THE ALGEBRA OF LINEAR FUNCTIONALS ON POLYNOMIALS, WITH APPLICATIONS TO PADE APPROXIMATION C. BREZINSKI
AND P. MARONI
y
Abstract. Some results about the algebra of linear functionals on the vector space of complex polynomials are given. These results have applications to Pade{type and Pade approximation. In particular an expression for the relative error is obtained. Key words. Orthogonal polynomials, Pade approximation. AMS(MOS) subject classi cations. 42C05, 41A21.
1. Introduction Let (ci) be a given sequence of complex numbers. We de ne the linear functional c on the space P of complex polynomial functions by
c(xi) = ci for i = 0; 1; : : : : In the sequel we shall make use of the notation of duality, that is we shall set c(xi) =< c; xi > : The numbers ci are called the moments of the functional c and the formal series f (t) =
1 X i=0
ci t i
its generating series. Since (1 ? xt)?1 = 1+ xt + x2t2 + , we shall make use of the notation f (t) =< c; (1 ? xt)?1 >. For a rigorous justi cation of this notation and for an extensive study of this formalism, see [11]. The family of polynomials fPk g, with Pk of exact degree k, such that, 8k
< c; xiPk (x) > = 0 for i = 0; : : : ; k ? 1 < c; xk Pk (x) > 6= 0 is called the family of formal orthogonal polynomials with respect to c. Each polynomial Pk is determined apart from a nonzero multiplying factor. Such polynomials satisfy most of the properties of the usual orthogonal polynomials on the real line (that is when c can be represented as an integral with respect to some positive measure) and, in particular, they satisfy a three{term recurrence relationship (in the regular case, that is when all the polynomials exist) and a Christo el{Darboux identity. See [3] for an account of these polynomials. Laboratoire d'Analyse Numerique et d'Optimisation, UFR IEEA - M3, Universite des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq cedex { France, e-mail:
[email protected] y Laboratoire d'Analyse Num erique, Universite Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris cedex 05 { France 1
Let Vk be an arbitrary polynomial of exact degree k and let P be the Hermite interpolation polynomial of the function x 7?! (1 ? xt)?1 at the zeros of Vk . Then < c; P (x) > is a rational function with a numerator of degree k ? 1 and a denominator of degree k such that its power series expansion (obtained by division in ascending powers of t) agrees with that of the series f as far as possible, that is up to the degree k inclusively. We set < c; P (x) >= (k ? 1=k)f (t). Such a rational approximant is called a Pade{type approximant [2] and we have, by construction (k ? 1=k)f (t) ? f (t) = O(tk ):
Vk is called the generating polynomial of the approximant. If we take Vk Pk , the preceding orthogonal polynomial, then < c; P (x) > is called a Pade approximant of f , it is denoted by [k ? 1=k]f (t) and we now have [k ? 1=k]f (t) ? f (t) = O(t2k): It is possible to construct Pade{type and Pade approximants with arbitrary degrees in the numerators and the denominators. Such approximants have very many applications in numerical analysis, applied mathematics and various other domains [1, 3, 6]. Thus, linear functionals play an important r^ole in the theory of such approximants and in that of formal orthogonal polynomials. This idea was exploited in [3] where it was showed, among other results, how to compute recursively any sequence of Pade approximants. The non{de nite case was extensively treated in [7]. Later, the algebra of linear functionals on P was put on a rm theoretical basis by Maroni in a series of papers [8, 9, 10, 11, 12], and by his students in the context of orthogonal polynomials. It was also used in [5] for showing its e ectiveness in Pade approximation problems. The aim of this paper is to exploit this approach more systematically.
