SOME PROPERTIES OF CHEBYSHEV SYSTEMS

SOME PROPERTIES OF CHEBYSHEV SYSTEMS RICHARD A. ZALIK Abstrct. We study Chebyshev systems defined on n intervl, whose constituent functions re either complex or rel vlued, nd focus on problems tht my hve hve n ppliction in the theory of differentil equtions nd cnnot be solved by mere rewording of existing proofs, specificlly those deling with the existence of n djoined function, the extension of the intervl of definition, nd the problem of embedding set of functions into n Extended Complete Chebyshev System. 1. Introduction A system of functions F = (f 0,f 1,...,f n ) of complex vlued functions defined on proper intervl I is clled Chebyshev system, or Tchebycheff system, or T system, if the determinnt (1) D(f 0,...f n ;t 0,...t n ) := det(f j (t k );0 j,k n) does not vnish for ny choice of points {t k ;0 k n} in I. It is clled Complete Chebyshev system or CT system or Mrkov system, if (f 0,f 1,...,f k ) is T system for ll k = 0,...,n. If the functions f j re sufficiently smooth, we cn extend the definition of D(f 0,...f n ;t 0,...t n ), so s to llow for equlities mongst the t k : if t 0 t n is ny set of points of I, then D (f 0,...f n ;t 0,...t n ) is defined to be the determinnt on the right hnd of (1), where for ech set of consecutivet k, the correspondingcolumnsrereplcedby the successivederivtives evluted t the point. For exmple, D (f 0,f 1,f 2 ;t 0,t 1,t 1 ) = f 0 (t 0 ) f 0 (t 1 ) f 0 (t 1) f 1 (t 0 ) f 1 (t 1 ) f 1(t 1 ) f 2 (t 0 ) f 2 (t 1 ) f 2 (t 1) nd D (f 0,f 1,f 2 ;t,t,t) = W(f 0,f 1,f 2 )(t). With this definition, the system F is clled n Extended Chebyshev system or ET system on I, provided tht for ny set t 0 t n of points of I, D (f 0,...f n ;t 0,...t n ) does not vnish, nd it is clled n Extended Complete Chebyshev system or ECT system on I, if (f 0,f 1,...,f k ) is n ET system on I for ll k = 0,...,n. Chebyshev systems re of considerble importnce in pproximtion theory, in prticulrin the study ofspline functions, swellsin the theoryoffinite moments. 2010 Mthemtics Subject Clssifiction. 30C15; 26A51; 26C10; 26E05; 34C07; 34C08. Key words nd phrses. Chebyshev systems; Extended Chebyshev systems; Extended complete Chebyshev systems. 1,

2 RICHARD A. ZALIK Exmples of T systems include eigenfunctions of Sturm Liouville opertors. These topics re discussed, for exmple, in Krlin nd Studden s clssicl monogrph [2]. Results on spline functions hve ppered in plethor of lter publictions. For more recent results in the theory of rel vlued T systems, the reder is referred to the rticle by Crnicer, Peñ nd the uthor [1], nd references thereof. Ltely, there hs been renewed interest in Chebyshev systems becuse of their pplictions in the theory of differentil equtions. For exmple P. Mrděsić in his memoir [3], which develops the theory of versl unfolding of cusps of order n, emphsizes the development of results on T systems for the study of unfolding singulrities of vector fields, wheres in [4] Mñoss nd Villdelprt use ECT systems in their study of the period functions of centers of potentil systems. It is therefore useful to study properties of T-systems tht my be pplied in the study of differentil equtions, nd tht my hve been previously overlooked. The following theorem is well known, lthough it is usully stted for rel vlued functions. Theorem 1. Let F = (f 0,f 1,...,f n ) be system of complex vlued functions defined on proper intervl I. Then (1) (f 0,f 1,...,f n ) is T system on I if nd only if ny nontrivil liner combintion of the functions of F hs t most n zeros. (2) (f 0,f 1,...,f n ) is n ET system on I if nd only if ny nontrivil liner combintion