Constructive Approximation

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

From Schoenberg Coefficients to Schoenberg Functions

  • Christian Berg
  • 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.


  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)CrossRefMATHGoogle Scholar
  2. 2.
    Bachoc, C.: Semidefinite Programming, Harmonic Analysis and Coding Theory. arXiv:0909.4767v2 (2010)
  3. 3.
    Barbosa, V.S., Menegatto, V.A.: Differentiable positive definite functions on two-point homogeneous spaces. J. Math. Anal. Appl. 434, 698–712 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Berg, C.: Corps convexes et potentiels sphériques, Mat.-Fys. Medd. Danske Vid. Selsk. 37(6), 64 pp (1969)Google Scholar
  5. 5.
    Berg, C.: Stieltjes–Pick–Bernstein–Schoenberg and their connection to complete monotonicity. In: Mateu, J., Porcu, E. (eds.) Positive Definite Functions: From Schoenberg to Space-Time Challenges, pp. 15–45. Department of Mathematics, University Jaume I, Castellón de laPlana (2008)Google Scholar
  6. 6.
    Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton, New Jersey (2007)MATHGoogle Scholar
  7. 7.
    Bingham, N.H.: Positive definite functions on spheres. Math. Proc. Camb. Philos. Soc. 73, 145–156 (1973)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bochner, S.: Hilbert distances and positive definite functions. Ann. Math. 42(3), 647–656 (1941)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Daley, D.J., Porcu, E.: Dimension walks and Schoenberg spectral measures. Proc. Am. Math. Soc. 142(5), 1813–1824 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. Springer, New York (2013)CrossRefGoogle Scholar
  11. 11.
    Dixmier, J.: Les \(C^*\)-algèbres et leurs représentations. Gauthier-Villars, Paris (1964)MATHGoogle Scholar
  12. 12.
    Gneiting, T.: Strictly and non-strictly positive definite functions on spheres. Bernoulli 19(4), 1327–1349 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gneiting, T.: Online supplemental materials to “Strictly and non-strictly positive definite functions on spheres”. doi: 10.3150/12-BEJSP06.
  14. 14.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 6th edn. Academic Press, San Diego (2000)MATHGoogle Scholar
  15. 15.
    Guella, J.C., Menegatto, V.A., Peron, A.P.: An Extension of a Theorem of Schoenberg to Products of Spheres. arXiv:1503.08174
  16. 16.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Corrected Reprint of the 1985 Original. Cambridge University Press, Cambridge (1985)Google Scholar
  17. 17.
    Munkres, J.R.: Topology, 2nd edn. Prentice Hall, Inc., New Jersey (2000)MATHGoogle Scholar
  18. 18.
    Müller, C.: Spherical Harmonics. Lecture Notes in Mathematics. Springer. Berlin (1966)Google Scholar
  19. 19.
    Møller, J., Nielsen, M., Porcu, E., Rubak, E.: Determinantal point process models on the sphere (submitted)Google Scholar
  20. 20.
    Porcu, E., Bevilacqua, M., Genton, M.G.: Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. J. Am. Stat. Assoc. doi: 10.1080/01621459.2015.1072541
  21. 21.
    Rudin, W.: Fourier Analysis on Groups. Interscience Publishers, New York, London (1962)MATHGoogle Scholar
  22. 22.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill Book Company, Singapore (1986)MATHGoogle Scholar
  23. 23.
    Sasvári, Z.: Positive Definite and Definitizable Functions. Akademie Verlag, Berlin (1994)MATHGoogle Scholar
  24. 24.
    Schoenberg, I.J.: Positive definite functions on spheres. Duke Math. J. 9, 96–108 (1942)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Shapiro, V.L.: Fourier Series in Several Variables with Applications to Partial Differential Equations. Chapman & Hall/CRC. Applied Mathematics and Nonlinear Science Series. CRC Press, Boca Raton, FL (2011)Google Scholar
  26. 26.
    Ziegel, J.: Convolution roots and differentiability of isotropic positive definite functions on spheres. Proc. Am. Math. Soc. 142(6), 2063–2077 (2014)MathSciNetCrossRefMATHGoogle Scholar

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

Personalised recommendations