Banach Journal of Mathematical Analysis

An extension of a theorem of Schoenberg to products of spheres

J. C. Guella, V. A. Menegatto, and A. P. Peron

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We present a characterization for the continuous, isotropic, and positive definite kernels on a product of spheres along the lines of a classical result of Schoenberg on positive definiteness on a single sphere. We also discuss a few issues regarding the characterization, including topics for future investigation.

Article information

Banach J. Math. Anal., Volume 10, Number 4 (2016), 671-685.

Received: 1 July 2015
Accepted: 19 December 2015
First available in Project Euclid: 31 August 2016

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Zentralblatt MATH identifier

Primary: 43A35: Positive definite functions on groups, semigroups, etc.
Secondary: 33C50: Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable 33C55: Spherical harmonics 42A10: Trigonometric approximation 42A82: Positive definite functions

positive definiteness spherical harmonics isotropy Gegenbauer polynomials addition formula


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), no. 4, 671--685. doi:10.1215/17358787-3649260.

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