On the "Favard theorem" and its extensions

On the \Favard theorem" and its extensions 1 Franiso Marellan a ; 2 and Renato  Alvarez-No darse b ; 3 a Departamento de Matematias. Universidad Carlos III de Madrid. Ave. de la Universidad 30, E-28911 Leganes, Madrid, SPAIN b Departamento de Analisis Matematio, Universidad de Sevil la, Apdo. 1160, E-41080 Sevil la, and Instituto Carlos I de Fsia Teoria y Computaional, Universidad de Granada, E-18071, Granada, SPAIN Abstrat In this pap er we present a survey on the \Favard theorem" and its extensions. Key words: Favard Theorem, reurrene relations 1 Intro dution. Given a sequene f P n g 1 n =0 of moni p olynomials satisfying a ertain reurrene relation, we are interested in nding a general inner pro dut, if one exists, suh that the sequene f P n g 1 n =0 is orthogonal with resp et to it. The original \lassial" result in this diretion is due to J. Favard [10℄ even though his result seems to b e known by dierent mathematiians. The rst who obtained a similar result was Stieltjes in 1894 [23℄. In fat, from the p oint of view of J  ontinued frations obtained from the ontration of an S  ontinued fration with p ositive o eÆients, Stieltjes proved the existene of a p ositive linear funtional suh that the denominators of the approximants are orthogonal with resp et to it [23, x 11℄. Later on, Stone gave another approah using the sp etral resolution of a self-adjoint op erator asso iated to a Jaobi matrix [24, Th. 10.23℄. In his pap er [21, page 454℄ Shohat laims \We 1 This work has b een partially supp orted by the Europ ean pro jet INTAS-93-219- ext and by the Spanish Direion General de Ense ~nanza Sup erior (DGES) grant PB-96-0120-C03-01. 2 Phone: +34 91 624 94 42, Fax: +34 91 624 91 51. E-mail: paomaring.u3m.es 3 Phone: +34 954 55 79 97, Fax: +34 95 455 79 72. E-mail: rania.es Preprint submitted to Elsevier Preprint 8 January 2000 have b een in p ossession of this pro of for several years. Reently J. Favard published an idential pro of in the Comptes Rendus ". Also I. P. Natanson in his b o ok [17, page 167℄ said \This theorem was also disovered (indep endent of Favard) by the author (Natanson) in the year 1935 and was presented by him in a seminar led by S. N. Bernstein. He then did not publish the result sine the work of Favard app eared in the meantime". The \same" theorem was also obtained by Perron [19℄, Wintner [28℄ and Sherman [20℄, among others. The Favard's result essenially means that if a sequene of moni p olynomials f P n g 1 n =0 satises a three-term reurrene relation xP n ( x ) = P n +1 ( x ) + a n P n ( x ) + b n P n  1 ( x ) ; (1.1) with a n ; b n 2 R , b n > 0, then there exists a p ositive Borel measure  suh that f P n g 1 n =0 is orthogonal with resp et to the inner pro dut h p; q i = Z R p q d: (1.2) This formulation is equivalent to the following: Given the linear op erator t : P ! P , p ( t ) ! tp ( t ), haraterize an inner pro dut suh that the op erator t is Hermitian with resp et to the inner pro dut. A rst extension of this problem is due to Chihara [5℄. If f P n g 1 n =0 satises a three-term reurrene relation like (1.1) with a n ; b n 2 C , b n 6 = 0, nd a linear funtional L dened on P , the linear spae of p olynomials with omplex o eÆients, suh that f P n g 1 n =0 is orthogonal with resp et to the general inner pro dut h p; q i = L [ pq ℄, where p; q 2 P . Notie that in the ase analyzed by Favard [10℄ the linear funtional has an integral representation L [ p ℄ = Z R p d: Favard's Theorem is an inverse problem in the sense that from information ab out p olynomials we an dedue what kind of inner pro dut indues orthogonality for suh p olynomials. The aim of this ontribution is to survey some extensions of the Favard Theorem when a sequene of moni p olynomials f P n g 1 n =0 satises reurrene relations of a dierent form than (1.1). In the rst plae, in [8℄ a similar problem is studied relating to p olynomials orthogonal with resp et to a p ositive Borel measure  supp orted on the unit irle, whih satisfy a reurrene relation  n ( z ) = z  n  1 ( z ) +  n (0)  n  1 ( z ) ; j  n (0) j < 1 ; (1.3) 2 where   n ( z ) = z n  n (1 =z ). Thus, a Favard Theorem means, in this ase, if we an identify an inner pro d- ut in P suh that f  n g 1 n =0 satisfying (1.3) is the orresp onding sequene of orthogonal p olynomials. The struture of the pap er is as