منابع مشابه
Strictly Hermitian Positive Definite Functions
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...
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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 reflection invariant functions
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...
متن کاملStrictly Positive Definite Functions on a Real Inner Product Space
Abstract. If f(t) = ∑∞ k=0 akt k converges for all t ∈ IR with all coefficients ak ≥ 0, then the function f(< x,y >) is positive definite on H ×H for any inner product space H. Set K = {k : ak > 0}. We show that f(< x,y >) is strictly positive definite if and only if K contains the index 0 plus an infinite number of even integers and an infinite number of odd integers.
متن کاملStrictly positive definite functions on the unit circle
We study strictly positive definite functions on the unit circle in the Euclidean space of dimension two. We develop several conditions pertaining to the determination of such functions. The major result is obtained by considering the set of real numbers as a vector space over the field of rational numbers and then applying the Kronecker approximation theorem and Weyl’s criterion on equidistrib...
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ژورنال
عنوان ژورنال: Journal of Approximation Theory
سال: 1996
ISSN: 0021-9045
DOI: 10.1006/jath.1996.0097