of the functions of F hs t most n zeros counting multiplicities. (3) (f 0,f 1,...,f n ) is n ECT system on I if nd only if for ny k, 0 k n, (f 0,f 1,...,f k ) is n ET system. Note tht if F = (f 0,...,f n ) is rel vlued T system on proper intervl I, continuity rgument shows tht, multiplying if needed f n by 1, there is no essentil loss of generlity if we ssume tht for ny set t 0 < < t n of points of I the determinnts D(f 0,...f n ;t 0,...t n ) re strictly positive. Moreover, if F is n ET system for which D(f 0,...f n ;t 0,...t n ) > 0 for ny set t 0 < < t n of points of I then, proceeding s in [2, pp. 6 8], we deduce tht for ny set t 0 t n of points of [,b], the determinnts D (f 0,...f n ;t 0,...t n ) re strictly positive. This in turn implies tht if F is n ECT system for which D(f 0,...f k ;t 0,...t k ) > 0 for ny 0 k n nd ny set t 0 < < t n of points of I then, for ny 0 k n nd ny set t 0 t n of points of I, the determinnts D (f 0,...f k ;t 0,...t k ) re strictly positive for 0 k n. We shll cll such systems positive. Thus we my spek of positive T systems, positive ET systems, nd positive ECT systems. Positive ECT systems, s we define them here, re the ECT systems of Krlin nd Studden [2]. They re clled full differentible ECT systems by Mrděsić [3]. We emphsize tht ll functions in positive systems, nd in prticulr positive ECT systems, re ssumed to be rel vlued. In the theory of rel vlued ECT systems defined on closed intervl [, b], the following theorem is of fundmentl importnce. A proof cn be found in [2, pp. 376 379]. We hve dpted the sttement to our definition of T systems. Theorem 2. Let f 0,f 1,...,u n be rel vlued functions of clss C n [,b]. The following two conditions re equivlent. (1) (f 0,...f n ) is positive ECT system on [,b].

SOME PROPERTIES OF CHEBYSHEV SYSTEMS 3 (2) W(f 0,...,f k )(t) > 0 on [,b] for 0 k n. If, in ddition, the functions f k stisfy the initil conditions (2) f (p) k () = 0, 0 p k 1; 1 k n, then () nd (b) re equivlent to (3) There re functions w k, strictly positive on [,b] nd of continuity clss C n k [,b], such tht f 0 (t) = w 0 (t) (3) f 1 (t) = w 0 (t) t w 1(s 1 )ds 1 f 2 (t) = w 0 (t) t w 1(s 1 ) s 1 w 2(s 2 )ds 2 ds 1.. f n (t) = w 0 (t) t w 1(s 1 ) s 1 w s 2(s 2 ) n 1 w n (s n )ds n ds 1. From [2, p. 380, (1.12) nd (1.13)] we lso know tht if (f 0,...f n ) hs the representtion (3), then (4) W(f 0,f 1, f k ) = w k+1 0 w k 1 w k, which implies tht (5) w 0 = f 0, w 1 = W(f 0,f 1 ) f0 2, w k = W(f 0,,f k )W(f 0, f k 2 ) [W(f 0, f k 1 )] 2, 2 k n. To prove tht (c) implies () in Theorem 2, Rolle s theorem is used. Thus, the proof is not vlid for complex vlued functions. The other prts of the sttement still hold. 2. Existence of Adjoined Functions In this section we discuss the existence of djoined functions i.e., given T system (f 0,...,f n ), whether there exists function f n+1 such tht (f 0,...,f n,f n+1 ) is T system. For dense subsets of open intervls this ws nswered in the ffirmtive by Zielke [7], nd for ny intervl by the uthor [5]. The question hs been rised of whether the sme is true for complex vlued T systems nd whether to T system of nlytic functions cn be djoined n nlytic function. Although the methods usully used for rel vlued functions cnnot be pplied in this setting, but we cn still give n nswer for rel nlytic functions. Theorem 3. Let (f 0,...,f n ) be n ECT system on proper intervl I. Assume, moreover, tht the functions f k re nlytic on n open region D tht contins I, nd tht they re rel vlued on I. Then there is function f n+1, nlytic on n open region D 1 tht contins I nd rel vlued on I, such tht (f 0,...,f n,f n+1 ) is n ECT system on I.

