Let A_{1} be the set of
nonnegative real functions f on [0,1] such that ∇_{h}^{1}f(x) = f(x) −f(j1j + h) ≦ 0,h > 0,
for [αj,X + h] ⊂ [0,1], and let A_{n},n > 1, be the set of functions belonging to A_{n−1}
such that ∇_{h}^{n}f(x) = ∇_{h}^{n−1}f(x) −∇_{h}^{n−1}f(x + h) ≦ 0 for
Since the sum of two functions in A_{n} belongs to A_{n} and since a nonnegative real
multiple of an A_{n} function is an A_{n} function, the set of A_{n} 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 A_{n} 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.
