Skip to main content
SpringerLink
Log in

Characterization of Strict Positive Definiteness on products of complex spheres

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

In this paper we consider Positive Definite functions on products \(\Omega _{2q}\times \Omega _{2p}\) of complex spheres, and we obtain a condition, in terms of the coefficients in their disc polynomial expansions, which is necessary and sufficient for the function to be Strictly Positive Definite. The result includes also the more delicate cases in which p and/or q can be 1 or \(\infty \). The condition we obtain states that a suitable set in \({\mathbb {Z}}^2\), containing the indexes of the strictly positive coefficients in the expansion, must intersect every product of arithmetic progressions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barbosa, V.S., Menegatto, V.A.: Strictly positive definite kernels on compact two-point homogeneous spaces. Math. Inequal. Appl. 19(2), 743–756 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Barbosa, V.S., Menegatto, V.A.: Strict positive definiteness on products of compact two-point homogeneous spaces. Integral Transforms Spec. Funct. 28(1), 56–73 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions. Graduate Texts in Mathematics, vol. 100. Springer, New York (1984)

    MATH  Google Scholar 

  4. Berg, C., Peron, A.P., Porcu, E.: Orthogonal expansions related to compact Gelfand pairs. Expos. Math. 36, 259–277 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berg, C., Peron, A.P.: Porcu, E: Schoenberg’s theorem for real and complex Hilbert spheres revisited. J. Approx. Theory 228, 58–78 (2018). arXiv:1701.07214

    Article  MathSciNet  MATH  Google Scholar 

  6. Berg, C., Porcu, E.: From Schoenberg coefficients to Schoenberg functions. Constr. Approx. 45(2), 217–241 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonfim, R.N., Menegatto, V.A.: Strict positive definiteness of multivariate covariance functions on compact two-point homogeneous spaces. J. Multivar. Anal. 152, 237–248 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, D., Menegatto, V.A., Sun, X.: A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Am. Math. Soc. 131(9), 2733–2740 (2003). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheney, E.W.: Approximation using positive definite functions. In: Approximation theory VIII, Vol. 1 (College Station, TX, 1995), vol. 6 of Ser. Approx. Decompos., World Sci. Publ., River Edge, NJ, pp. 145–168 (1995)

  10. Cheney, W., Light, W.: A course in approximation theory. Graduate Studies in Mathematics, vol. 101. American Mathematical Society, Providence, RI, 2009, reprint of the 2000 original

  11. Christensen, J.P.R., Ressel, P.: Positive definite kernels on the complex Hilbert sphere. Math. Z. 180(2), 193–201 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gneiting, T.: Strictly and non-strictly positive definite functions on spheres. Bernoulli 19(4), 1327–1349 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Godement, R.: Introduction aux travaux de A. Selberg. In: Séminaire Bourbaki, vol. 4, Soc. Math. France, Paris, pp. Exp. No. 144, 95–110 (1995)

  14. Guella, J., Menegatto, V.A.: Strictly positive definite kernels on the torus. Constr. Approx. 46(2), 271–284 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guella, J.C., Menegatto, V.A.: Strictly positive definite kernels on a product of spheres. J. Math. Anal. Appl. 435(1), 286–301 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guella, J.C., Menegatto, V.A.: Schoenberg’s theorem for positive definite functions on products: a unifying framework. J. Fourier Anal. Appl. (2018). https://doi.org/10.1007/s00041-018-9631-5

  17. Guella, J.C., Menegatto, V.A.: Unitarily invariant strictly positive definite kernels on spheres. Positivity 22(1), 91–103 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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(4), 671–685 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guella, J.C., Menegatto, V.A., Peron, A.P.: Strictly positive definite kernels on a product of spheres II. SIGMA Symmetry Integrability Geom. Methods Appl. 12 Paper No. 103, 15 (2016)

  20. Guella, J.C., Menegatto, V.A., Peron, A.P.: Strictly positive definite kernels on a product of circles. Positivity 21(1), 329–342 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hannan, E.J.: Multiple Time Series. Wiley, New York (1970)

    Book  MATH  Google Scholar 

  22. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990). corrected reprint of the 1985 original

