REPRESENTATION OF COMPLETELY CONVEX FUNCTIONS BY THE EXTREME-POINT METHOD

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1 REPRESENTATION OF COMPLETELY CONVEX FUNCTIONS BY THE EXTREME-POINT METHOD Autor(en): Objekttyp: Berg, Christian Article Zeitschrift: L'Enseignement Mathématique Band (Jahr): 23 (1977) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Persistenter Link: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 REPRESENTATION OF COMPLETELY CONVEX FUNCTIONS BY THE EXTREME-POINT METHOD by Christian Berg 0. Introduction A function/: ]0, 1[ -* R is called completely convex, if it is C and i) y ( 2fc ) o for ail A: 0. A completely convex function /is called mini malif f(x)? a sin (tix) is not completely convex for any number a > 0. ( Widder showed (cf. [s]) that a completely convex function can be extended to an entire holomorphic function, and in the paper [6] he proved that a minimal completely convex function can be expanded in a Lidstone séries. This indicates that the Lidstone polynomials lie on the extrême rays of the cône W of completely convex functions. The purpose of the présent paper is to treat the completely convex functions by the extreme-point method and to obtain the expansion in Lidstone séries as a spécial case of the Choquet représentation theorem. We will proceed as follows : In the topology of point-wise convergence the set Wof completely convex functions is a closed, metrizable convex cône. We prove directly that the extrême rays of W are generated by certain and the function essentially the Lidstone polynomials polynomials sin (nx). The occurrence of the extrême ray generated by sin (jzx) is related to the fact that only minimal completely convex functions can be expanded in Lidstone séries. The cône W has a compact convex base B, and the extrême points of B are determined. It turns out that B is a Bauer simplex, i.e. B is a simplex and the extrême points form a closed set. The author wants to acknowledge Widder's paper [6] as a source of inspiration. The reason for writing this paper is that we felt it natural to investigate the cône W by the extreme-point method. Recently Mugler [2] showed that real part of the holomorphic extension of /e Wto the strip Reze]o, 1[ is completely convex on each segment {x + iy 0 < x < 1 }. We give a very short proof of this resuit.??

3 1. Completely convex functions / Let 1 dénote an open interval. A function/ : -> R is called completely 00 (2k) and convex, if it is oon/ for k 0. The set of completely convex functions is a convex cône denoted W = W (/). We always equip W with the topology of pointwise convergence, i.e., with the topology induced by the product space R 7. (-lff COOC If I Lemma 1.1. is unbounded W (/) consists of the non-negative affine functions, and W (R) consists of the non-negative constants. Proof Assume first that since it (-1)*/ -/' < inf/=? 00. Then every fe W is decreasing few is non-negative and concave. For k 0 and we hâve 2fc >e W and consequently (_i)*/(2*+d g 0. This shows that also 6W and then affine. The case sup = -/" 0, but by définition /" 0 and therefore/is / oo is treated in a similar manner. Finally, every non negativeconcave function on R is constant. Remark. Completely convex séquences are non-negative and affine. a séquence a= (a 0, a u...) of real numbers we define Aatobz the 1 k 0, and A a is defîned as A (A k a) for séquence (Ad) n? n+1? am n 1, where A a k séquence ais called completely convex if (? l) ka 2k a 0 for /c 0. The same method as in Lemma 1.1 leads to the 0 and A 2 a conclusion that every completely convex séquence satisfies Aa 0. the therefore The = exactly completely convex séquences are séquences 0. a n =an+ /?, where a, /? This is an answer to a remark by Boas [1] : "Nothing seems to be known about completely convex séquences". For ' an+la =a.a In the following we will always assume that /is bounded, and for the sake convenience wechoose/tobe/ = ]0, 1 [. We simply write Wîov W (]O, 1[). of is deflned by W we hâve ~f" e W and e W, where For (x) is itself. affine W The of onto isomorphism an = x). mapping/h>/*? /* /* /g /(l Lemma 1.2. Let following holds : /:]O, (i) /(x)^2/(y2) (ii) /(x) -/(x 0) 71 1[ -> R /or sin (jtx) Z? /* non-negative and concave. Then the xe]o,l[. /or x, x 0x 0 e]o, l[.