2. The algebra of linear functionals
Let us begin by reminding some de nitions and giving some results about the algebra of linear functionals on P . 2.1 Multiplication of a functional by a polynomial The right{multiplication of a functional by a polynomial is a polynomial. The value of the polynomial cxn at the point t is, by de nition (cxn) (t) =
n X i=0
cn?i ti =< c; (tn+1 ? xn+1)(t ? x)?1 > :
We make the convention that ci = 0 if i < 0. If p(x) = a0 + + ak xk , then we have k k X X j (cp) (t) = t aici?j j =0 i=j
=< c; (tp(t) ? xp(x))(t ? x)?1 > : 2
(1)
2.2 Product of two functionals The product of two linear functionals c and d is a linear functional [10]. If we denote by c~ the application from P into P such that c~ : xn 7?! cxn, then the transpose of c~ allows to de ne the product cd as < cd; xn > = < c; dxn > n X = < c; dn?i xi > = =
Thus cd = dc
n X i=0 n X
i=0
cidn?i = (cd)n < c; xi > : < d; xn?i > :
i=0 Pn since = i=0 dicn?i . It follows that < cd; (1 ? xt)?1 >=< c; (1 ? xt)?1 > : < d; (1 ? xt)?1 Pn i=0 ci dn?i
> which means that the Cauchy product of two generating series is equal to the generating series of the product of the two linear functionals. is the unit element for this product since, from (1), (p) (x) = p(x) for every polynomial p and < c; p >=< c; p >=< c; p >. 2.3 Inverse of a functional If c and d are two linear functionals such that cd = , it follows immediately that d = c?1. exists if and only if c0 6= 0 [10]. Let us now show how to compute the moments of c?1. We have
c?1
((cd)xn ) (t) = Thus d = c?1, the inverse of c, if
n X
(cd)n?i ti:
i=0 ((cd)xn) (t) = tn that
is if
(cd)0 = 1; (cd)n = 0; 8n 1 or, in other words, if n X i=0
c0d0 = 1; cidn?i = 0; 8n 1:
Thus, c?1 exists if and only if c0 6= 0. Let now
g(x) =
1 X i=0
3
dixi
be the formal reciprocal series of f de ned by f (x)g(x) = 1. It is easy to see that the coecients di of g are given by the same relations as those de ning the moments of c?1. Thus, c?1 is the linear functional whose moments are the coecients of the reciprocal series g of f . We have
< c; (1 ? xt)?1 > : < c?1; (1 ? xt)?1 >=< cc?1; (1 ? xt)?1 >= 1:
2.4 Multiplication of a polynomial by a functional by
The left{multiplication of a functional c by a polynomial p is the linear functional de ned
< p(x)c; xn >=< c; xnp(x) > : It must be noticed that p(x)(cd) 6= (p(x)c) d. Let q(x) = a0 + a1x + a2x2 + + ak xk , we have < q(x)(cd); p(x) > = < (a0 + a1x + + ak xk )(cd); p(x) > = a0 < cd; p(x) > +a1 < x(cd); p(x) > +a2 < x2(cd); p(x) > + + ak < xk (cd); p(x) > : But, for n 1 < xn(cd); p(x) > = < cd; xnp(x) >=< x(cd); xn?1p(x) > = < (xc)d; xn?1p(x) > +c0 < xd; xn?1p(x) > : Thus, it follows that < xn (cd); p(x) >=< xc; dxn?1 p(x) > +c0 < d; xnp(x) >; which can also be written as xn(cd) = xn?1(xc)d + c0xnd: By linear combination of this relation for various values of n, and de ning x?1c by < x?1c; p >=< c; (p(x) ? p(0))x?1 > we obtain < q(x)(cd); p(x) >=< (xc)d; x?1p(x)q(x) > +c0 < d; p(x)q(x) >; which can also be written as cd = x?1(xc)d + c0d: Thus, the relations given above are also valid for n = 0 which can also be checked directly by using the properties given in [12]. 4
We have, 8n
< x?1(cd); xn > = < cd; xn?1 >= c0dn?1 + + cn?1 d0; n X < (x?1c)d; xn > = < x?1c; dxn >=< x?1c; dn?i xi > i=0 = dn c?1 + dn?1 c0 + + d0cn?1: Since c?1 = 0 and by the commutativity of c and d, it follows that x?1(cd) = (x?1c)d = (x?1d)c:
2.5 Expression of the product of two functionals Let us now prove a result that will be useful later. We have < (xk c)d; xi > = < xk c; dxi >=< xk c; di + + d0xi > = < c; dixk + + d0xi+k >= dick + + d0ci+k = < d; ci+k + + ck xi > : But, for k 1 ci+k + + ck xi = ck+i + + ck xi + ck?1xi+1 + + c0xi+k ? (ck?1 xi+1 + +c0xi+k ) = ci+k (x) ? xi+1 ck?1 (x): Thus, we nally obtain, for k 1 < (xk c)d; xi >=< d; ci+k (x) > ? < d; xi+1 ck?1 (x) > : The formulae above still remain valid for k = 0 since the last terms in the right hand sides are zero. 2.6 Associated polynomials Let fPk g be the family of orthogonal polynomials with respect to c. Their associated polynomials fQk g are de ned by ! P k (x) ? Pk (t) Qk (t) = c : x?t Qk is a polynomial of exact degree k ? 1. These polynomials satisfy the same recurrence relationship as the polynomials Pk but with di erent initializations. Their properties can be found in [3, pp. 105 .]. In particular, we have < c?1; xi+2Qk (x) >= 0 for i = 0; : : : ; k ? 2 that is < x2c?1; xiQk (x) >= 0 for i = 0; : : :; k ? 2; which shows that the family fQkg is orthogonal with respect to the linear functional x2c?1. Another proof of this result can be found in [4]. 5
