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

SIGMA 14 (2018), 112, 14 pages      arXiv:1803.03105      https://doi.org/10.3842/SIGMA.2018.112

### Strictly Positive Definite Functions on Compact Two-Point Homogeneous Spaces: the Product Alternative

Rafaela N. Bonfim a, Jean C. Guella b and Valdir A. Menegatto b
a) DEMAT-Universidade Federal de São João Del Rei, Praça Frei Orlando, 170, Centro, 36307-352 São João del Rei - MG, Brazil
b) Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos - SP, Brazil

Received March 08, 2018, in final form October 10, 2018; Published online October 16, 2018

Abstract
For two continuous and isotropic positive definite kernels on the same compact two-point homogeneous space, we determine necessary and sufficient conditions in order that their product be strictly positive definite. We also provide a similar characterization for kernels on the space-time setting $G \times S^d$, where $G$ is a locally compact group and $S^d$ is the unit sphere in $\mathbb{R}^{d+1}$, keeping isotropy of the kernels with respect to the $S^d$ component. Among other things, these results provide new procedures for the construction of valid models for interpolation and approximation on compact two-point homogeneous spaces.

Key words: strict positive definiteness; spheres; product kernels; linearization formulas; isotropy.

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