4 (iii) ([6], Lemma 7.1) Ifthere exists x0 e]o, 1[ and a > 0 such that f(x0) <a sin (nx 0 ) then f(x) <* an for xe ]0, I[. Proof. (i). For x e ]0, %] we hâve that/(*) lies below the Une through (i/2,/(i/2)) and (1, 0) and (i) follows for xe]o, %]. The interval [%, l[ is treated similarly. (ii). Let x 0x 0 e ]0, I[. For xe ]0, xo]x 0 ] we hâve and for xe [xo,x 0, I[we hâve (iii). If/(x0) > tf7r the inequality (ii) implies that /(.y) > a sin (%x) for xe]o,l[. Since every/e W ca.n be extended to an entire holomorphic function ail derivatives of/have finite limits at 0 and 1. This can also be established in an elementary way from the property (? l) fc/ (2/c) 0 for k 0. We will therefore freely use/ (/c) (x) for x=o,las the limit of/ (/c) (x) at thèse points. Lemma 1.3. The cône W is a closed and metrizable subset ofr r. Proof. The set of concave functions/ / : -? R is a closed and metrizable subset of R r, and therefore it suffices to prove that the pointwise limit/of a séquence (/?) from W belongs to W. It follows by Lemma 1.2 (i) that there exists a constant A such that /? A for ail n 1 ). The dominated convergence theorem then shows that for ail q>e@qo, l[), so (/?) converges to / weakly in the distribution sensé. Therefore (- 1)*/< 2 *> 0 for ail k oin the distribution sensé, and COOC this implies that/is 00 and hence fe W.!) In fact, A = 2 sup/^/a) can be used. It is finite because lim/^a) exists. «->OO

5 2. Determination of the extreme rays of W Inspired by [6] we consider the Green's function If q> is a continuous function on [0, 1] the unique solution fe o C2C 2 (]O, 1[) to the équations (2.1) C ([O, 1]) 1S (2.2) The successive itérâtes of G are defmed for x, t e [o,l] by the équations It is clear that G n (x, /) 0 for x, te [0, I]. We derme recursively a séquence of polynomials (A n ) n o1)o requirement (2.3) 1 ) by the The polynomial A n is of degree (2n + 1), and we clearly hâve (2.4) It follows that A n oon [0, 1] for ail n, and since (- l) k A (2k) n = AA n _ k for k nwe see that A n ew. We recall that a ray R+x of a cône C is called extrême, if an équation *=/+9 w itn /> e is possible only if/, e R + x, cf. [3]. x ) Our terminology is différent from that of [6] ;(?VfAn polynomial of [4] and [6]. is equal to the n'th Lidstone

6 Proposition 2.1. The polynomials À n9 n 0, lie on extrême rays of W. Proof. If A o =f+g with /, gew we hâve 0 =/" + #", but since /" and g" are both 0, we conclude that/and g are affine. Furthermore, since /(O) = #(0) = 0, we conclude that / and g are proportional to A o. Suppose now that AA n _ u n 1, lies on an extrême ray of W, and assume that A n =/+ g where f,gew. Then /!?_! = -/" + (-^")» and the induction hypothesis implies that?/" and? g" are proportional to A-i- Therefore we hâve/ = l/1,, (x) +ax+b for certain numbers X 0, a, Z?. Since 0 A / n, we have/(0) = /(l) =0 which implies that a=6=o. This proves that / (and similarly g) are proportional to A n which then lies on an extrême ray of W. Since/i->/* is an affine isomorphism of Wthe polynomials A* also lie on extrême rays of W. The following resuit is a spécial case of [6], Theorem 1.1. Proposition 2.2. Every function few can for n Ibe written as where Proof. For n = 1 the formula is équivalent with (2.5) which follows directly from (2.2), and it is clear that R 1 e W. Suppose now the formula holds for some «1. Applying (2.5) to (-1)Y (2/i) ê Wwq get which substituted in the expression for R n because of (2.4). To see that R n e W we notice that yields the formula for n+\

7 The following lemma is easy to establish, but instead of giving the proof hère we refer to [6]. Lemma 2.3. There exists a constant M>o such that Proposition 2.4. The only functions fe W satisfying /(2/c)/ (2/c) = f {2k) (1) =0 for ail k 0 are f{x) =a sin {nx) with a 0. (0) Pn?6>/. Suppose fe W satisfies /(2/c)/ (2/c) (0) = /(2/c)/ (2/c) (1) =0 for ail k 0. Defining a= sup {a 0 /?a sin {nx) e W}, g=f?a belongs to W because Wis closed in Rl.R1. Furthermore sin {nx) Let s>obe given. Since (p =#-8 sin (ttx) there exist k 0 and x 0x 0 e ]0, 1[ such that (- 1) k cp (2k) {x0)x 0) <0, hence By Lemma 1.2 (iii) applied to (? l) k g (2k) we get and therefore by Proposition 2.2 and Lemma 2.3 for 0 < x < 1 This proves that g is identically zéro. Proposition ze extrême rays of W are precisely the rays generated by A n and A*>, where n 0, and sin {nx). Proof We flrst show that sin {nx) lies on an extrême ray. If sin {nx) (1) = 0. = f+g where fgew, we hâve /(0) = /(l) =g (0) =g Diiferentiating 2& times we similarly get (2/c) (0) - f {2k) (1) =# (2/c) (0) /(2/c)/ = g(2fc)(l)g (2fc) (l) = 0, and it follows by Proposition 2.4 that / and g are pro portionalto sin {nx). We flnally hâve to show that an arbitrary extrême ray is generated by one of the above functions.

