {S}^d)\) of continuous functions \(f:[-1,1] \rightarrow \mathbb {R}\) such that \(f(\cos \theta (\xi ,\eta ))\) is positive definite on the product space \(\mathbb {S}^d \times \mathbb {S}^d\), with \(\mathbb {S}^d\) being the unit sphere of \(\mathbb {R}^{d+1}\) and \(\theta (\xi ,\eta )\) being the great circle distance between \(\xi ,\eta \in \mathbb {S}^d\). In the present paper, we consider the product space \(\mathbb {S}^d \times G\), for G a locally compact group, and define the class \(\mathcal P(\mathbb {S}^d, G)\) of continuous functions \(f:[-1,1]\times G \rightarrow \mathbb {C}\) such that \(f(\cos \theta (\xi ,\eta ), u^{-1}v)\) is positive definite on \(\mathbb {S}^d \times \mathbb {S}^d \times G \times G\). This offers a natural extension of Schoenberg’s theorem. Schoenberg’s second theorem corresponding to the Hilbert sphere \(\mathbb {S}^\infty \) is also extended to this context. The case \(G=\mathbb {R}\) 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.


Positive definite Space-time covariances Spherical harmonics 

Mathematics Subject Classification

Primary 43A35 Secondary 33C55 



This work was initiated during the visit of the first author to Universidad Técnica Federico Santa Maria, Chile. The visit and E.P. have been supported by Proyecto Fondecyt Regular. The authors want to thank two independent referees for useful suggestions and references.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  2. 2.Department of MathematicsUniversidad Técnica Federico Santa MariaValparaisoChile

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