4 RICHARD A. ZALIK Proof. The hypotheses imply tht the Wronskins W(f 0,...,f k ), 1 k n do not vnish on I. Multiplying the functions f k by 1 if necessry, we my ssume without essentil loss of generlity tht these Wronskins re strictly positive on I. Let < b be points in I. Subtrcting if necessry from ech function f k suitble liner combintion of its predecessors we obtin system (u 0,...,u n ) tht stisfies the initil conditions (2). Thus, from Theorem 2 we know tht (u 0,...,u n ) hs representtion of the form (3) on [,b]. It follows from (5) tht the functions w k re strictly positive on I nd nlytic on some open region D 1 tht contins I. Let w n+1 be n entire function strictly positive on I (eg. e t2 ), nd define u n+1 (t) := w 0 (t) t w 1 (s 1 ) sn 1 w n (s n ) sn Clerly u n+1 is nlytic on D 1. From (4) we deduce tht W(u 0,u 1, u n+1 ) = w n+2 0 w n+1 1 w n+1 > 0 w n+1 (s n+1 )ds n+1 ds n ds 1. on I, nd by nother ppliction of Theorem 2 we deduce tht (u 0,...,u n,u n+1 ) is n ECT system on [,b]. Since nd b re rbitrry, the ssertion follows. 3. Extending the Domin of Definition The problem of extending the domin of definition of T system hs been studied extensively (see [1]). Here we consider the problem of extending the domin of definition of n ECT system of complex vlued functions. Theorem 2 cnnot be pplied in this cse, which mkes the rguments more involved. Theorem 4. Let F = (f 0,...,f n ) be n ECT system of complex vlued functions defined on proper intervl I with endpoints nd b. Assume, moreover, tht the functions f k re of clss C n (α,β), where α < < b < β. If I there is c < such tht F is n ECT system on (c,) I, wheres if b I there is d > b such tht F is ECT system on I (b,d). Proof. It suffices to ssume tht I: the other cse redily follows by the chnge of vribles t t. Let I k denote the set of integers from 0 to k. A prtition of I k is fmily {S r ;0 m} of sets of integers such tht (1) m r=0 S r = I k. (2) If α is the lrgest number in S r nd β is the smllest number in S r+1,then β = α+1. The preceding definition implies tht the S r re sets of consecutive integers. A simple inductive rgument shows tht there re 2 k+1 different prtitions of I k. If P is prtition of I k nd S is set in P, then S is clled component of P. A set of integers t 0 t 1 t k is clled configurtion ssocited with P if, whenever α nd β belong to the sme component of P, t α = t β, nd whenever α nd β belong to different components, then t α t β. Thus, ny set t 0 t 1 t k of points of I k belongs to one of 2 k+1 configurtions. For ech configurtion, D (f 0,...f k ;t 0,...t k ) is continuous function of the free vribles involved. For exmple, if t 0 < t 1 < t 2, then D (f 0,f 1,f 2 ;t 0,t 1,t 2 ) is continuous function of t 1, t 2 nd t 3, wheres if t 0 < t 1 = t 2, then D (f 0,f 1,f 2 ;t 0,t 1,t 1 ) is continuous function of t 0 nd t 1. It follows tht for n

