Decomposition of conditionally positive definite functions on commutative hypergroups

In this paper we consider bounded, conditionally positive definite functions on commutative hypergroups. We show an integral representation that allows to decompose conditionally positive definite functions as differences of positive definite functions.     

Sine functions on compact commutative hypergroups

Recently, Fechner and Székelyhidi introduced sine functions on hypergroups. They conjectured that on a compact hypergroup, all sine functions are constant zero. We prove this conjecture for compact commutative hypergroups by Fourier analysis.     

On the Fourier transformation of positive,positive definite measures on commutative hypergroups,and dual convolution structures

We show that the support of the Fourier transform of a positive, positive definite measure on a commutative hypergroupK contains a positive character. This generalizes the known fact that the support of the Plancherel measure π contains a positive character (which in general is not the identity character1). It follows that contains a positive character for whenever a dual convolution exists. In particular, if1∈supp π, then1 is this character. We also give some further general results about the support of dual convolution products in terms ofsupp π. Some examples associated with Gelfand pairs and, in particular, non-compact Riemannian symmetric spaces of rank 1 are discussed.     

Moment functions and exponential monomials on commutative hypergroups

Aequationes mathematicae - The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its...     

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.     

Interpolation of scattered data: Distance matrices and conditionally positive definite functions

Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.     

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     

Decomposition of positive definite functions defined on a neighbourhood of the identity

LetV be a symmetric open neighbourhood of the identity of a topological groupG. We show that every positive definite functionf onV can be written asf=f _c +f _s wheref _c andf _s are positive definite functions onV, f _c is continuous andf _s averages to zero. IfG is locally compact with Haar measurem _G andf ism _G -measurable thenf _s =0m _G -almost everywhere.     

Exponential polynomials on commutative hypergroups

Polynomials and exponential polynomials play a fundamental role in the theory of spectral analysis and spectral synthesis on commutative groups. Recently several new results have been published in this field [2–4,6]. Spectral analysis and spectral synthesis has been studied on some types of commutative hypergroups, as well. However, a satisfactory definition of exponential monomials on general commutative hypergroups has not been available so far. In [5,7,8] and [9], the authors use a special concept on polynomial and Sturm–Liouville-hypergroups. Here we give a general definition which covers the known special cases.     

Spectral analysis on commutative hypergroups

The problem of spectral analysis is formulated on commutative hypergroups and is solved for finite dimensional varieties.     

Weinstein positive definite functions

Positivity - In this paper, we introduce the notion of the Weinstein positive definite functions and we state a version of Bochner’s theorem. Furthermore, we study the strictly Weinstein...     

Extension of positive definite functions

Let Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ be an open, connected subset of Rⁿ${\mathbb{R}}^n$, and let F:Ω−Ω→C$F:\Omega -\Omega \to \mathbb{C}$, where Ω−Ω={x−y:x,y∈Ω}$\Omega -\Omega =\{x-y:x,y\in \Omega \}$, be a continuous positive definite function. We give necessary and sufficient conditions for F   to have an extension to a continuous positive definite function defined on the entire Euclidean space Rⁿ${\mathbb{R}}^n$. The conditions are formulated in terms of existence of a unitary representations of Rⁿ${\mathbb{R}}^n$ whose generators extend a certain system of unbounded Hermitian operators defined on a Hilbert space associated to F. Different positive definite extensions correspond to different unitary representations.     

Factoring of positive definite functions on semigroups

No abstract. June 15,1999     

Strictly Hermitian positive definite functions

LetH be any complex inner product space with inner product <·,·>. We say thatf: ℂ→ℂ is Hermitian positive definite onH if the matrix

Extensions of positive definite functions on free groups

An analogue of Krein's extension theorem is proved for operator-valued positive definite functions on free groups. The proof gives also the parametrization of all extensions by means of a generalized type of Szegö parameters. One singles out a distinguished completion, called central, which is related to quasi-multiplicative positive definite functions. An application is given to factorization of noncommutative polynomials.