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Let A1 be the set of
nonnegative real functions f on [0,1] such that ∇h1f(x) = f(x) −f(j1j + h) ≦ 0,h > 0,
for [αj,X + h] ⊂ [0,1], and let An,n > 1, be the set of functions belonging to An−1
such that ∇hnf(x) = ∇hn−1f(x) −∇hn−1f(x + h) ≦ 0 for
![[x,x + nh] ⊂ [0,1].](a200x.png)
Since the sum of two functions in An belongs to An and since a nonnegative real
multiple of an An function is an An function, the set of An functions forms 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, to prove that they form a closed set in a
compact convex set that does not contain the origin but meets every ray of the cone,
and to show that for the functions of the cone an integral representation in terms of
extremal elements is possible. The intersection of the An cones is the class of
functions alternating of order∞. Thus, the set of these functions, which will be
denoted by A∞, forms a convex cone also. The extremal elements for the convex cone
A∞ are given too.
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