Pólya-type criteria for conditional strict positive definiteness of functions on spheres. (English) Zbl 1446.41001
The authors provide new sufficient and also some necessary conditions for functions to be conditionally strictly positive definite on spheres \(\mathbb{S}^{d-1}\), \(2 < d\in \mathbb{N}\). This is done by using only monotonicity properties. In the case of strictly positive definite and conditionally negative definite functions on spheres a characterization is given solely in terms of monotonicity properties. Moreover, it is proven that multiply monotone functions are positive definite on spheres up to a certain dimension.
Reviewer: Peter Massopust (München)
MSC:
| 41A05 | Interpolation in approximation theory |
| 42A82 | Positive definite functions in one variable harmonic analysis |
Keywords:
spherical basis functions; isotropic kernels; dot product kernels; completely monotone functions; interpolation of scattered data; conditionally positive definite kernelsSoftware:
DLMFReferences:
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