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Extreme convex set functions with many nonnegative differences

✍ Scribed by František Matúš

Publisher
Elsevier Science
Year
1994
Tongue
English
Weight
827 KB
Volume
135
Category
Article
ISSN
0012-365X

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✦ Synopsis

Where N is a finite set of the cardinality n and 9 the family of all its subsets, we study real functions on B having nonnegative differences of orders n -2, n-1 and n. Nonnegative differences of zeroth order, first-order, and second-order may be interpreted as nonnegativity, nonincreasingness and convexity, respectively. If all differences up to order n of a function are nonnegative, the set function is called completely monotone in analogy to the continuous case. We present a discrete Bernstein-type theorem for these functions with Miibius inversion in the place of Laplace one. Numbers of all extreme functions with nonnegative differences up to the orders n, n -1 and n -2, which is the most sophisticated case, and their Mabius transforms are found. As an example, we write out all extreme nonnegative nondecreasing and semimodular functions to the set N with four elements. KclcL a;f= c f(I),

KcLcN.

KclcL

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