SIGMA 11 (2015), 014, 13 pages      arXiv:1410.4125      https://doi.org/10.3842/SIGMA.2015.014

Generalized Convolution Roots of Positive Definite Kernels on Complex Spheres

Victor S. Barbosa and Valdir A. Menegatto
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos - SP, Brasil

Received October 16, 2014, in final form February 10, 2015; Published online February 13, 2015

Abstract
Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the kernel can be recovered as a generalized convolution root of an equally positive definite and zonal kernel.

Key words: positive definiteness; zonal kernels; recovery formula; convolution roots; Zernike or disc polynomials.

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References

1. Bezubik A., Strasburger A., On spherical expansions of smooth ${\bf SU}(n)$-zonal functions on the unit sphere in ${\mathbb C}^n$, J. Math. Anal. Appl. 404 (2013), 570-578, arXiv:1112.0648.
2. Estrade A., Istas J., Ball throwing on spheres, Bernoulli 16 (2010), 953-970, arXiv:0810.4004.
3. Ferreira J.C., Menegatto V.A., Eigenvalues of integral operators defined by smooth positive definite kernels, Integral Equations Operator Theory 64 (2009), 61-81.
4. Gneiting T., Strictly and non-strictly positive definite functions on spheres, Bernoulli 19 (2013), 1327-1349, arXiv:1111.7077.
5. Koornwinder T.H., The addition formula for Jacobi polynomials. II. The Laplace type integral representation and the product formula, Math. Centrum Amsterdam, Report TW133, 1976.
6. Levesley J., Kushpel A., A multiplier version of the Bernstein inequality on the complex sphere, J. Comput. Appl. Math. 240 (2013), 184-191, arXiv:1204.6147.
7. Menegatto V.A., Oliveira C.P., Annihilating properties of convolution operators on complex spheres, Anal. Math. 31 (2005), 13-30.
8. Menegatto V.A., Peron A.P., Positive definite kernels on complex spheres, J. Math. Anal. Appl. 254 (2001), 219-232.
9. Platonov S.S., Approximations on compact symmetric spaces of rank $1$, Sb. Math. 188 (1997), 753-769.
10. Platonov S.S., On some problems in the theory of the approximation of functions on compact homogeneous manifolds, Sb. Math. 200 (2009), 845-885.
11. Quinto E.T., Injectivity of rotation invariant Radon transforms on complex hyperplanes in ${\bf C}^n$, in Integral Geometry (Brunswick, Maine, 1984), Contemp. Math., Vol. 63, Amer. Math. Soc., Providence, RI, 1987, 245-260.
12. Rudin W., Function theory in the unit ball of ${\mathbb C}^n$, Classics in Mathematics, Springer-Verlag, Berlin, 2008.
13. Schaback R., A unified theory of radial basis functions: Native Hilbert spaces for radial basis functions II, J. Comput. Appl. Math. 121 (2000), 165-177.
14. Schreiner M., Locally supported kernels for spherical spline interpolation, J. Approx. Theory 89 (1997), 172-194.
15. Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
16. Thirulogasanthar K., Saad N., Honnouvo G., $2D$-Zernike polynomials and coherent state quantization of the unit disc, arXiv:1303.5483.
17. Torre A., Generalized Zernike or disc polynomials: an application in quantum optics, J. Comput. Appl. Math. 222 (2008), 622-644.
18. Wünsche A., Generalized Zernike or disc polynomials, J. Comput. Appl. Math. 174 (2005), 135-163.
19. Ziegel J., Convolution roots and differentiability of isotropic positive definite functions on spheres, Proc. Amer. Math. Soc. 142 (2014), 2063-2077, arXiv:1201.5833.