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

SIGMA 14 (2018), 112, 14 pages      arXiv:1803.03105      https://doi.org/10.3842/SIGMA.2018.112

Strictly Positive Definite Functions on Compact Two-Point Homogeneous Spaces: the Product Alternative

Rafaela N. Bonfim a, Jean C. Guella b and Valdir A. Menegatto b
a) DEMAT-Universidade Federal de São João Del Rei, Praça Frei Orlando, 170, Centro, 36307-352 São João del Rei - MG, Brazil
b) Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos - SP, Brazil

Received March 08, 2018, in final form October 10, 2018; Published online October 16, 2018

Abstract
For two continuous and isotropic positive definite kernels on the same compact two-point homogeneous space, we determine necessary and sufficient conditions in order that their product be strictly positive definite. We also provide a similar characterization for kernels on the space-time setting $G \times S^d$, where $G$ is a locally compact group and $S^d$ is the unit sphere in $\mathbb{R}^{d+1}$, keeping isotropy of the kernels with respect to the $S^d$ component. Among other things, these results provide new procedures for the construction of valid models for interpolation and approximation on compact two-point homogeneous spaces.

Key words: strict positive definiteness; spheres; product kernels; linearization formulas; isotropy.

pdf (350 kb)   tex (19 kb)

References

  1. Askey R., Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975.
  2. 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.
  3. Barbosa V.S., Menegatto V.A., Strict positive definiteness on products of compact two-point homogeneous spaces, Integral Transforms Spec. Funct. 28 (2017), 56-73, arXiv:1605.07071.
  4. Berg C., Porcu E., From Schoenberg coefficients to Schoenberg functions, Constr. Approx. 45 (2017), 217-241, arXiv:1505.05682.
  5. Bingham N.H., Positive definite functions on spheres, Proc. Cambridge Philos. Soc. 73 (1973), 145-156.
  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. De Iaco S., Myers D.E., Posa D., On strict positive definiteness of product and product-sum covariance models, J. Statist. Plann. Inference 141 (2011), 1132-1140.
  8. De Iaco S., Myers D.E., Posa D., Strict positive definiteness of a product of covariance functions, Comm. Statist. Theory Methods 40 (2011), 4400-4408.
  9. De Iaco S., Posa D., Strict positive definiteness in geostatistics, Stoch. Environ. Res. Risk Assess. 32 (2018), 577-590.
  10. Faraut J., Fonction brownienne sur une variété riemannienne, in Séminaire de Probabilités, VII (Univ. Strasbourg, année universitaire 1971-1972), Lecture Notes in Math., Vol. 321, Springer, Berlin, 1973, 61-76.
  11. 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.
  12. Gasper G., Linearization of the product of Jacobi polynomials. I, Canad. J. Math. 22 (1970), 171-175.
  13. Gasper G., Linearization of the product of Jacobi polynomials. II, Canad. J. Math. 22 (1970), 582-593.
  14. 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.
  15. Guella J.C., Menegatto V.A., A limit formula for semigroups defined by Fourier-Jacobi series, Proc. Amer. Math. Soc. 146 (2018), 2027-2038.
  16. Guella J.C., Menegatto V.A., Unitarily invariant strictly positive definite kernels on spheres, Positivity 22 (2018), 91-103.
  17. Guella J.C., Menegatto V.A., Schoenberg's theorem for positive definite functions on products: a unifying framework, J. Fourier Anal. Appl., to appear.
  18. Helgason S., Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs, Vol. 83, Amer. Math. Soc., Providence, RI, 2000.
  19. Horn R.A., Johnson C.R., Matrix analysis, Cambridge University Press, Cambridge, 1990.
  20. Hylleraas E.A., Linearization of products of Jacobi polynomials, Math. Scand. 10 (1962), 189-200.
  21. Koornwinder T., Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula, J. London Math. Soc. 18 (1978), 101-114.
  22. Menegatto V.A., Strictly positive definite kernels on the Hilbert sphere, Appl. Anal. 55 (1994), 91-101.
  23. 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.
  24. Menegatto V.A., Peron A.P., Positive definite kernels on complex spheres, J. Math. Anal. Appl. 254 (2001), 219-232.
  25. Schoenberg I.J., Positive definite functions on spheres, Duke Math. J. 9 (1942), 96-108.
  26. Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
  27. Wang H.-C., Two-point homogeneous spaces, Ann. of Math. 55 (1952), 177-191.
  28. Wolf J.A., Spaces of constant curvature, 6th ed., AMS Chelsea Publishing, Providence, RI, 2011.
  29. Wünsche A., Generalized Zernike or disc polynomials, J. Comput. Appl. Math. 174 (2005), 135-163.

Previous article  Next article   Contents of Volume 14 (2018)