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. y(k)(t) = f y(s)Gk(t s)ds, where Gk(s) = (k/^/2Ï)exp(-V2k2 a2). The set of functions convex with respect to the system {y0,... ,y„] 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 {_y,}"=0 be a system of continuous functions of bounded variation on an interval [a,b] such that y0 = 1, and {.yj^o " a WT-system thereon for r — 1, ..., n. If P is the set of points of (a, b) at which all the functions y¡ are differentiable, the system {y'¡}"= i is a WT-system on P. Proof. If the functions {y¡}"=0 are linearly dependent, the assertion is obvious. Otherwise, from the basic composition formula (cf. [1, pp. 14, 15]), we know that [yW}"=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 {[y}k'/y(jc']'}1=\ is an ECT-system. However, bearing in mind that y0 = I, from [2, Chapter X, Exercise 9], we conclude that iimk^x[y\k\t)/y^(t)] = y'¡(t) on P, for /' = 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. © American Mathematical Society 1976 11« License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GENERALIZED CONVEX FUNCTIONS 119 Lemma 2. Let [yQ,y\,. ■. ,y„+\} be a WT-system on the interval (a,b). Then: (a) If n > 1 and {y0,... ,y„) is a CT-system on (a,b), then yn+x is continuous on (a, b). (b) If n > 2 and {y0,... ,yn) is an ECT-system on [a,b], then yn+x has a continuous derivative on (a, b). Proof. The hypotheses imply that without any loss of generality, we can assume y0 m 1. It will suffice to carry out the proof for every interval of the form (a,a'), a < a' < b. Let D(y0,... ,yr/t0,... ,tr) denote the determinant of the matrix (y¡(tj); i,j = 0,... ,/•). Let a' < i0 < • • • < s„ < b, and define «,(/) = D(y0,... ,y„/s0,... ,s¡_x,t,si+x,.. .,s„), i = 0, ..., n, and un+x(t) = D(y0,... ,yn+}/s0,... ,sn,t). Since n D(u0,..., un/s0, ...,*„) = II ",(*,) > 0, ;=o it is easily seen that {u0,...,u„) is a T-system on (a,b), and that un+x E C(u0,... ,un) = C(y0,... ,yn) (cf. [4, Lemma 2]). It will suffice to prove the assertions for the function un+x. Let a < /0 < /, < t2 < a'. Then 0 < D(u0,... ,un+x/t0,tx,t2,s2,... ,sn) = D(u0,ux,(-l)" u„+x,u2,... ,u„/t0,tx,t2,s2,... ,s„) = in ui(s.iûixuo,ifi£T'i)n~}um.l/tQ*tî,tt). We therefore conclude that {«o»"i>(_0"~lMn+i} is a WT-system on (a, a'). In similar fashion we see that {u0, ux) is a CT-system thereon, i.e. ux/u0 is strictly increasing. Thus the function h = (-1)"~ (w„+i/"o) ° (Mi/Mo)_1 's convex, and thus continuous, whence the proof of (a) follows. In order to prove (b), note that, since h is convex, the function un+x = (-l)"~x[h ° (u2/ux)]u0 admits of a representation of the form (0 un+x(t) = un+x(c) + u0(t)jc q(s)ds, a < c < a', on (a,a'), where q is left continuous on (a,a'), and continuous on a dense subset P thereof. Since un+x E C(y0,... ,yn) and is of bounded variation on every closed subinterval of (a,a'), from Lemma 1 we conclude that {y\, y'2,... ,y'„,u'„+x) is a WT-system on P. Since u'n+x = q on P, and q is left continuous, we conclude that [y\,y'2,... ,y'n,q] is a WT-system on (a,a'). The hypotheses imply that {y\ ,y'2,... ,y'n) is a CT-system on (a,b). Thus part (a) of this lemma implies that q is continuous on (a,a'), and the conclusion follows from (1). Q. E. D. Theorem. Assume that { j>,}"=q is an extended complete Tchebycheff system on [a, b], and let y E C(y0,... ,y„). Then: License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use