SOME PROPERTIES OF CHEBYSHEV SYSTEMS 5 rbitrry k, 0 k n, if P is prtition of I k hving m sets nd S is configurtion ssocited with P, then D (f 0,...f k ;t 0,...t k ) is continuous nonvnishing function in the m fold crtesin product of I with itself. Therefore there is number c k (P) < such tht D (f 0,...f k ;t 0,...t k ) 0 whenever t 0 t 1 t k is configurtion ssocited with P nd the points t k re in (c k (P),) I. Setting c k to be the lrgest of the c k (P) nd c to be the lrgest of the c k, the ssertion follows. 4. Embedding Given finite set of functions, the embedding problem consists in finding necessry nd sufficient conditions for the existence of T system whose liner spn contins them. For single rel vlued function, this problem ws solved by the uthor in [6], wheres in [4, Proposition 2.2 nd Proposition 2.3] Mñoss nd Villdelprt show how to embed n nlytic function into n ECT system of nlytic functions defined on n intervl. The problem in its full generlity remins unsolved. We cn give n nswer in prticulr cse, but first we need to prove some uxiliry propositions. Lemm 5. Let (f 0,,f n ), n 1, be positive ECT system on closed intervl [,b] such tht the functions f k stisfy (3), nd let (c(k);0 k m) be strictly incresing sequence with 0 c(0) < c(n) n. Then (f c(k) ;0 k m) is positive ECT system on (, b]. Proof. Let D 0 = f/w 0, D r f(t) := d dt ( ) f(t), 1 r k, w r (t) nd L r f(t) := D r D r 1 D 0 f(t). Weproceedbyinduction. Thessertionisobviousforn = 1. Toprovetheinductive step we proceed s follows: Let α = c 0. Since (L α f k ;α + 1 k n) hs representtion of the form (3), the inductive hypothesis implies tht (L α f c(k) ;c 1 k m) is positive ECT system on (,b]. By repeted ppliction of the inverse opertorsdα 1, D 1 α 1...D 1 0 to(l α f c(k) ;c 1 k m)ndusingrolle stheoremt ech step, we deduce tht (f c(k) ;0 k m) is positive ECT systemon (,b]. Lemm 6. Let (f 0,,f n ), n 1, be positive ECT system on n intervl I(,b) hving endpoints < b, let c I(,b), ssume tht the functions f k stisfy initil conditions of the form (2) t the point c, tht f 0 (t) > 0 on I(,b), nd let g k (t) := (t c)f k (t). Then (g 0,,g n) is positive ECT system on I(,b]. Proof. We my ssume, without loss of generlity, tht I(, b) is closed intervl. Assume first tht c =. Let k be rbitrry but fixed nd f := (f 0, f k ) T. Then W(g 0 )(t) > 0, nd for k > 0 W(g 0,,g k )(t) = det((t )f (t)+f(t),,(t )f (k) (t)+f (k 1) (t)). Let c = (c(r);0 r k) be sequence of zeros nd ones, let Λ denote the set of ll such sequences, nd for 0 r k let q r (1,t) := (t )f (r+1) (t), q r (2,t) := (r+1)f (r) (t). Then W k (t) := det(q 0 (1,t)+q 0 (2,t),,q k (1,t)+q k (2,t))(t) =

