PROPERTIES OF MULTIVARIATE HOMOGENEOUS ORTHOGONAL POLYNOMIALS Brahi Benouahane y Annie Cuyt? Keywords Abstract It is well-known that the denoinators of Pade approxiants can be considered as orthogonal polynoials with respect to a linear functional. This is usually shown by dening Pade-type approxiants fro socalled generating polynoials and then iproving the order of approxiation by iposing orthogonality conditions on the generating polynoials. In the ultivariate case, a siilar construction is possible when dealing with the ultivariate hoogeneous Pade approxiants introduced by the second author. Moreover it is shown here, that several wellknown properties of the zeroes of classical univariate orthogonal polynoials, in the case of a denite linear functional, generalize to the ultivariate hoogeneous case. For the ultivariate hoogeneous orthogonal polynoials, the absence of coon zeroes is translated to the absence of coon factors. y This research was carried out while the author was visiting the University of Antwerp as a guest researcher? Research Director FWO-Vlaanderen, Departent of atheatics and coputer science, Universiteit Antwerpen (UIA), Universiteitsplein, B{6 Wilrijk, Belgiu, eail: cuyt@uia.ua.ac.be
. The univariate situation. It is well-known for a long tie that denoinators of Pade approxiants can be considered as orthogonal polynoials with respect to a linear functional. This is usually shown by dening Pade-type approxiants fro so-called generating polynoials and then iproving the order of approxiation by iposing orthogonality conditions on the generating polynoials. Assue you are given a series developent f(t) By dening the linear functional c acting on the space of univariate polynoials, as c i t i f(t) can forally be rewritten as c(x i ) c i f(t) c? xt Now take any polynoial V (t) of degree and dene its associated polynoial V (x)? V (t) W (t) c x? t which is then a polynoial of degree?. Then for the Pade-type approxiation conditions ~V (t) t V (t? ) ~W (t) t? W (t? ) (f ~ V? ~ W )(t) i d i t i hold. If we do not choose V randoly, but ipose the conditions c(x i V (x)) i ; : : : ;? () then V (t) is called the orthogonal polynoial of degree with respect to the functional c (the conditions under which V can be coputed fro () are well-known [3]). For V (t) satisfying () the Pade approxiation conditions (f ~ V? ~ W )(t) hold (here V (t) is noralized such that it is onic). The Pade-type and Pade approxiants for f of degree? in the nuerator and in the denoinator are usually denoted by (? ) f and [? ] f respectively. i d i t i
The construction of the Pade-type and Pade approxiants ( + k) f and [ + k] f with k? explained next, follows the sae lines. The series for f(t) can be written as with If we dene the functional c by and the polynoials then W f(t) k f k (t) c i t i + t k+ f k (t) c k++i t i c (x i ) c k++i (t) c V (x)? V (t) x? t ~W (t) V ~ (t) k (f ~ V? ~ W )(t) c i t i + t k+ t? W (t? ) i+k+ and so everything reains valid with the functional c replaced by c. Note that ( + k) f W (t) ~ (t) ~V k d i t i c i t i + t k+ (? ) f k (t) The additional conditions on V necessary for the construction of the Pade approxiant [ + k] f are and we shall denote polynoials V satisfying () by V c (x i V (x)) i ; : : : ;? () so that [ + k] f ~ W V ~. The ratio W ~ V ~ which was introduced for the special case k?, will usually be denoted by W ~ () V ~ (). The sae holds for the functional c that can be denoted by c (). 3
