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

SIGMA 12 (2016), 103, 15 pages      arXiv:1605.09775      https://doi.org/10.3842/SIGMA.2016.103

Strictly Positive Definite Kernels on a Product of Spheres II

Jean C. Guella, Valdir A. Menegatto and Ana P. Peron
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos - SP, Brazil

Received June 01, 2016, in final form October 24, 2016; Published online October 28, 2016

Abstract
We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result complements similar results previously obtained for strict positive definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169] and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016), 286-301, arXiv:1505.03695].

Key words: positive definite kernels; strictly positive definiteness; isotropy; covariance functions; sphere; circle.

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