Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 103, 15 pages      arXiv:1605.09775      https://doi.org/10.3842/SIGMA.2016.103

Strictly Positive Definite Kernels on a Product of Spheres II

Jean C. Guella, Valdir A. Menegatto and Ana P. Peron
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos - SP, Brazil

Received June 01, 2016, in final form October 24, 2016; Published online October 28, 2016

Abstract
We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result complements similar results previously obtained for strict positive definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169] and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016), 286-301, arXiv:1505.03695].

Key words: positive definite kernels; strictly positive definiteness; isotropy; covariance functions; sphere; circle.

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References

  1. Barbosa V.S., Menegatto V.A., Strictly positive definite kernels on compact two-point homogeneous spaces, Math. Inequal. Appl. 19 (2016), 743-756, arXiv:1505.00591.
  2. Barbosa V.S., Menegatto V.A., Strict positive definiteness on a product of compact two-point homogeneous spaces, Integral Transform. Spec. Funct., to appear, arXiv:1605.07071.
  3. Beatson R.K., zu Castell W., Dimension hopping and families of strictly positive definite zonal basis functions on spheres, arXiv:1510.08658.
  4. Beatson R.K., zu Castell W., One-step recurrences for stationary random fields on the sphere, SIGMA 12 (2016), 043, 19 pages, arXiv:1601.07743.
  5. Berg C., Porcu E., From Schoenberg coefficients to Schoenberg functions, Constr. Approx., to appear, arXiv:1505.05682.
  6. Chen D., Menegatto V.A., Sun X., A necessary and sufficient condition for strictly positive definite functions on spheres, Proc. Amer. Math. Soc. 131 (2003), 2733-2740.
  7. Emonds J., Führ H., Strictly positive definite functions on compact abelian groups, Proc. Amer. Math. Soc. 139 (2011), 1105-1113, arXiv:1002.3017.
  8. Gangolli R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy's Brownian motion of several parameters, Ann. Inst. H. Poincaré Sect. B 3 (1967), 121-226.
  9. Gneiting T., Nonseparable, stationary covariance functions for space-time data, J. Amer. Statist. Assoc. 97 (2002), 590-600.
  10. Gneiting T., Strictly and non-strictly positive definite functions on spheres, Bernoulli 19 (2013), 1327-1349, arXiv:1111.7077.
  11. Guella J.C., Menegatto V.A., Strictly positive definite kernels on a product of spheres, J. Math. Anal. Appl. 435 (2016), 286-301, arXiv:1505.03695.
  12. Guella J.C., Menegatto V.A., Peron A.P., An extension of a theorem of Schoenberg to products of spheres, Banach J. Math. Anal. 10 (2016), 671-685, arXiv:1503.08174.
  13. Guella J.C., Menegatto V.A., Peron A.P., Strictly positive definite kernels on a product of circles, Positivity, to appear, arXiv:1505.01169.
  14. Menegatto V.A., Oliveira C.P., Peron A.P., Strictly positive definite kernels on subsets of the complex plane, Comput. Math. Appl. 51 (2006), 1233-1250.
  15. Müller C., Analysis of spherical symmetries in Euclidean spaces, Applied Mathematical Sciences, Vol. 129, Springer-Verlag, New York, 1998.
  16. Ron A., Sun X., Strictly positive definite functions on spheres in Euclidean spaces, Math. Comp. 65 (1996), 1513-1530.
  17. Schoenberg I.J., Positive definite functions on spheres, Duke Math. J. 9 (1942), 96-108.
  18. Shapiro V.L., Fourier series in several variables with applications to partial differential equations, Chapman \rm & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011.
  19. Sun X., Strictly positive definite functions on the unit circle, Math. Comp. 74 (2005), 709-721.
  20. Szegő G., Orthogonal polynomials, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, RI, 1975.

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