. The ultivariate hoogeneous situation. In the ultivariate case, a siilar construction is possible to obtain the ultivariate hoogeneous Pade approxiants [ + k] f H introduced by Cuyt in [5]. We give a dierent and slightly ore elegant presentation than the one in [, 8, ]. We restrict our description to the bivariate case only for the reason of notational siplicity. Assue you are given a bivariate series developent f(t; s) i;j c ij t i s j For copleteness we repeat that the ultivariate hoogeneous Pade approxiant [? ] f H is dened as the irreducible for of P?; Q?; with P?; (t; s) Q?; (t; s) (fq?;? P?; ) (t; s) (?)+? i+j(?) (?)+ i+j(?) a ij t i s j b ij t i s j i+j(?)+ d ij t i s j (3) Note that the nuerator and denoinator polynoials P?; (t; s) and Q?; (t; s) start with ters of degree (? ) instead of with a constant ter. When coputing [? ] f H, in other words taking the irreducible for of P?; Q?;, the nuerator and denoinator polynoials of [? ] f H ay start with a constant ter but still this is not guaranteed. If we denote the order of the denoinator polynoial of [? ] f H (lowest hoogeneous degree of its ters) by then (? ) and we can show that the order of the nuerator polynoial of [? ] f H is at least. By dening the linear functional C acting on the space of bivariate polynoials, as i + j C(x i y j ) c ij j the bivariate series can forally be rewritten as f(t; s) C? xt? ys 4
By introducing the notations (t; s) ( u; u) t; s; u C ( ; ) C jjjj p C i (t; s) c i () a i () b i () d i () i j i j i j i j i j c i?j;j t i?j s j c i?j;j i?j j a i?j;j i?j j b i?j;j i?j j d i?j;j i?j j where jj:jj p is one of the Minkowski-nors on C, we can rewrite the series developent as and (3) as f(t; s) C i (t; s) c i ()u i P?; ( u; u) Q?; ( u; u) (fq?;? P?; ) ( u; u) (?)+? i(?) (?)+ i(?) i(?)+ With the introduction of the functional? acting on the variable z, as a i ()u i b i ()u i d i ()u i (4) the series can now forally also be viewed as?(z i ) c i () f(t; s) f( u; u)?? zu This new view on the ultivariate proble in which the cartesian coordinates (t; s) are replaced by the coordinates ( ; ) and u, with jjjj, will turn out to be a powerful tool in the sequel of the text. It is strongly linked to the following two features of hoogeneous ultivariate Pade approxiants: 5 (5)
the ost striking eleent in the denition (4) of the hoogeneous ultivariate Pade approxiant [ + k] f H is that this denition coincides with that of the univariate Pade approxiant if you discard the shift in the nuerator and denoinator degrees and replace the hoogeneous expressions by onoials [4, 6]; the hoogeneous Pade approxiants apparently satisfy a very strong projection property that we want to exploit here, reducing to univariate Pade approxiants on every straight line through the origin [6], naely [ + k] f H ( t; t) [ + k] f ; (t) with f ; (t) f( t; t) In order to clarify the presentation and underline the siilarity with the univariate case we again start with the construction of the hoogeneous Pade-type approxiants (?) f H and shall only afterwards deal with the ore general ( + k) f H. Let us rst introduce soe notations. We denote by C [u] the linear space of polynoials in the variable u with coplex coecients, by C [ ; ] the linear space of bivariate polynoials in and with coplex coecients, by C ( ; ) the coutative eld of rational functions in and with coplex coecients, by C ( ; )[u] the linear space of polynoials in the variable u with coecients fro C ( ; ) and by C [ ; ][u] the linear space of polynoials in the variable u with coecients fro C [ ; ]. For chosen and dened as above, take any function V (t; s) of the for V (t; s) V (u) B +?i() B +?i()u i +?i j b +?i?j;j +?i?j j which is a polynoial of degree in u with hoogeneous polynoial coecients fro C [ ; ] and dene V (z)? V (u) W (t; s) W (u)? z? u which is then of the for W (t; s) W (u) A +??i()???i j A +??i()u i B +??i?j()c j () Note that V (t; s) and W (t; s) do not necessarily belong to C [t; s] anyore because the hoogeneous degree in and doesn't equal the degree in u. Instead they belong to C [ ; ][u]. In the sequel of the text we will use both the notations V (t; s) and V (u) interchangeably to refer to (6a) and analogously 6 (6a) (6b)
