Extreme convex set functions with many nonnegative differences

Where N is a nite set of the cardinality n and P the family of all its subsets, we study real functions on P having nonnegative diierences of orders n?2, n?1 and n. Nonnegative diierences of zeroth order, rst order, and second order may be interpreted as nonnegativity, nonincreasingness and convexity, respectively. If all diierences 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 MM obius inversion in the place of Laplace one. Numbers of all extreme functions with nonnegative diierences up to the orders n, n ? 1 and n ? 2, which is the most sophisticated case, and their MM obius transforms are found. As an example, we write out all extreme nonnegative nondecreasing and semimodular functions to the set N with four elements.