follows. In Setion 2 we present a survey of results surrounding the Favard Theorem when a sequene of p olynomials satises a linear relation like (1.1). In partiular, we show that the interlaing prop erty for the zeros of two onseutive p olynomials gives basi information ab out the preeding ones in the sequene of p olynomials. In Setion 3, an analogous approah is presented in the ase of the unit irle in a more general situation when j  n (0) j 6 = 1. Furthermore, an integral representation for the orresp onding inner pro dut is given. The onnetion with the trigonometri moment problem is stated when we assume that the n th p olynomial  n is oprime with   n . In Setion 4, we present some reent results ab out a natural extension of the ab ove Favard theorems taking into aount their interpretation in terms of op erator theory. Indeed, the multipliation by t is a Hermitian op erator with resp et to (1.2) and a unitary op erator with resp et to the inner pro dut h p; q i = Z R p ( e i ) q ( e i ) d (  ) : (1.4) Thus we are interested in haraterizing inner pro duts suh that the multipliation by a xed p olynomial is a Hermitian or a unitary op erator. The onnetion with matrix orthogonal p olynomials is stated, and some examples relating to Sob olev inner pro duts are given. 2 The Favard theorem on the real line. 2.1 Preliminaries. In this subsetion we summarize some denitions and preliminary results that will b e useful throughout the work. Most of them an b e found in [5℄. Denition 2.1 Let f  n g 1 n =0 be a sequene of omplex numbers (moment sequene) and L a funtional ating on the linear spae of polynomials P with omplex oeÆients. We say that L is a moment funtional assoiated with f  n g 1 n =0 if L is linear, i.e., for al l polynomials  1 and  2 and any omplex 3 numbers  1 and  2 L [  1  1 +  2  2 ℄ =  1 L [  1 ℄ +  2 L [  2 ℄ ; and L [ x n ℄ =  n ; n = 0 ; 1 ; 2 ; ::: : Denition 2.2 Given a sequene of polynomials f P n g 1 n =0 , we say that f P n g 1 n =0 is a sequene of orthogonal polynomials (SOP) with respet to a moment fun- tional L if for al l nonnegative integers n and m the fol lowing onditions hold: (1) P n is a polynomial of exat degree n , (2) L [ P n P m ℄ = 0 ; m 6 = n , (3) L [ P 2 n ℄ 6 = 0 . Usual ly, the last two onditions are replaed by L [ x m P n ( x )℄ = K n Æ nm ; K n 6 = 0 ; 0  m  n ; where Æ nm is the Kroneker symbol. The next theorems are diret onsequenes of the ab ove denition [5, Chapter I, x 2,3, pages 8-17℄. Theorem 2.1 Let L be a moment funtional and f P n g 1 n =0 a sequene of polynomials. Then the fol lowing are equivalent: (1) f P n g 1 n =0 is an SOP with respet to L . (2) L [  P n ℄ = 0 for al l polynomials  of degree m<n , while L [  P n ℄ 6 = 0 if the degree of  is n . (3) L [ x m P n ( x )℄ = K n Æ nm , where K n 6 = 0 , for m = 0 ; 1 ; :::; n . Theorem 2.2 Let f P n g 1 n =0 be an SOP with respet to L . Then, for every polynomial  of degree n  ( x ) = n X k =0 d k P k ( x ) ; where d k = L [  P k ℄ L [ P 2 k ℄ ; k = 0 ; 1 ; :::; n: (2.1) A simple onsequene of the ab ove theorem is that an SOP is uniquely determined if we imp ose an additional ondition that xes the leading o eÆient k n of the p olynomials ( P n ( x ) = k n x n + lower order terms). When k n = 1 for all n = 0 ; 1 ; 2 ; ::: the orresp onding SOP is alled a moni SOP (MSOP). If we ho ose k n = ( L [ P 2 n ℄)  1 2 , the SOP is alled an orthonormal SOP (SONP). 4 The next question whih obviously arises is the existene of an SOP. To answer this question, it is neessary to intro due the Hankel determinants  n ,  n =                0  1   n  1  2   n +1 . . . . . . . . . . . .  n  n +1   2 n               : Theorem 2.3 Let L be a moment funtional assoiated with the sequene of moments f  n g 1 n =0 . Then, the sequene of polynomials f P n g 1 n =0 is an SOP with respet to L if and only if  n 6 = 0 for al l nonnegative n . Moreover, the leading oeÆient k n of the polynomial P n is given by k n = K n  n  1  n . Denition 2.3 A moment funtional L is al led positive denite if for every nonzero and nonnegative real polynomial  , L [  ℄ > 0 . The following theorem haraterizes the p ositive denite funtionals in terms of the moment sequenes f  n g 1 n =0 . The pro of is straightforward. Theorem 2.4 A moment funtional L is positive denite if and only if their moments are real and  n > 0 for al l n  0 . Using the ab ove