for (6b). For ~V (t; s) ~ V (u) u + V (u? ) B +i()u +i +i j ~W (t; s) ~ W (u) b +i?j;jt +i?j s j u +? W (u? )?? A +i()u +i +i j a +i?j;jt +i?j s j the Pade-type approxiation conditions f V ~? W ~ (t; s) f V ~? W ~ (u) i+ i+ d i ()u i @ i d i?j;j t i?j s j A hold, where as in (6) the subscripted function d i () is a hoogeneous function of degree i in and. We reark here that ~ V (t; s) and ~ W (t; s) again belong to C [t; s] contrary to V (t; s) and W (t; s). As in (), if the function V (t; s), say the polynoial V (u), is not chosen randoly, but if it satises the additional orthogonality conditions j?(z i V (z)) i ; : : : ;? (7) then the Pade approxiation conditions f V ~ (t; s)? W ~ (t; s) f V ~? W ~ (u) i+ i+ d i ()u i @ i d i?j;j t i?j s j A are satised and W ~ (u)~ V (u) equals the hoogeneous Pade approxiant [? ] f H []. As in the univariate case the orthogonality conditions (7) only deterine V (u) up to a kind of noralization: 7 j
+ polynoial coecients B +?i() ust be deterined fro conditions. How this is solved, is explained below. With the c i () we now dene the polynoial Hankel deterinants H () () c () c? ().... c (). c? () c? () H () () generalizing the classical Hankel deterinants as dened in [7]. We also call the functional? denite if H () () 6 In the sequel of the text we shall assue that V (u) satises (7) and that? is a denite functional. Also we shall assue that V (u) as given by (6a) is priitive, eaning that its polynoial coecients B +?i() are relatively prie. This last condition can always be satised, because for a denite functional? a solution of (7) is given by [] V (u) p () () c () c? () c ().... c + () c? () c? () u u V (u) (8) where the polynoial p () () is a polynoial greatest coon divisor of the polynoial coecients of the powers of u. Clearly (8) copletely deterines V (u) and consequently V (t; s). As in the univariate situation the functional? can be dened by? (z i ) c k++i () and? can be replaced by? for the construction of hoogeneous Pade approxiants [ + k] f H with k?. The shift in the nuerator and denoinator degrees of [ + k] f H then satises ( + k) and the nuerator and denoinator of [+k] f H are respectively denoted by W ~ (t; s) W ~ (u) and V ~ (t; s) V ~ (u). We denote by p () a polynoial greatest coon divisor of the polynoial coecients of u i ; i ; : : : ; in the deterinant c k+ () c k+ () c k++ ().... c k++ () c k+ () c k+ () u u 8
and identify W V and? respectively with W () V () and? (). It is then easy to check that for V given by V (u) p () c k+ () c k+ () c k++ ().... c k++ () c k+ () c k+ () u u V (u) (9) one has? u V (u)? B @ p () p () H + () p () c k+ () c k+ () c k++ ().... c k++ () c k+ () c k+ () u u + u c k+ () c k+ () c k++ ()..... c k+ () c k+ () c k++ () c k++ () c k++ () C A () To conclude this section we suarize the ost iportant results. Suary: (a) For the bivariate series f(t; s) and for k? holds [ + k] f H (t; s) ~ W (t; s) ~V (t; s) (b) For the onic univariate polynoial V (u) satisfying () and for the bivariate polynoial V (t; s) V (u) given by (8) with (t; s) ( u; u) holds H () ( ; )V (u) p () ( ; )V ( u; u) p () ( ; )V (u) This last property can be seen as a projection property. 9
3. Properties of the hoogeneous orthogonal polynoials. Let us now generalize the well-known univariate property [3, p. 57] that for a denite functional c as in () the polynoials V (t) and V + (t) have no coon zeroes. The sae is true in the univariate case for the polynoials W (t) and W + (t), and the polynoials V (t) and W (t). Before we can forulate the ultivariate generalization, we rst need a nuber of leas and theores. In the ultivariate discussion we shall often switch between the coordinates (t; s) and the coordinate u in the one-diensional subspaces spanned by the vectors. Reeber that