6 RICHARD A. ZALIK l (c(r),t);0 l k) = c Λdet(q D c (t). c Λ For given 0 l k 1 there re four possibilities: If c(l) = 1 nd c(l+1) = 2 then q l (c(l),t) = (t )f (l+1) (t) nd q l+1 (c(l+1),t) = (l+2)f (l+1) (t). (This implies tht D c (t) = 0). If c(l) = 2 nd c(l+1) = 1 then q l (c(l),t) = (l+1)f (l) (t) nd q l+1 (c(l+1),t) = (t )f (l+2) (t). If c(l) = 1 nd c(l+1) = 1 then q l (c(l),t) = (t )f (l+1) (t) nd q l+1 (c(l+1),t) = (t )f (l+2) (t). If c(l) = 2 nd c(l+1) = 2 then q l (c(l),t) = (l)f (l) (t) nd q l+1 (c(l+1) = (l+1)f (l+1) (t). In summtion, there re constnts α > 0 nd β 0, nd sequence 0 1 k such tht D c (t) = α(t ) β det(f (0) (t),f (1) (t), f ( k) (t)), whence Lemm 5 implies tht D c (t) 0. In prticulr, if c(l) = 2 for ll l, then ( ) D c (t) = det (l+1)f (l) (t);0 l k = (k +1)!W(f 0,,f k )(t) > 0. Thus W(g 0,,g k )(t) > 0 on I for 0 k n. If c = b, the ssertion follows by the chnge of vrible t t. In the generl cse,whethert < cort c, theprecedingdiscussioninsuresthtw(g 0,,g k )(t) > 0 on I for 0 k n. Theorem 7. Let 0 k n nd ssume tht the functions f k+1,...,f n re in C n (,b), tht for k+1 r n, ny liner combintion of the functions f k+1,...,f r hs t most r zeros counting multiplicities nd there is t lest one liner combintion of these functions hving exctly r zeros counting multiplicities. Assume, moreover, tht f (p) r () = 0, k +1 r n, 0 p r 1. Then there re functions f 0,...,f k such tht (f 0,...,f n ) is n ECT system on [,b]. Proof. Without essentillossofgenerlitywemy ssumetht ech function f r hs exctly r zeros counting multiplicities. We proceed by induction. Assume first tht n = 1. The hypotheses imply tht f 1 (t) = (t )g(t), where g(t) is nonvnishing nd continuously differentible on [,b]. Setting f 0 (t) := g(t), the ssertion follows. To prove the inductive step, let g r 1 (t) := (t ) 1 f r (t). Then for k r n 1 every nontrivil liner combintion of the functions g k,...,g r hs t most r zeros countingmultiplicities. Byinductivehypothesisthererefunctionsg 0,...g k 1 such tht (g 0,,g n 1 ) is n ECT system on I. Since W(exp(α )f 0,...,exp(α )f r )(t) = exp(r α t)w(f 0,...,f r )(t), multiplying if necessry the functions f r by exp(α t) with α sufficiently lrge we my ssume, without loss of generlity tht g 0 is strictly positive on I. If f l(t) := (t )g l+1, 1 l k, we see tht (f 1,,f n) is positive ECT system on I. Defining f 0 (t) := 1 we see tht for 1 k n, W(f 0,,f k )(t) = W(f 1,,f k)(t) > 0,

SOME PROPERTIES OF CHEBYSHEV SYSTEMS 7 whence the ssertion follows from Theorem 2. References [1] J. M. Crnicer, J. M. Peñ nd R.A. Zlik, Strictly Totlly Positive Systems, J. Approx. Theory 92 (1998) 411 441. [2] S. Krlin nd W. Studden, Tchebycheff Systems: With Applictions in Anlysis nd Sttistics, Interscience, New York, 1966. [3] P. Mrděsić, Chebyshev systems nd the versl unfolding of the cusps of order n, Trvux en Cours 57. Hermnn, Pris, 1998. [4] F. Mñoss nd J. Villdelprt, Criteri to bound the number of Criticl Periods, J. Differentil Equtions 246 (2009), 2415 2433. [5] R. A. Zlik, Existence of Tchebycheff Extensions, J. Mth. Anl. Appl. 51 (1975). [6] R. A. Zlik, Embedding Function into Hr Spce, J. Approx. Theory 55 (1988), 61 64. [7] R. Zielke, Alterntion Properties of Tchebyshev Systems nd the Existence of Adjoined Functions, J. Approx. Theory 10 (1974), 172 184. Deprtment of Mthemtics nd Sttistics, Auburn University, AL 36849 5310. E-mil ddress: zlik@uburn.edu