Strictly positive definite functions on spheres in Euclidean spaces
نویسندگان
چکیده
منابع مشابه
Strictly positive definite functions on spheres in Euclidean spaces
In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are act...
متن کاملStrictly Positive Definite Functions on Spheres
In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are act...
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Let H be any complex inner product space with inner product < ·, · >. We say that f : | C → | C is Hermitian positive definite on H if the matrix ( f(< z,z >) )n r,s=1 (∗) is Hermitian positive definite for all choice of z, . . . ,z in H, all n. It is strictly Hermitian positive definite if the matrix (∗) is also non-singular for any choice of distinct z, . . . ,z in H. In this article we prove...
متن کاملMultivariate positive definite functions on spheres
In 1942 I.J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a positive linear combination of the Gegenbauer polynomials. In this paper we extend Schoenberg’s theorem for multivariate Gegenbauer polynomials. This extension derives new positive semidefinite constraints for the distance distribution which can be applied for spherical codes.
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Strictly positive definite functions are used as basis functions for approximation methods in various contexts. Using a group theoretic interpretation of Bochner’s Theorem we give a sufficient condition for strictly positive definite functions on a semi-direct product which are invariant under the natural action of a given subgroup. As an application strictly positive definite, reflection invar...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1996
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-96-00780-6