V (t; s) V (u) and W (t; s) W (u) do not belong to C [t; s] but to C [ ; ][u]. Lea : Let the functional? which is dened for k? be denite and let the polynoials fv (u)g satisfy (7). Then the fv (u)g are linearly independent in C ( ; )[u]. Proof: Suppose we have coecients (); (); : : : C ( ; ) such that 8u C : i ()V i (u) Then we also have for j that i ()? u j V i (u) Taking (7) into account, we obtain for j For j this reduces to j i ()? u j V i (u) ()? V (u) which results in () because? V (u) that j () is by induction. Theore : Let the functional? and p recurrence relations c k+ () H () 6. For j > the proof which is dened for k? be denite and let the polynoials V (u) (u)g and fw (u)g satisfy the () be dened as in (9). Then the polynoials fv V + (t; s) + () W + (t; s) (u? + ())V V? (t; s) V (t; s) + () (u? + ())W W? (t; s)? W (t; s) (t; s)? + ()V? (t; s) (t; s)? + ()W? (t; s)
with + p () H () p + () + ()? u + () () h V (t; s) H? hv (t; s) i i +? () () p p () H + () H () () c k+ () Proof: The polynoial uv (u) uv (t; s) as dened in (6a) can be written as a linear cobination where the i V j + uv (t; s) i ()V i (t; s) () are rational functions of the variable. We ultiply left and right hand side with (t; s) and apply the linear functional? to obtain On the other hand we have so that consequently i () i ; : : : ;?? ()? () Using () and the fact that + ()? V uv? (t; s)v (t; s)? (V? (t; + s)) (t; s))? u(v? (V uv (t; s)) (t; s)v + (t; s) + ()? (V + (t; s)) (u) V H () p ()? (V (u)) (t; s) H () p () u + : : : + () H H () + () p + () p ()? u V (u)
the expression for + () is obtained. For the associated polynoials W (u) we have, because? V (z), W + (u) + ()? (u? + ())V (u)? V (z) u? z? + which gives the desired result. The starting value for () is easy to verify. p? ()V (u)? V? (z) u? z Theore : Let the functional? which is dened for k? be denite and let the polynoials V () be dened as in (9). Then the polynoials fv (u)g and fw V (u)w + (u)? W (u)v + (u) V (t; s)w h H + () i + p ()p + () Proof: For siplicity we oit writing the arguents (t; s) in V. The proof akes use of the previous recurrence relations: W V + + V W + + By subtracting these expressions one obtains V W +? W (u? + (u? + V + + + : : : h H + i p p + and W (t; s)? W (u) and (u)g satisfy the identity (t; s)v + (t; s) and () in ; and )V W? + V? W )W V? + W? V V W? W V Let us now take a closer look at the factorisation of the orthogonal polynoials V (u) and their associated polynoials W (u) in irreducible factors. This factorisation is unique in C [ ; ][u] except for ultiplicative constants fro C which are the unit ultiples in C [ ; ] and except for the order of the factors. This is because C [ ; ][u] is a unique factorization doain. Theore 3: Let the functional? which is dened for k? be denite and let the polynoials V () be dened as in (9). Let W (u) be given by (6b). Then p (a) V (b) W (c) V (u) and V + (u) and W + (u) and W (u) have no coon factor (u) have no coon factor (u) have no coon factor (u) and Proof: We only give the proof for (a) since the proof for (b) and (c) is copletely siilar. The proof is by!
contradiction. Assue that V (u) and V + (u) have a coon factor. Then, because of theore, it is necessarily a polynoial in, dierent fro a coplex constant if it is a true coon factor. Hence the polynoials V (u) and V + (u) are not priitive, which is a contradiction. Let us now restrict ourselves to all variables and coecients being real and turn to soe results for positive denite functionals. The functional? is called positive denite if 8 IR : H () > Lea : For a positive denite functional? and for any polynoial P(u) IR[ ; ][u] holds?? P (u) > where the functional? acts on the variable u as dened above. Proof: Every polynoial P(u) of degree in IR[ ; ][u] can be written in the for P(u) i ()V i (u) where the i () IR( ; ) are rational functions of the variable with real coecients. Fro the orthogonality conditions satised by V (u) we obtain?? P (u) i ()? (V i (u)) i () H j ()H j+ () p j > Theore 4: For a positive denite functional?, the polynoials V factors in IR[ ; ][u] of ultiplicity larger than. () (u) satisfying (7) have no irreducible Proof: Assue V (u) has an irreducible factor F(u) of ultiplicity ` >. Then we can write V (u) F `(u)z(u) where Z(u) is a polynoial in u of degree?