theorem, we an dene a p ositive denite moment funtional L entirely in terms of the determinants  n . In other words, a moment fun- tional L is alled p ositive denite if all its moments are real and  n > 0 for all n  0. Notie also that for a MSOP, it is equivalent to say that K n > 0 for all n  0. This, and the fat that an SOP exists if and only if  n 6 = 0, leads us to dene more general moment funtionals: the so-alled quasi-denite moment funtionals. Denition 2.4 A moment funtional L is said to be quasi-denite if and only if  n 6 = 0 for al l n  0 . We an write the expliit expression of the MOP in terms of the moments of the orresp onding funtional: P n ( x ) = 1  n  1                    0  1   n  1  2   n +1 . . . . . . . . . . . .  n  1  n   2 n  1 1 x  x n                   ;   1  1 ; n = 0 ; 1 ; 2 ; ::: : (2.2) 5 One of the simplest harateristis of orthogonal p olynomials is the so-alled three-term reurrene relation (TTRR) that onnets every three onseutive p olynomials of the SOP. Theorem 2.5 If f P n g 1 n =0 is a MSOP with respet to a quasi-denite moment funtional, then the polynomials P n satisfy a three-term reurrene relation P n ( x ) = ( x   n ) P n  1 ( x )   n P n  2 ( x ) ; n = 1 ; 2 ; 3 ; ::: ; (2.3) where f  n g 1 n =0 and f  n g 1 n =0 are given by  n = L [ x P 2 n  1 ℄ L [ P 2 n  1 ℄ ; n  1 ; and  n = L [ xP n  1 P n  2 ℄ L [ P 2 n  2 ℄ = L [ P 2 n  1 ℄ L [ P 2 n  2 ℄ ; n  2 ; respetively, and P  1 ( x )  0 , P 0 ( x )  1 . The pro of of the ab ove theorem is a simple onsequene of the orthogonality of the p olynomials and Theorem 2.2. A straightforward alulation shows that (  1 = L [1℄)  n +1 = K n K n  1 =  n  2  n  2 n  1 ; n = 1 ; 2 ; 3 ; :::; and   1  1. From Theorem 2.4 and Denition 2.4 it follows that, if  n 6 = 0, then L is quasi-denite whereas, if  n > 0, then L is p ositive denite. Notie also that from the ab ove expression we an obtain the square norm K n  L [ P 2 n ℄ of the p olynomial P n as K n  L [ P 2 n ℄ =  1  2   n +1 : (2.4) A useful onsequene of Theorem 2.5 are the Christoel-Darb oux identities. Theorem 2.6 Let f P n g 1 n =0 be a MSOP whih satises (2.3) with  n 6 = 0 for al l nonnegative n . Then n X m =0 P m ( x ) P m ( y ) K m = 1 K n P n +1 ( x ) P n ( y )  P n +1 ( y ) P n ( x ) x  y ; n  0 ; (2.5) and n X m =0 P 2 m ( x ) K m = 1 K n [ P 0 n +1 ( x ) P n ( x )  P n +1 ( x ) P 0 n ( x )℄ ; n  0 : (2.6) For an arbitrary normalization (not neessarily the moni one) of the p olynomials P n , the three-term reurrene relation b eomes xP n  1 ( x ) =  n P n ( x ) +  n P n  1 ( x ) +  n P n  2 ( x ) : (2.7) 6 In this ase, the o eÆients  n and  n an b e obtained omparing the o ef- ients of x n and x n  1 , resp etively, in b oth sides of (2.7) and  n is given by L [ xP n  1 P n  2 ℄ L [ P 2 n  2 ℄ . This leads to  n = k n  1 k n ;  n = b n  1 k n  1  b n k n ;  n = k n  2 k n  1 K n  1 K n  2 ; (2.8) where k n is the leading o eÆient of P n and b n denotes the o eÆient of x n  1 in P n , i.e., P n ( x ) = k n x n + b n x n  1 +  . Notie also that knowing two of the o eÆients  n ,  n , and  n , one an nd the third one using (2.7) provided, for example, that P n ( x 0 ) 6 = 0 for some x 0 (usually x 0 = 0) and for all n = 1 ; 2 ; 3 ; ::: . The ab ove TTRR (2.7) an b e written in matrix form, x P n  1 = J n P n  1 +  n P n ( x ) e n ; (2.9) where P n  1 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 P 0 ( x ) P 1 ( x ) P 2 ( x ) . . . P n  2 ( x ) P n  1 ( x ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; J n = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4  1  1 0 ::: 0 0  2  2  2 ::: 0 0 0  3  3 ::: 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 : : :  n  1  n  1 0 0 0 : : :  n  n 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; e n = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 0 0 . . . 0 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : (2.10) Denoting by f x n;j g 1  j  n the zeros of the p olynomial P n , we see from (2.9) that eah x n;j is an eigenvalue of the orresp onding tridiagonal matrix of order n and [ P 0 ( x n;j ) ; :::; P n  1 ( x n;j )℄ T is the asso iated eigenvetor. From the ab ove representation many useful prop erties of zeros of orthogonal p olynomials an b e found. 