Smoothness properties of generalized convex functions

We present a concise and elementary proof of a theorem of Karlin and Studden concerning the smoothness properties of functions belonging to a generalized convexity cone. In [1, Chapter XI], Karlin and Studden showed that a function which is convex with respect to an extended complete Tchebycheff system has a continuous derivative of order n 1, a fact which is of considerable importance in the theory of generalized convexity. Since their original proof is rather technical, we present here an alternative one which has the advantage of being very concise and elementary. Given a function y y(k) will denote its convolution with the Gauss kernel, i.e. Jb y(k)(t) a y(s)Gk(t s)ds, where Gk(s) = (k//2w)exp(-?2k2 a2). The set of functions convex with respect to the system { Yo, .. ,yn} will be denoted by C(y0,...,yn). The abbreviations T, CT, WT and ECT will respectively stand for Tchebycheff, complete Tchebycheff, weak Tchebycheff and extended complete Tchebycheff. For the definition of other terms and symbols employed, the reader is referred to the monograph by Karlin and Studden [1, Chapters I and XI]. LEMMA 1. Let { yi}n 0 be a system of continuous functions of bounded variation on an interval [a, b] such that yo 1, and { y .}7= is a WT-system thereon for r = 1, . . ., n. If P is the set of points of (a, b) at which all the functions yi are differentiable, the system { Y7ni I is a WT-system on P. PROOF. If the functions { jy1}=0 are linearly dependent, the assertion is obvious. Otherwise, from the basic composition formula (cf. [1, pp. 14, 15]), we know that {y [k)}"0 is an ECT-system for any natural number k > 0. From [1, Chapter XI, Theorem 1.2 and Remark 1.2], we conclude that also the reduced system {[flk)/y'k)] }=I is an ECT-system. However, bearing in mind that Yo = 1, from [2, Chapter X, Exercise 9], we conclude that lim1k [y(k)(t) /yk)(t)]' = y'(t) on P, for i = 1, ..., n, whence the conclusion follows. Q.E.D. REMARK. Lemma 1 furnishes a much shorter proof of [3, Theorem 3] in a more general framework. Received by the editors February 20, 1975. AMS (MOS) subject classifications (1970). Primary 26A51; Secondary 41A50. X) American Mathematical Society 1976