Strictly positive definite functions on compact two-point homogeneous spaces: the product alternative. (English) Zbl 1403.43004
Summary: 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.
MSC:
| 43A35 | Positive definite functions on groups, semigroups, etc. |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42A82 | Positive definite functions in one variable harmonic analysis |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
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