2.2 The zeros of orthogonal polynomials. Denition 2.5 Let L be a moment funtional. The support of the funtional L is the largest interval ( a; b )  R where L is positive denite. The following theorem holds. Theorem 2.7 Let ( a; b ) be the support of the positive denite funtional L , and let f P n g 1 n =0 be the MSOP assoiated with L . Then, 7 (1) Al l zeros of P n are real, simple, and loated inside ( a; b ) . (2) Two onseutive polynomials P n and P n +1 have no ommon zeros. (3) Let f x n;j g n j =1 denote the zeros of the polynomial P n , with x n; 1 < x n; 2 <  < x n;n . Then, x n +1 ;j < x n;j < x n +1 ;j +1 ; j = 1 ; 2 ; 3 ; :::; n : The last property is usual ly al led the interlaing prop erty . Pro of: Notie that, in the ase when the SOP is an SNOP, i.e, K n = 1 for all n , then the matrix J n is a symmetri real matrix ( J n = J T n , where J T n denotes the transp osed matrix of J n ). So its eigenvalues, and thus, the zeros of the orthogonal p olynomials are real. To prove that all zeros are simple, we an use the Christoel-Darb oux identity (2.6). Let x k b e a multiple zero of P n , i.e., P n ( x k ) = P 0 n ( x k ) = 0. Then (2.6) gives 0 < n X m =0 P 2 m ( x k ) K m = 1 K n [ P 0 n +1 ( x k ) P n ( x k )  P n +1 ( x k ) P 0 n ( x k )℄ = 0 : This ontradition proves the statement. Let f x k g p k =1 b e the zeros of P n inside ( a; b ). Then, P n ( x ) Q p k =1 ( x  x k ) do es not hange sign in ( a; b ) and L [ P n ( x ) Q p k =1 ( x  x k )℄ 6 = 0, so p = n , i.e., all the zeros of P n are inside ( a; b ). Thus, the statement 1 is proved. To prove 2, we use the TTRR. In fat, if x k is a zero of P n and P n +1 , then it must b e a zero of P n  1 . Continuing this pro ess by indution, we get that x k must b e a zero of P 0 ( x )  1, whih is a ontradition. Before proving the interlaing prop erty 3 we will prove a theorem due to Cauhy [22, page 197℄. Theorem 2.8 Let B be a prinipal ( n  1)  ( n  1) submatrix of a real symmetri n  n matrix A , with eigenvalues  1   2       n  1 . Then, if  1   2       n are the eigenvalues of A ,  1   1   2       n  1   n : Pro of: Let A b e the n  n matrix A = 0 B  B a a T b 1 C A ; and assume that the theorem is not true, i.e.,  i >  i or  i +1 >  i (sine the matrix A is real symmetri, all its eigenvalues are real). Let i b e the rst suh index. If  i >  i (the other ase is similar), there exists a real numb er  suh that  i >  >  i . Then, B   I n  1 , where I k denotes the identity k  k matrix, 8 is nonsingular (det ( B   I n  1 ) 6 = 0), and the matrix H = 0 B  B   I n  1 0 0 b    a T ( B   I n  1 )  1 a 1 C A = 0 B  I n  1 0  a T ( B   I n  1 )  1 1 1 C A 0 B  B   I n  1 a a T b   1 C A 0 B  I  ( B   I n  1 )  1 a 0 1 1 C A ; is ongruent to A   I n . Then, by the inertia theorem, the matrix H has the same numb er of p ositive eigenvalues as A   I n , i.e., i  1. But H has at least as many p ositive eigenvalues as B   I n  1 , i.e., i . The ontradition proves the theorem. Obviously, the interlaing prop erty 3 an b e obtained as a simple orollary of the Cauhy Theorem, sine the matrix J n asso iated with the SONP is a real symmetri matrix and we an ho ose as A the matrix J n +1 whose zeros are the zeros of the p olynomial P n +1 and then, the prinipal submatrix B is the matrix J n whose eigenvalues oinide with the zeros of P n . This ompletes the pro of of Theorem 2.7. 2.3 The Favard Theorem and some appliations. In this subsetion we will prove the so-alled Favard Theorem. Theorem 2.9 Let f  n g 1 n =0 and f  n g 1 n =0 be two arbitrary sequenes of omplex numbers, and let f P n g 1 n =0 be a sequene of polynomials dened by the relation P n ( x )=( x   n ) P n  1 ( x )   n P n  2 ( x ) ; n = 1 ; 2 ; 3 ; ::: ; (2.11) where P  1 ( x ) = 0 and P 0 ( x )=1 . Then, there exists a unique moment fun- tional L suh that L [1℄ =  1 ; L [ P n P m ℄=0 if n 6 = m: Moreover, L is quasi-denite and f P n g 1 n =0 is the orresponding MSOP if and only if  n 6 = 0 , and L is positive denite if and only if  n are real numbers and  n > 0 for al l n = 1 ; 2 ; 3 ; ::: . Pro of: To prove the theorem, we will dene the funtional L by indution on P n , the linear subspae of p olynomials with degree at most n . We put L [1℄ =  0 =  1 ; L [ P n ℄ = 0 ; n = 1 ; 2 ; 3 ; ::: : (2.12) 9 If  n ( z ) = z n +  n; 1 z n  1 +  +  n;n  1 z + a n , we dene  n =   n; 1  n  1       n;n  1  1  a n  0 . Notie that this means that h  1 ( z ) ; 1 i = 0. On the other hand, for 1  k  n  1, using the reurrene relation (i) D  n ( z ) ; z k E = D  n  1 ( z ) ; z k  1 E + a n D   n  1 ( z ) ; z k E = 0 ; where the last term in the ab ove sum vanishes sine D   n  1 ( z ) ; z k E = D z n  k  1 ;  n  1 ( z ) E : Finally, using (3.4), we have h  n ( z ) ; z n i = det T n det T n  1 = (1  j  n (0) j 2 ) det T n  1 det T n  2 ; and thus, b eause of the indution hyp othesis, h  n ( z ) ; z n i 6 = 0. Corollary 3.1 The funtional L is positive denite if and only if j  n (0) j < 1 , for n = 1 ; 2 ; ::: . 