Chebyshev系统凸性的一个新刻画
A new characterization of convexity with respect to Chebyshev systems
摘要：
通过给定实值函数的$(n+1)$阶除差的非负性，定义了$n$阶凸性的概念.根据已知均差的递推公式，$(n+1)$均差的非负性等价于$k$阶差的$(n-k-1)$阶凸性，从而给出了$n$阶凸性的一个特征.本文的目的是将高阶均差的概念应用于卡林在1968年引入的切比雪夫系统的凸性。利用Sylvester的一个行列式恒等式，我们建立了一个广义均差公式，从而得到了Chebyshev系统凸性的一个新的刻画.我们的结果推广了Wessersowicz在2006年得到的结果。作为应用，我们得到了函数可表示为两个函数之差关于给定Chebyshev系统凸的一个必要条件。
英文摘要：
The notion of $n$th order convexity in the sense of Hopf and Popoviciu is defined via the nonnegativity of the $(n+1)$st order divided differences of a given real-valued function. In view of the well-known recursive formula for divided differences, the nonnegativity of $(n+1)$st order divided differences is equivalent to the $(n-k-1)$st order convexity of the $k$th order divided differences which provides a characterization of $n$th order convexity. The aim of this paper is to apply the notion of higher-order divided differences in the context of convexity with respect to Chebyshev systems introduced by Karlin in 1968. Using a determinant identity of Sylvester, we then establish a formula for the generalized divided differences which enables us to obtain a new characterization of convexity with respect to Chebyshev systems. Our result generalizes that of Wąsowicz which was obtained in 2006. As an application, we derive a necessary condition for functions which can be written as the difference of two functions convex with respect to a given Chebyshev system.
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