Conditionally positive definite dot product kernels

This paper deals with conditionally positive definite kernels on Euclidean spaces. The focus here is on dot product kernels, that is, those depending on the inner product of the variables. Among the results, we include some properties relating conditional positive definiteness and standard convolution in the line and also results related to the characterization of the conditionally positive definite dot product kernels with respect to finite-dimensional polynomial spaces. We also introduce and characterize two large classes of strictly conditionally positive definite dot product kernels.     

On conditionally positive definite dot product kernels

Let m and n be fixed, positive integers and P a space composed of real polynomials in m variables. The authors study functions f ： R →R which map Gram matrices, based upon n points of R^m, into matrices, which are nonnegative definite with respect to P Among other things, the authors discuss continuity, differentiability, convexity, and convexity in the sense of Jensen, of such functions     

Conditionally positive definite kernels on Euclidean domains

This paper characterizes several classes of conditionally positive definite kernels on a domain Ω of either or . Among the classes is that composed of strictly conditionally positive definite kernels. These kernels are known to be useful in the solution of variational interpolation problems on Ω. Our study covers the case in which Ω is the sphere S^l−1 of or a similar manifold. Among other things, our results imply that the characterization of (strict) conditional positive definiteness on Ω can be obtained from a characterization of (strict) positive definiteness on Ω. The bi-zonal strictly conditionally positive definite kernels on S^l−1, l?3, are described.     

On the Lévy-Khintchine decomposition of conditionally positive definite functions

Summary The main aim of the paper is to prove still another version of the Lévy--Khintchine decomposition of conditionally positive definite functions on a general locally compact Abelian group. The exposition is based on the two-cones theorem proved by N. Drumm in 1976. Application of the main result to the Euclidean group shows the novelty of the approach.     

Approximation with Constraints by Conditionally Positive Definite Functions

This paper deals with interpolation and approximation satisfying constraints. We consider approximation by conditionally positive definite functions in norms which are associated with the conditionally positive definite functions. The theory of reproducing kernels is used to transform the approximation problems to quadratic optimization problems. Then we can give the existence, characterization and uniqueness results for the solutions. The methods of optimization theory can be used in order to determine solutions.     

Laplace-Beltrami differentiability of positive definite kernels on the sphere

This contribution gives results on the action of the Laplace-Beltrami derivative on sufficiently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions.     

Positive definite matrices and differentiable reproducing kernel inequalities

Let I⊆R be a interval and be a reproducing kernel on I. By the Moore-Aronszajn theorem, every finite matrix k(xi,xj) is positive semidefinite. We show that, as a direct algebraic consequence, if k(x,y) is appropriately differentiable it satisfies a 2-parameter family of differential inequalities of which the classical diagonal dominance is the order 0 case. An application of these inequalities to kernels of positive integral operators yields optimal Sobolev norm bounds.     

Coxeter图的正定性

用线性代数的方法证明了李代数中正定的Coxeter图仅是Aι，Bι，Dι，Eι，F4和Gz六种类型。