3.2 The zeros of the orthogonal polynomials. In the following we will analyze the existene of an integral representation for a moment funtional. First, we will onsider the ase of p ositive deniteness. Prop osition 3.1 [12℄ If  is a zero of  n ( z ) , then j  j < 1 . Pro of: Let  n ( z ) = ( z   ) q n  1 ( z ), where q n  1 is a p olynomial of degree n  1. Then, 0 < h  n ( z ) ;  n ( z ) i = h ( z   ) q n  1 ( z ) ;  n ( z ) i = h z q n  1 ( z ) ;  n ( z ) i = h z q n  1 ( z ) ; z q n  1 ( z )  q n  1 ( z ) i = h q n  1 ( z ) ; q n  1 ( z ) i   h z q n  1 ( z ) ; q n  1 ( z ) i = h q n  1 ( z ) ; q n  1 ( z ) i   [ h  n ( z ) ; q n  1 ( z ) i +  h q n  1 ( z ) ; q n  1 ( z ) i ℄ = (1  j  j 2 ) h q n  1 ( z ) ; q n  1 ( z ) i ; and the result follows. Corollary 3.2 (Montaner & Alfaro [16℄) If  is a zero of   n ( z ) , then j  j > 1 . 16 Remark: Notie that, in the quasi-denite ase, we only an guarantee that j  j 6 = 1. Next, we will dene an absolutely ontinuous measure suh that the indued inner pro dut in P n agrees with the restrition to P n of our inner pro dut asso iated with the p ositive denite linear funtional. In order to do this, we need some preliminary result. Lemma 3.1 [8℄ Let  n be the n th orthonormal polynomial with respet to a positive denite linear funtional. Then, 1 2  2  Z 0  k ( e i )  j ( e i ) d j  n ( e i ) j 2 = Æ j;k ; 0  j  k  n < 1 : Pro of: Notie that 1 2  2  Z 0  n ( e i )  n ( e i ) d j  n ( e i ) j 2 = 1 ; (3.5) and, for j < n , 1 2  2  Z 0  n ( e i )  j ( e i ) d j  n ( e i ) j 2 = 1 2  2  Z 0 "  j ( e i )  n ( e i ) # d = 1 2  2  Z 0 e i ( n  j )    j ( e i )   n ( e i ) d = 1 2  i Z T z n  j  1   j ( z )   n ( z ) d z = 0 ; (3.6) b eause of the analytiity of the funtion in the last integral (see Corollary 3.2). Then,  n ( z ) is the n th orthonormal p olynomial with resp et to b oth, a p ositive linear funtional and the absolutely ontinuous measure d n = d j  n ( e i ) j 2 . Using the bakward reurrene relation (Theorem 3.3, (ii)) for the orthonormal ase, the p olynomials f  j g n  1 j =0 , whih are uniquely dened by this reurrene relation, are orthogonal with resp et to b oth, the linear funtional and the measure d  n . Thus, the result follows. Remark: In [8℄, an indution argument is used in order to prove the previous 17 result. Indeed, assuming that for a xed k  n , 1 2  2  Z 0  k ( e i )  j ( e i ) d j  n ( e i ) j 2 = Æ j k ; 0  j  k ; they proved that 1 2  2  Z 0  k  1 ( e i )  l ( e i ) d j  n ( e i ) j 2 = Æ k  1 l ; 0  l  k  1 : Notie that the n th orthogonal p olynomial denes in a unique way the previous ones; thus, the pro of of the seond statement (the indution) is not neessary. Of ourse, here we need not do this sine we are using the bakward reurrene relation for the orthogonal p olynomials  n . Notie also that the measure d n = d j  n ( e i ) j 2 denes a MSOP f  n g 1 n =0 suh that  m ( z ) = z m  n  n ( z ), for m  n , where  n is the moni p olynomial orresp onding to  n . Moreover, the sequene of reetion o eÆients orresp onding to this MSOP f  n g 1 n =0 is f  1 (0) ; :::;  n (0) ; 0 ; 0 ; ::: g . Usually, in the literature of orthogonal p olynomials, this measure d n is alled a Bernstein-Szego measure (see [25℄). In Setion 2, Theorem 2.10, we proved that the interlaing prop erty for the zeros of two p olynomials P n  1 and P n of degree n  1 and n , resp etively, means that they are the ( n  1)st and n th orthogonal p olynomials of a MSOP. Indeed, the three-term reurrene relation for a MSOP plays a entral role in the pro of. In the ase of the unit irle, we have an analogous result, whih is known in the literature as the Shur-Cohn-Jury riterion [4℄. Theorem 3.5 A moni polynomial p of degree n has its n zeros inside the unit irle if and only if the family of parameters f a k g n k =0 dened by the fol lowing bakward algorithm q n ( z ) = p ( z ) ; q n (0) = a n ; q k ( z ) = q k +1 ( z )  a k +1 q  k +1 ( z ) z (1  j a k +1 j 2 ) ; a k = q k (0) ; k = n  1 ; n  2 ; :::; 0 ; satises j a k j < 1 , k = 1 ; 2 ; :::; n: Pro of: Notie that the p olynomials f q k g n k =1 , q 0 = 1, satisfy a bakward re- urrene relation like the p olynomials orthogonal on the unit irle with trun- ated Shur parameters f a k g 1 k =1 . Beause f a 1 ; a 2 ; :::; a n ; 0 ; 0 ; ::: g is indued by 18 the measure d  n = d j q n ( e i ) j 2 = d j p ( e i ) j 2 , up to a onstant fator, then p = q n ( z ) is the n th moni orthogonal p olynomial with resp et to the measure d  n . A- ording to Prop osition 3.1 its zeros are lo ated inside the unit disk. Conversely, if the p olynomial p has its zeros inside the unit disk, then j a n j = j q n (0) j < 1. On the other hand, sine q n  1 ( z ) = q n ( z )  a n q  n ( z ) z (1  j a n j 2 ) ; if  is a zero of q n  1 with j  j  1, then q n (  ) = a n q  n (  ), and 0 < j q n (  ) j < j q  n (  ) j . This means that      q n (  ) q  n (  )      < 1, but this is in ontradition with the fat that the zeros of q n ( z ) are inside the unit disk and thus, by the maximum mo dulus priniple,      q n ( z ) q  n ( z )       1 if j z j < 1, whih is equivalent to      q n ( z ) q  n ( z )       1 for j z j  1. The same pro edure applied to all 1  k  n  2 leads to the result. Remark: The ab ove riterion is a very useful qualitative result in the stability theory for disrete linear systems [4℄. In fat, given the harateristi p olynomial of the matrix of a linear system, we do not need to alulate its zeros (the eigenvalues of the matrix) in order to prove that they are lo ated inside the unit disk, and then to prove the stability of the system. 3.3 The trigonometri moment problem revisited. Next, we an state our main result. Theorem 3.6 [8℄ Let f a n g 1 n =1 be a sequene of omplex numbers suh that j a n j < 1 , n = 1 ; 2 ; ::: . Let  0 ( z )=1 ;  n ( z ) = z  n  1 ( z ) + a n   n  1 ( z ) ; n  1 : Then, there exists a unique positive and nite Borel measure  supported on T suh that f  n g 1 n =0 is the orresponding MSOP. In other words, the positive denite linear funtional assoiated with the reetion oeÆients f a n g 1 n =0 an be represented as L [ p ( z )℄ = 2  Z 0 p ( e i ) d (  ) : 19 Pro of: Let  n (  ) =  Z 0 d n ( t ) =  Z 0 d t j  n ( e i t ) j 2 ; where  n denotes the n th orthonormal p olynomial with resp et to L . The funtion  n is monotoni inreasing in [0 ; 2  ℄ and aording to Lemma 3.1, j  n (  ) j  2  Z 0 d j  n ( e i ) j 2  2  d 0 < + 1 8 n 2 N ;  2 [0 ; 2  ℄ : From Helly's seletion priniple (see e.g. [5℄) there exists a subsequene f  n k g 1 n k =0 and a monotoni inreasing funtion  suh that lim n k !1  n k (  ) =  (  ). Furthermore, for every ontinuous funtion f on T , lim n k !1 1 2  2  Z 0 f ( e i ) d n k (  ) = 1 2  2  Z 0 f ( e i ) d (  ) : Finally, 1 2  2  Z 0  k ( e i )  j ( e i ) d (  ) = lim n l !1 1 2  2  Z 0  k ( e i )  j ( e i ) d n l (  ) = Æ j;k ; taking n l > max f k ; j g . To onlude the study of the p ositive denite ase, we will show an analog of Theorem 2.11 of Setion 2 in the following sense. Theorem 3.7 [1℄ Let f z n g 1 n =1 be a sequene of omplex numbers suh that j z n j < 1 . Then, there exists a unique sequene of moni polynomials  n orthogonal with respet to a positive denite moment funtional suh that  n ( z n ) = 0 . Pro of: Sine  1 ( z ) = z +  1 (0) = z  z 1 , then  1 (0) =  z 1 , and j  1 (0) j < 1. Using indution, assume that z n  1 is a zero of  n  1 and j  n  1 (0) j < 1. Let  n ( z ) = z  n  1 ( z ) +  n (0)  n  1 ( z ), for n > 1, and z n b e a zero of  n . Then, substituting z n in the ab ove expression, we dedue z n  n  1 ( z n ) =   n (0)  n  1 ( z n ) : But   n  1 ( z n ) 6 = 0 (otherwise z n would b e a zero of  n  1 , whih is a ontradition). Thus,  n (0) =  z n  n  1 ( z n )   n  1 ( z n ) ; but then j  n (0) j = j z n j       n  1 ( z n )   n  1 ( z n )      < j z n j < 1 ; sine      n  1 ( z n )   n  1 ( z n )     < 1 by the maximum mo dulus priniple (see the pro of of Theorem 3.5). Then, the sequene f z n g 1 n =1 denes uniquely a sequene of 20 omplex numb ers f a n g 1 n =1 , with a n =  n (0), and this sequene, aording to Theorem 3.6, uniquely denes a sequene of orthogonal p olynomials f  n g 1 n =0 with reetion parameters a n suh that  n ( z n ) = 0. In the quasi-denite ase, as we already p ointed out after Prop osition 3.1, if  n is the n th orthonormal