` @ u F < where @ u F is the degree of F(u) as a polynoial in u. If ` > is even then because of lea? Z(u)V (u)?? Z (u)f `(u) which is ipossible because of the orthogonality conditions satised by V. If ` > is odd then which is also a contradiction.? F(u)Z(u)V (u) >?? Z (u)f `+ (u) > 3
4. Coon zeroes instead of coon factors. Fro the previous section it is clear that our orthogonal polynoials fv (u)g IN do not have any irreducible factors in coon in C [ ; ][u]. Since each of these irreducible factors uniquely deterines a zero curve, it is also clear that the fv (t; s)g IN do not have any zero curves in coon. But since their coecients belong to the unique factorization doain C [ ; ], we can use a well-known theore to detect isolated zeroes for which for instance V vanish siultaneously. (u) V (t; s) and V n Lea 3: Let the functional? which is dened for k? be denite. Let the polynoials satisfy (7). Then V R() V (u) v i ()u i (u) and V n (u) have a coon zero for ( ; ) satisfying () : : : v ()...... v () : : : v () n () : : : v nn ()...... v n () : : : v nn () v v 9 > >; n ties 9 > >; ties (u) V n (t; s) Proof: The (n + ) (n + ) deterinant R() is the resultant of the polynoials V (u) and V n (u) and this proves the lea [9, pp. 3{3]. We can illustrate this procedure with a siple exaple. Consider the functional The orthogonal polynoials V () V () (u)? ( + ) + u? () (z i ) c i () i! (i + i? + : : : + i ) () (u) and V (u) satisfying (7) are then given by V () (u)? (4 + 3 + 5 + 3 + 4 )+ + 6 (3 + 5 + 5 + 3 )u? ( + 3 + )u The resultant of V () () (u) and V (u) equals R()? 64 (3 + p 5) + (3? p 5) + 4
Consequently V () (t; s) V () () (u) and V (t; s) V () (u) have a coon zero for satisfying ( R() This is for and or in ters of t and s, for and () () @? 3 @? 3 jjjj s s s 6 3 + p 5 ; s 6 3? p 5 ; 3 + p 5A 6 3? p 5A 6 (t; s) (t () ; s () )? + p 5 9 + 3 p 5 ; + p 5 6 (t; s) (t () ; s () )?? p 5 9? 3 p 5 ;? p 5 6!! Let us at the sae tie illustrate that W ~ () (u) V ~ () (u) W ~ () (t; s) V ~ () (t; s) equals the hoogeneous Pade approxiant [] f H for the series f(t; s) Fro V () () (u) V (t; s) we copute ~V () (t; s) u 4 ~ V () (u? ) c i ()u i t exp(t)? s exp(s) t? s i;j? (t4 + t 3 s + 5t s + ts 3 + s 4 )+ (i + j)! ti s j + 6 (t3 + 5t s + 5ts + s 3 )? (t + 3ts + s ) and (t; s)? V () () (z)? V (u) z? u W () ~W () (t; s) u 3 W () (u? )!? (t + 3ts + s )? 6 (t3 + 7t s + 7ts + s 3 ) to obtain [] f H. 5
References [] S. Arioka. Pade-type approxiants in ultivariables. Appl. Nuer. Math., 3:497{5, 987. [] B. Benouahane. Approxiants de Pade \hoogenes" et polyn^oes orthogonaux a deux variables. Rend. Mat. (7), :673{689, 99. [3] C. Brezinski. Pade type approxiation and general orthogonal polynoials. ISNM 5, Birkhauser Verlag, Basel, 98. [4] A. Cuyt. A coparison of soe ultivariate Pade approxiants. SIAM J. Math. Anal., 4:95{, 983. [5] A. Cuyt. Pade approxiants for operators: theory and applications. LNM 65, Springer Verlag, Berlin, 984. [6] A. Cuyt. How well can the concept of Pade approxiant be generalized to the ultivariate case? J. Coput. Appl. Math., 5:5{5, 999. [7] P. Henrici. Applied and coputational coplex analysis I. John Wiley, New York, 974. [8] S. Kida. Pade-type and Pade approxiants in several variables. Appl. Nuer. Math., 6:37{39, 89/9. [9] R.J. Walker. Algebraic Curves. Dover Publications, New York, 95. 6