8 Assume that/e W générâtes an extrême ray. If/ (2/c) (2/c) /(2/c)(l)/ (0) = (l) = 0 owe already know by Proposition 2.4 that/is proportional to for ail k sin (nx). Otherwise let nbe the smallest number 0 for which/ (2/ (0) 0 or/ (2" } (l) / 0. By Proposition 2.2 we then hâve but since/lies on an extrême ray ail three terms on the right-hand side lie on this ray. If/ (2 ' (o) 0 this shows that (- l)y (2n) (l)yl n and R n +1 are proportional to A* n. Therefore f (2n \l) = 0 and R {2 "± 2) = /( 2 'I +2> is proportional to (^*)(2«+2) = Oj so that/ (2n+2) = 0 and hence R n +1 = 0 (cf. Proposition 2.2). If/(2») (i) owe similarly get/ (2n) (0) =0 and R n+i=o. This shows that /lies on the ray generated by either A* or A n. 3. Determination of a base for W There are several ways of determining a base for W. We choose the following set By Lemma 1.2 (ii) we get for fe B and x0 e ]0, 1[ that so the functions in B are uniformly bounded by 2n. It is therefore clear that B is a compact convex base for W. The extrême points of B are exactly the intersections between B and the extrême rays of W. We see that 2 sin (ttx) e B. We claim that the following formulas hold, cf. [4]: (3.1) (3.2)

9 Formula (3.2) follows immediately from (3.1). For n = 0 (3.1) is the familiar formula Suppose that (3.1) holds for n replacée! by n-1 for some n 1. Denoting the right-hand side of (3.1) by /?, we have/ n (0) =fn (1) =0 and which is equal to -A*_ t by the induction hypothesis. It follows by (2.3) that fn = A*, and (3.1) is proved. From (3.1) and (3.2) it follows that 2n 11 + n A n and n 2n + 1 A* n eb.we also get lim n 2n + 11A n (x) = lim n 2n + la* (x) n-+co n-+oo? 2 sin (nx), We hâve now established the following resuit: Proposition 3.1. The set B is a compact convex base for W and the extrême points of B are 2 sin (nx), n 2n n + la* (x), n A n (x), n 0, which form a closed subset of B. By /+ we dénote the set of séquences (oc n ) n^ o of non-negative numbers 00 such that Yj a n < o o By the Choquet représentation theorem or just by the Krein-Milman theorem we get the following, cf. [3] : Theorem 3.2. For every few there exist a 0 and séquences (a n ), (p n)ell such that (3.1) The fonctions in B are uniformly bounded by 2n, and therefore the séries (3.1) is uniformly convergent. If we differentiate the séries in (3.1) two times and change sign we get the séries

10 00 00 which also converges uniformly on ]0, 1[ because ] oc n+l + X! +i < 00. It follows that the following formula holds : (3.2) for o<x<l,k 0 and furthermore (3.3) This proves that the séquences (a w ), (/??) and hence also a are uniquely determined by/. We hâve thus shown that B is a simplex. The extrême points of i? form a closed subset of i? as remarked in Proposition 3.1 so we can formulate the following Corollary 3.3. The base B for W is a Bauer simplex Whittaker proved in [4] that the séries in (3.1) in fact converges uniformly over arbitrary compact subsets of the complex plane. This also proves that/ can be extended to an entire holomorphic function which we also call f. For x g ]0, 1 [ and jeßwe then hâve hence which shows that x h> Ref (x + iy) belongs to W for ail y e R, as sum of the functions which ail belong to the closed cône W. This gives a short proof of the récent resuit of Mugler [2].

11 REFERENCES [1] Boas, Jr., R. P. Signs of derivatives and analytic behavior. Amer. Math. Monthly 78 (1971), [2] Mugler, D. H. Completely convex and positive harmonie functions. SIAM J. Math. Anal. 6 (1975), [3] Phelps, R. R. Lectures on Choquet's theorem. Van Nostrand, Princeton, N.J., [4] Whittaker, J. M. On Lidstone's séries and two-point expansions of analytic functions. Proc. London Math. Soc. 36 (1934), [5] Widder, D. V. The Laplace transform. Princeton University Press, Princeton, N.J., [6] Completely convex functions and Lidstone séries. Trans. Amer. Math. Soc. 51 (1942), (Reçu le 26 avril 1977) Christian Berg Matematisk Institut Universitetsparken Kobenhavn 0 Danmark