p olynomial with resp et to a quasi-denite moment funtional L , then the p olynomials z  n ( z ) and   n ( z ) have no zeros in ommon. They are oprime, and by the Bezout identity [4℄, there exist p olynomials r ( z ) and s ( z ) suh that z r ( z )  n ( z ) + s ( z )   n ( z ) = 1 ; or, equivalently, if u ( z ) = z r ( z ), i.e., u (0) = 0, u ( z )  n ( z ) + s ( z )   n ( z ) = 1 : The next result is analogous to that stated in Lemma 3.1. Theorem 3.8 [3℄ There exists a unique real trigonometri polynomial f (  ) of degree at most n , suh that 1 2  2  Z 0  n ( e i ) e  ik  f (  ) d = 0 ; 0  k  n  1 ; (3.7) 1 2  2  Z 0     n ( e i )    2 f (  ) d = 1 ; (3.8) if and only if there exist u; v 2 P n , with u (0) = 0 , suh that u ( z )  n ( z ) + v ( z )   n ( z )=1 . Furthermore, f (  ) = j u ( e i ) j 2  j v ( e i ) j 2 : Pro of: If f satises (3.7) and (3.8), onsider the funtion g (  ) = f (  )  n ( e i ), whih is a trigonometri p olynomial of degree at most 2 n . The onditions mean that the Fourier o eÆients ^ g ( k ) of g (  ) are ^ g ( j ) = 0, j = 0 ; 1 ; :::; n  1, and ^ g ( n )   n (0) = 1. Then, there exist p olynomials u; v 2 P n , suh that u (0) = 0, v (0))   n (0) = 1 and g (  ) = e in v ( e i )  u ( e i ). In fat, u ( z ) =  n X j =1 ^ g (  j ) z j and v ( z ) = n X j =0 ^ g ( j + n ) z j : Now we intro due the trigonometri p olynomial of degree at most 3 n , h (  ) =  n ( e i ) f (  )  n ( e i ). Notie that h (  ) =   n ( e i ) v ( e i )  u ( e i )  n ( e i ) ; 21 and h is a real-valued funtion. Then,   n ( e i ) v ( e i )  u ( e i )  n ( e i ) =   n ( e i ) v ( e i )  u ( e i )  n ( e i ) ; or, equivalently, s (  ) = u ( e i )  n ( e i ) + v ( e i )   n ( e i ) 2 R : This means that the algebrai p olynomial of degree at most 2 n , q ( z ) = u ( z )  n ( z ) + v ( z )   n ( z ) ; is real-valued on the unit irle, and thus ^ q ( j ) = ^ q (  j ) = 0, i.e., q ( z ) = q (0) = u (0)  n (0) + v (0)   n (0) = 1 : This yields our result. Conversely, assume there exist p olynomials u; v 2 P n with u (0) = 0, suh that u ( z )  n ( z ) + v ( z )   n ( z )=1 : (3.9) Let f (  ) = v ( e i ) v ( e i )  u ( e i ) u ( e i ), a trigonometri p olynomial of degree at most n . We will prove that the orthogonality onditions (3.7) and (3.8) hold. Indeed, let g (  ) = f (  )  n ( e i ). Taking into aount (3.9), we have u ( e i )  n ( e i )+ v ( e i ) e  in  n ( e i ) = 1 ; i.e. ; e in = u ( e i )   n ( e i )+ v ( e i )  n ( e i ) : Then, using (3.9) as well as the last expression, we obtain g (  ) =  n ( e i ) h v ( e i ) v ( e i )  u ( e i ) u ( e i ) i = e in v ( e i )  u ( e i ) ; (3.10) whih yields our orthogonality onditions ^ g ( j ) = 0 ; j = 0 ; 1 ; :::; n  1 ; and ^ g ( n )   n (0) = 1 : In order to prove uniqueness of f , notie that if u; v 2 P n , satisfy (3.9) together with u (0) = 0, then f (  ) = u ( e i )  n ( e i ) f (  ) + v ( e i )   n ( e i ) f (  ). By (3.10), we get f (  )  n ( e i ) = e in v ( e i )  u ( e i ) ; and f (  )   n ( e i ) = v ( e i )  e in u ( e i ) : Thus, f (  ) = j v ( e i ) j 2  j u ( e i ) j 2 . The uniqueness of f follows from the uniqueness of u; v . 22 To onlude this setion, we will show with two simple examples how to nd the funtion f expliitly. Example 3.1 Let  3 ( z )=2 z 3 + 1. Notie that b eause the zeros are inside the unit irle, we are in a p ositive denite ase. Moreover,   3 ( z ) = z 3 + 2. Using the Eulidean algorithm for z  3 ( z ) and   3 ( z ), we nd 2 z 4 + z = 2 z ( z 3 + 2)  3 z ; and z 3 + 2 =  3 z (  1 3 z 2 ) + 2 : Thus, 1 6 z 2 (2 z 4 + z )+( z 3 + 2)( 1 2  1 3 z 3 ) = 1 ; and u ( z ) = 1 6 z 3 ; v ( z ) = 1 2  1 3 z 3 : Then f (  ) =    1 2  1 3 e 3 i    2  1 36 = 1 3 (1  os 3  ) = 1 6    e 3 i  1    2  0 : Example 3.2 Let  3 ( z ) = z ( z 2 + 4). Notie that now there are two zeros outside the unit irle. In this ase,   3 ( z )=4 z 2 + 1. An analogous pro edure leads to z  3 ( z ) = z 4 + 4 z 2 = 1 4 z 2 (4 z 2 + 1) + 15 4 z 2 ;   3 ( z ) = 16 15 ( 15 4 z 2 )+1 : Thus  16 15 z 2 ( z 2 +4)+( 4 15 z 2 + 1)(4 z 2 + 1) = 1 ; u ( z ) =  16 15 z 2 ; v ( z ) = 4 15 z 2 + 1 ; so f (  ) =    4 15 e 2 i  1    2  256 225 =  1 15 (1 + 8 os 2  ) ; whih gives rise to a nonp ositive ase, i.e., to a signed measure on [   ;  ℄. 