  23. Koornwinder, T.: Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula. J. Lond. Math. Soc. (2) 18(1), 101–114 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. Koornwinder, T.H.: The addition formula for Jacobi Polynomials II. The Laplace type integral representation and the product formula. Math. Centrum Amsterdam, report TW133 (1972)

  25. Laurent, M.: Équations exponentielles-polynômes et suites récurrentes linéaires. II. J. Number Theory 31(1), 24–53 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Light, W.A., Cheney, E.W.: Interpolation by periodic radial basis functions. J. Math. Anal. Appl. 168(1), 111–130 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Massa, E., Peron, A.P., Porcu, E.: Positive definite functions on complex spheres and their walks through dimensions. SIGMA Symm. Integrab. Geom. Methods Appl. 13, 088 (2017). arXiv:1704.01237

    MathSciNet  MATH  Google Scholar 

  28. Menegatto, V.A.: Strict positive definiteness on spheres. Analysis (Munich) 19(3), 217–233 (1999)

    MathSciNet  MATH  Google Scholar 

  29. Menegatto, V.A., Oliveira, C.P., Peron, A.P.: Strictly positive definite kernels on subsets of the complex plane. Comput. Math. Appl. 51(8), 1233–1250 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Menegatto, V.A., Peron, A.P.: A complex approach to strict positive definiteness on spheres. Integral Transforms Spec. Funct. 11(4), 377–396 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Menegatto, V.A., Peron, A.P.: Positive definite kernels on complex spheres. J. Math. Anal. Appl. 254(1), 219–232 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Musin, O.R.: Multivariate positive definite functions on spheres. In: Discrete Geometry and Algebraic Combinatorics, vol. 625. Contemp. Math., Amer. Math. Soc., Providence, RI, pp. 177–190 (2014)

  33. Pinkus, A.: Strictly Hermitian positive definite functions. J. Anal. Math. 94, 293–318 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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. 111(514), 888–898 (2016)

    Article  MathSciNet  Google Scholar 

  35. Ramos-López, D., Sánchez-Granero, M.A., Fernández-Martínez, M., Martínez-Finkelshtein, A.: Optimal sampling patterns for Zernike polynomials. Appl. Math. Comput. 274, 247–257 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Schoenberg, I.J.: Positive definite functions on spheres. Duke Math. J. 9, 96–108 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  37. Szegö, G.: Orthogonal polynomials. American Mathematical Society Colloquium Publications, vol. 23. Revised ed. American Mathematical Society, Providence, RI (1959)

  38. Torre, A.: Generalized Zernike or disc polynomials: an application in quantum optics. J. Comput. Appl. Math. 222(2), 622–644 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wünsche, A.: Generalized Zernike or disc polynomials. J. Comput. Appl. Math. 174(1), 135–163 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xu, Y., Cheney, E.W.: Strictly positive definite functions on spheres. Proc. Am. Math. Soc. 116(4), 977–981 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions. Basic Results. Springer Series in Statistics, vol. I. Springer, New York (1987)

    Book  Google Scholar 

Download references

Acknowledgements

Mario H. Castro was supported by: Grant \(\#\)APQ-00474-14, FAPEMIG and CNPq/Brazil. Eugenio Massa was supported by: Grant \(\#\)2014/25398-0, São Paulo Research Foundation (FAPESP) and Grant \(\#\)303447/2017-6, CNPq/Brazil. Ana P. Peron was supported by: Grants \(\#\)2016/03015-7, \(\#\)2016/09906-0 and \(\#\)2014/25796-5, São Paulo Research Foundation (FAPESP).

Author information

Authors and Affiliations

  1. Departamento de Matemática, UFU, Uberlândia, MG, Brazil

    Mario H. Castro

  2. Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, São Carlos, SP, 13560-970, Brazil

    Eugenio Massa & Ana Paula Peron

Authors
  1. Mario H. Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eugenio Massa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ana Paula Peron
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ana Paula Peron.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Castro, M.H., Massa, E. & Peron, A.P. Characterization of Strict Positive Definiteness on products of complex spheres. Positivity 23, 853–874 (2019). https://doi.org/10.1007/s11117-018-00641-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-018-00641-5

Keywords

Mathematics Subject Classification

Search

Navigation