Skip to main content
SpringerLink
Log in

Unitarily invariant strictly positive definite kernels on spheres

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We present a Fourier characterization for the continuous and unitarily invariant strictly positive definite kernels on the unit sphere in \({\mathbb {C}}^{q}\), thus adding to a celebrated work of I. J. Schoenberg on positive definite functions on real spheres.

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. Aharmim, B., Amal, E.H., Fouzia, E.W., Ghanmi, A.: Generalized Zernike polynomials: operational formulae and generating functions. Integral Transforms Spec. Funct. 26(6), 395–410 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

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

  4. Boyd, J.N., Raychowdhury, P.N.: Zonal harmonic functions from two dimensional analogs of Jacobi polynomials. Appl. Anal. 16, 243–259 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Dreseler, B., Hrach, R.: Summability of Fourier expansions in terms of disc polynomials. In :“Functions, Series, Operators”, Vol. I,II (Budapest, 1980), pp. 375–384. North Holland, Amsterdam, New York (1983)

  7. Everest, G., van der Poorten, A., Shparlinski, I., Ward, T.: Recurrence sequences. In: Mathematical Surveys and Monographs, 104. American Mathematical Society, Providence, RI (2003)

  8. 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 

  9. 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 

  10. 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 

  11. Guella, J.C., Menegatto, V.A., Peron, A.P.: Strictly positive definite kernels on a product of spheres II. SIGMA Symmetry Integr. Geom. Methods Appl. 12, 103 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Haagerup, U., Schlichtkrull, H.: Inequalities for Jacobi polynomials. Ramanujan J. 33(2), 227–246 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Janssen, A.J.E.M.: New analytic results for the Zernike circle polynomials from a basic result in the Nijboer-Zernike diffraction theory. J. Eur. Opt. Soc. Rapid Publ. 6, 11028 (2011). doi:10.2971/jeos.2011.11028

  14. Koornwinder, T.H.: The addition formula for Jacobi polynomials II. In: The Laplace Type Integral Representation and the Product Formula. Mathematisch Centrum Amsterdam, Report TW133 (1972)

  15. Koornwinder, T.H.: The addition formula for Jacobi polynomials III. In: Completion of the Proof, Mathematisch Centrum Amsterdam, Report TW135 (1972)

  16. Lakshminarayana, T.H., Fleck, A.: Zernike polynomials: a guide. J. Mod. Opt. 58(7), 545–561 (2001)

    Article  Google Scholar 

  17. 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 

  18. Menegatto, V.A., Peron, A.P.: Strict positive definiteness on spheres via disk polynomials. Int. J. Math. Math. Sci. 31(12), 715–724 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Rudin, W.: Function theory in the unit ball of \({\mathbb{C}}^n\). In: Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Science), vol. 241. Springer, New York, Berlin (1980)

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Szegö, G.: Orthogonal polynomials, 4th edn. In: Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, RI (1975)

  23. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ICMC-USP - São Carlos, Caixa Postal 668, São Carlos, SP, 13560-970, Brazil

    J. C. Guella & V. A. Menegatto

Authors
  1. J. C. Guella
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. A. Menegatto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Menegatto.

Additional information

The second author recognizes a partial support from FAPESP, Grant 2016/09906-0.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guella, J.C., Menegatto, V.A. Unitarily invariant strictly positive definite kernels on spheres. Positivity 22, 91–103 (2018). https://doi.org/10.1007/s11117-017-0502-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-017-0502-0

Keywords

Mathematics Subject Classification

Search

Navigation