ON RUBIN’S HARMONIC ANALYSIS AND ITS RELATED POSITIVE DEFINITE FUNCTIONS

In this paper,a new formulation of the Rubin’s q-translation is given,which leads to a reliable q-harmonic analysis.Next,related q-positive definite functions are introduced and studied,and a Bochner’s theorem is proved.     

Strictly positive definite functions on a compact group

We recognize a result of Schreiner, concerning strictly positive definite functions on a sphere in an Euclidean space, as a generalization of Bochner's theorem for compact groups.

     

Strictly positive definite functions by integral transforms

We study the continuity and strict positive definiteness of positive definite functions on quasi-metric spaces given by integral transforms. We apply some of our findings to positive definite functions on the Euclidean space $R^m$ which are given by cosine transforms ($m=1$) and Fourier–Bessel transforms ($m>1$). We also apply the results to positive definite functions on a general quasi-metric space realized as extensions of certain real Laplace transforms defined by conditionally negative definite functions on the quasi-metric space itself.     

On interpolation with products of positive definite functions

In this paper we consider the problem of scattered data interpolation for multivariate functions. In order to solve this problem, linear combinations of products of positive definite kernel functions are used. The theory of reproducing kernels is applied. In particular, it follows from this theory that the interpolating functions are solutions of some varational problems.     

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.     

On positive definite solution of a nonlinear matrix equation

In this paper, some necessary and sufficient conditions for the existence of the positive definite solutions for the matrix equation X + A^*X^?αA = Q with α ∈ (0, ∞) are given. Iterative methods to obtain the positive definite solutions are established and the rates of convergence of the considered methods are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Elements of harmonic analysis related to the third basic zero order Bessel function

This paper is devoted to the study of some q-harmonic analysis related to the third q-Bessel function of order zero. We establish a product formula leading to a q-translation with some positive kernel. As an application, we provide a q-analogue of the continuous wavelet transform related to this harmonic analysis.     

On differentiability and analyticity of positive definite functions

We derive a set of differential inequalities for positive definite functions based on previous results derived for positive definite kernels by purely algebraic methods. Our main results show that the global behavior of a smooth positive definite function is, to a large extent, determined solely by the sequence of even-order derivatives at the origin: if a single one of these vanishes then the function is constant; if they are all non-zero and satisfy a natural growth condition, the function is real-analytic and consequently extends holomorphically to a maximal horizontal strip of the complex plane.     

Cardinal Hermite interpolation using positive definite functions

In this paper, we study cardinal Hermite interpolation by using positive definite functions. Among other things, we establish a procedure that employs the multiquadrics for cardinal Hermite interpolation.     

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.     

On infinite divisibility of positive definite functions arising from operator means

For a (point-wisely non-negative) positive definite function a certain criterion for its infinite divisibility (i.e., all its fractional powers are also positive definite) is obtained. This criterion enables us to show infinite divisibility for many positive definite functions appearing naturally in study of operator means. In particular, we determine when the function

     

Piecewise polynomial,positive definite and compactly supported radial functions of minimal degree

We construct a new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support. For given smoothness and space dimension it is proved that they are of minimal degree and unique up to a constant factor. Finally, we establish connections between already known functions of this kind.     

Gradient estimates for positive smooth f-harmonic functions

For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classical ones of Yau (i.e., when f is constant).     

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.     

Extension and separation properties of positive definite functions on locally compact groups

Continuing earlier work, we investigate two related aspects of the set of continuous positive definite functions on a locally compact group . The first one is the problem of when, for a closed subgroup of , every function in extends to some function in . The second one is the question whether elements in can be separated from by functions in which are identically one on .

     

On asymptotics of q-Gamma functions

In this paper we derive some asymptotic formulas for the q-Gamma function Γq(z) for q tending to 1.     

On a new condition for strictly positive definite functions on spheres

Recently, Xu and Cheney (1992) have proved that if all the Legendre coefficients of a zonal function defined on a sphere are positive then the function is strictly positive definite. It will be shown in this paper that, even if finitely many of the Legendre coefficients are zero, the strict positive definiteness can be assured. The results are based on approximation properties of singular integrals, and provide also a completely different proof of the results of Xu and Cheney.

     

流形上矩阵方程AX=B的正定及半正定阵的反问题

文[1-5]中研究了对称、对称半正定及流形上的对称半正定的反问题,并说明了其应用背景.本文研究线性流形上的正定及半正定阵的反问题,说明了文[1-3]中的一些结果为本文的特例.