On Quotients of Positive Definite Functions

For a class of real analytic functions on R, we give a condition for a function to be a quotient of two positive definite functions.     

Positive Definite Functions and Operator Inequalities

We construct several examples of positive definite functions,and use the positive definite matrices arising from them toderive several inequalities for norms of operators. 1991 MathematicsSubject Classification 42A82, 47A63, 15A45, 15A60.     

Complex Variable Positive Definite Functions

In this paper we develop an appropriate theory of positive definite functions on the complex plane from first principles and show some consequences of positive definiteness for meromorphic functions.     

Inequalities for Positive Definite Functions

A characterization of bounded sets in Banach spaces in terms of asymptotic cones and the Hausdorff deviations of sets from them homothetic images is obtained. Similar results for generalizations of the notion of boundedness are presented. Boundedness criteria have previously been known only for recessively compact sets.

     

On the Extremal Rays of the Cone of Positive, Positive Definite Functions

The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on ℝ ^d . Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there are many other extremals than the Gaussians, thus disproving a conjecture of G. Choquet, and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain.     

半群上条件正定函数的扩张

讨论半群上条件正定函数的扩张问题.得到一个扩张定理.作为应用,得到有界扩张定理和Pontryagin空间上量子力学的一个基本定理.     

Continuation of Hermitian Positive Definite Functions and Related Questions

We study the intimate relation of the continuation problem for hermitian positive definite functions with direct and inverse spectral problems for canonical systems and strings and with the extrapolation problem for second order stochastic processes.     

Exchangeable Random Ordered Trees by Positive Definite Functions

We prove that the set of exchangeable random totally ordered trees is a Bauer simplex whose extreme points are the generalized paint-box processes. The method is based on semigroups, analogous to Hirth and Ressel,^{(2, 3)} and is readily also applied to inverse trees.     

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.     

D_x-广义正定矩阵

讨论Dx广义正定阵的问题,给出了Dx广义正定阵的一些等价性的刻划,同时还讨论了Dx广义正定阵的若干性质.     

Positive Definite Functions as an Instrument of Mathematical Analysis

For the subject in question, the paper describes its connections that are close to the author’s interests with branches of functional analysis. The present survey may be suitable as a basis for a special course.     

关于广义正定矩阵的进一步推广

受到Johnson、佟文廷、夏长富的启发,进一步推广了广义正定矩阵的定义,并得出了更广义正定矩阵P∈PA+的若干性质.     

关于实方阵的正定性

本文研究一般实方阵的正定性 ,给出了方阵正定的一些充分必要条件     

On Hankel Positive Definite Perturbations of Hankel Positive Definite Sequences and Interrelations to Orthogonal Matrix Polynomials

We derive a representation of orthogonal matrix polynomials on the real line as product of certain Schur complements and obtain matricial generalizations of well-known formulas for orthogonal scalar polynomials associated to a polynomial modified moment sequence.     

A Note on Factoring of Positive Definite Functions on Semigroups

The problem of characterizing those semigroups of class ℳ︁ such that every positive definite function factors via the greatest C ‐separative ∗︁‐homomorphic image is shown to be equivalent to several other problems, including that of characterizing flat semigroups. If the latter could be solved then the problem of characterizing perfect semigroups would be reduced to characterizing quasi‐perfect semigroups.     

Strictly Positive Definite Functions on a Real Inner Product Space

If converges for all with all coefficients , then the function is positive definite on H×H for any inner product space H. Set K={k: a _k >0}. We show that 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.     

二元函数正定性的三种常见判别方法

研究了二元函数正定性的判别法,通过对二元函数定义和性质的讨论,得到了三个判别二元函数正定性的方法.     

On Separation of Points from Additive Subgroups of Banach Spaces by Continuous Characters and Positive Definite Functions

Let G be an additive subgroup of a normed space X. We say that a point is weakly separated (resp. -separated) from G if it can be separated from G by a continuous character (resp. by a continuous positive definite function). Let T : X → Y be a continuous linear operator. Consider the following conditions: (ws) if , then x is weakly separated from G; (ps) if , then x is -separated from G; (wp) if Tx is -separated from T(G), then x is weakly separated from G. By (resp. , ) we denote the class of operators T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all and all subgroups G of X. The paper is an attempt to describe the above classes of operators for various Banach spaces X, Y. It is proved that if X, Y are Hilbert spaces, then is the class of Hilbert-Schmidt operators. It is also shown that if T is a Hilbert-to-Banach space operator with finite ℓ-norm, then .