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arXiv · 1607.04026

A new characterization of convexity with respect to Chebyshev systems

Zsolt Páles·Éva Székelyné Radácsi
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Abstract

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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Zsolt Páles, Éva Székelyné Radácsi. 2018-03-17. A new characterization of convexity with respect to Chebyshev systems. https://doi.org/10.7153/jmi-2018-12-46

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