Vol. 43, No. 3, 1972

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The convex cone of n-monotone functions

Roy Martin Rakestraw

Vol. 43 (1972), No. 3, 735–752

A reformulation of the Krein-Milman Theorem is used to obtain an integral representation of each function in a certain class of real monotonic functions defined on [0,1]. Let {i1,i2,is,} denote a fixed sequence all of whose terms are either 0 or 1, and let M1 be the set of real nonnegative functions f on [0,1] such that

(− 1)(i1)Δ1hf(x) = (− 1)(i1)[f(x+ h)− f (x)] ≧ 0,

h > 0, for [x,x + h[ [0,1]. Let Mn,n > 1, be the set of functions belonging to Mn1 such that

(− 1)(in)Δnhf(x) = (− 1)(in)[Δnh−1f(it+ h)− Δnh−1f(x )] ≧ 0

for [x,x + nh] [0,1]. If f Mn, then f is said to be an n-monotone function. Since the sum of two n-monotone functions is in Mn and since a nonnegative real multiple of an n-monotone function is an n-monotone function, the set Mn is a convex cone. It is the purpose of this paper to give the extremal elements (i.e., the generators of extreme rays) of this cone, and to show that for the n-monotone functions an integral representation in terms of extremal elements is possible.

Mathematical Subject Classification 2000
Primary: 46A99
Secondary: 46E05
Received: 15 March 1971
Published: 1 December 1972
Roy Martin Rakestraw