4 The Favard Theorem for nonstandard inner pro duts. To onlude this work, we will survey some very reent results onerning the Favard theorem for Sob olev-typ e orthogonal p olynomials. First of all, we want to p oint out that the Favard Theorem on the real line an b e b e onsidered in a funtional-analyti framework as follows. Theorem 4.1 (Duran [6℄) Let P be the linear spae of real polynomials and B an inner produt on P . Then, the fol lowing onditions are equivalent: (1) The multipliation operator t , i.e., the operator t : P ! P , p ( t ) ! t p ( t ) , is Hermitian for B , that is, B ( t f ; g ) = B ( f ; t g ) for every polynomial f , g . 23 (2) There exists a nondisrete positive measure  suh that B ( f ; g ) = R f ( t ) g ( t ) d ( t ) . (3) For any set of orthonormal polynomials ( q n ) with respet to B the fol lowing three-term reurrene holds: tq n ( t ) = a n +1 q n +1 ( t ) + b n q n ( t ) + a n q n  1 ( t ) ; n  0 ; (4.1) with q  1 ( t )=0 , q 0 ( t )=1 and f a n g 1 n =0 , f b n g 1 n =0 real sequenes suh that a n > 0 for al l n . Notie that from the three-term reurrene relation (4.1) we get t 2 q n ( x ) = a n +2 a n +1 q n +2 ( t )+( b n +1 a n +1 + b n a n +1 ) q n +1 ( t ) +( a 2 n +1 + a 2 n + b 2 n ) q n ( t )+( a n b n + a n b n  1 ) q n  1 ( t ) + a n a n  1 q n  2 ( t ) ; i.e., the sequene f q n g 1 n =0 satises a ve-term reurrene relation, whih is a simple onsequene of the symmetry of the op erator t 2  t  t . Here we are interested in the onverse problem, whih is a natural extension of the Favard Theorem: To haraterize the real symmetri bilinear forms suh that the op erator t 2 is a Hermitian op erator. A nonstandard example of suh an inner pro duts is B ( f ; g ) = Z f ( t ) g ( t ) d ( t ) + M f 0 (0) g 0 (0) ; f ; g 2 P ; for whih t 2 is Hermitian, i.e., B ( t 2 f ; g ) = B ( f ; t 2 g ). Theorem 4.2 Let B be a real symmetri bilinear form on the linear spae P . Then the fol lowing onditions are equivalent: (1) The operator t 2 is Hermitian for B , that is, B ( t 2 f ; g ) = B ( f ; t 2 g ) for every polynomial f , g . (2) There exist two funtions  and  suh that B ( f ; g ) = Z f ( t ) g ( t ) d ( t ) + 4 Z f 0 ( t ) g 0 ( t ) d ( t ) ; (4.2) where f 0 and g 0 denote the odd omponents of f and g , respetively, i.e., f 0 ( t ) = f ( t )  f (  t ) 2 ; g 0 ( t ) = g ( t )  g (  t ) 2 : Moreover, if we put  n = R t n d ( t ) and  n = 4 R t n d ( t ) , then the matrix a n;k = 8 > < > :  n + k if n or k are even ;  n + k +  n + k otherwise ; 24 is positive denite if and only if B is an inner produt. In this ase the set of orthonormal polynomials with respet to an inner produt of the form (4.2) satises a ve-term reurrene relation t 2 q n ( x ) = A n +2 q n +2 ( t ) + B n +1 q n +1 ( t ) + C n q n ( t ) + B n q n  1 ( t ) + A n q n  2 ( t ) ; n  0 ; (4.3) where f A n g 1 n =0 , f B n g 1 n =0 , and f C n g 1 n =0 are real sequenes suh that A n 6 = 0 for al l n . Also we get a generalization of the Favard Theorem. Theorem 4.3 Let f q n g 1 n =0 be a set of polynomials satisfying the initial onditions q  1 ( t ) = q  2 ( t )=0 , q 0 ( t )=1 and the ve-term reurrene relation (4.3). Then, there exist two funtions  and  suh that the bilinear form (4.2) is an inner produt and the polynomials f q n g 1 n =0 are orthonormal with respet to B . Remark: The ab ove theorem do es not guarantee the p ositivity of the measures  and  . In fat in [6℄ some examples of inner pro duts of typ e (4.2) where b oth measures annot b e hosen to b e p ositive, or  is p ositive and  annot b e hosen to b e p ositive, are shown. All the previous results an b e extended to real symmetri bilinear forms suh that the op erator \multipliation by h ( t )", where h is a xed p olynomial, is Hermitian for B , i.e., B ( h f ; g ) = B ( f ; h g ). The basi idea onsists in the hoie of an adequate basis of P whih is asso iated with the p olynomial h . Assume that deg h = N , and let E h = span[1 ; h; h 2 ; ::: ℄; then P = E h  t E h      t N  1 E h : If  k denotes the pro jetor op erator in t k E h , then  k ( p ) = t k q [ h ( t )℄. We introdue a new op erator ~  k : P ! P , p ! q , where q denotes a p olynomial suh that  k ( p ) = t k q [ h ( t )℄. Then we obtain the following extension of Theorem 4.2: Theorem 4.4 Let B be a real symmetri bilinear form in P . Then the fol lowing statements are equivalent: (1) The operator \multipliation by h " is Hermitian for B , i.e., B ( h f ; g ) = B ( f ; h g ) for every polynomial f , g , where h is a polynomial of degree N . 25