3. Application to Pade approximation In the introduction, we saw that (k ? 1=k)f (t) =< c; P (x) > where P is the Hermite interpolation polynomial of x 7?! (1 ? xt)?1 at the zeros of a given polynomial Vk of exact degree k called the generating polynomial of the approximant (k ? 1=k). In [5], an expression for the linear functional d(Vk ), depending on Vk , such that (k ? 1=k)f (t) =< d(Vk ); (1 ? xt)?1 > was given. We shall now look for the linear functional e such that
d(Vk ) = ce: Let us mention that, since all the Pade{type and Pade approximants with a numerator and a denominator of arbitrary degrees can be constructed from the approximants (k ? 1=k), it is enough to consider these approximants. We have
di =< ce; xi >=< c; exi >= e0ci + + eic0; that is
d0 = e0c0 d1 = e0c1 + e1c0
dk?1 = e0ck?1 + + ek?1c0 dk+i = e0ck+i + + ek?1ci+1 + ek ci + + ek+ic0: But di = ci for i = 0; : : : ; k ? 1 and it follows that e0 = 1; e1 = 0; : : : ; ek?1 = 0. Moreover, if Vk Pk (that is, in the Pade case), then ek = = e2k?1 = 0. Let e(t) be the formal generating series of the ei's de ned by
e(t) = 1 + ek tk + ek+1tk+1 + : In the Pade case, we have
e(t) = 1 + e2kt2k + e2k+1t2k+1 + : Since < ce; (1 ? xt)?1 >=< c; (1 ? xt)?1 > : < e; (1 ? xt)?1 >, we have (k ? 1=k)f (t) = f (t)e(t): It must be noticed that the linear functional e depends on k and Vk . 6
Let us now show how to compute the ei's. It is known (see [5, prop.5]) that 8i 0; < d(Vk ); xiVk (x) >= 0 that is < ce; xiVk (x) >=< c; exiVk (x) >= 0. If we set Vk (x) = v0 + + vk xk , then exiVk (x) = v0exi + + vk exi+k = v0
i X
j =0
ei?j xj + + vk
i+k X
ei+k?j xj
j =0 i x ) + + v
k+1 + e xk + + e xi+k ) = v0(ei + + e0 k (ei+k + + ei?1x i 0 i k ? 1 = e0x Vk (x) + + eiVk (x) + ei+1(v1 + + vk x ) + + ei+k (vk ): With the convention that < c; xi >= 0 for i < 0, we see that v1 + + vk xk?1 can be represented by < c; x?1Vk (x) >, that v2 + + vk xk?2 can be represented by < c; x?2Vk (x) >, and so on. An important point that must be clearly understood is that such representations must not be used in the course of a proof since they do not possess the properties of the usual duality bracket as discussed above. They are only a more compact notation used for simplifying the computations. With this notation, it follows that < c; exiVk (x) > = < c; (e0xi + + ei + ei+1x?1 + + ei+k x?k )Vk (x) > = < c; x?k (ei+k + + e0xi+k )Vk (x) >= 0 and nally we have < c; (ei+k x?k + + e0xi)Vk (x) >= 0: Since e0 = 1; e1 = = ek?1 = 0 and < c; x?k Vk (x) >= vk c0, we obtain 3
2
i?1 X 1 i 4 ei+k = ? v c < c; x Vk (x) > + ek+j < c; xi?k?j Vk (x) >5 k 0 j =0
with the convention that the sum is zero if i = 0. Of course, it is necessary to assume that c0 6= 0. vk 6= 0 since Vk has exactly the degree k. If < c; xiVk (x) >= 0 for i = 0; : : :; k ? 1 (that is, in the Pade case) then the preceding relation when i = 0 leads to ek = 0. When i = 1, we obtain ek+1 = 0 and so on until i = k ? 1 which gives e2k?1 = 0. Thus a necessary and sucient condition that < c; xiVk (x) >= 0 for i = 0; : : : ; k ? 1 is that ek+i = 0 for i = 0; : : : ; k ? 1. Pade{type approximants of higher order (that is such that (k ? 1=k)f (t) ? f (t) = O(tk+m) with 0 m k ? 1, see [3, pp. 32 .]) can also be taken into account in a similar way. We have (k ? 1=k)f (t) ? f (t) = f (t)(ek tk + ek+1tk+1 + ) = f (t)tk < e; xk(1 ? xt)?1 >; that is < ce ? c; (1 ? xt)?1 > = tk < c; (1 ? xt)?1 > : < e; xk (1 ? xt)?1 > = tk < (xk e)c; (1 ? xt)?1 > 7
which is an expression for the relative error in Pade{type approximation. We have
< ce ? c; (1 ? xt)?1 >= tk < (xk e)c; 1 + xt + + xk?1tk?1 + xk tk (1 ? xt)?1 > : Thus, in order to increase the order of approximation, we have to nd a linear functional e (that is, a generating polynomial Vk ) such that
< (xk e)c; xi >= 0 for i = 0; : : : ; k ? 1: By using the relation proved at the end of subsection 2.5, it is easy to see that these conditions are equivalent to the conditions < c; xiVk (x) >= 0 for i = 0; : : : ; k ? 1 which are the usual orthogonality conditions. Thus Vk is identical to the orthogonal polynomial Pk and the Pade approximants are recovered. Let us now nd a closed expression for the series e. We have e(t) = 1 + tk gk (t) with gk (t) = ek + ek+1t + . Using the above expressions for ek ; ek+1; : : :, we obtain
?c0vk (ek +ek+1t+ ) =< c; Vk (x)(1?xt)?1 > +t < c; x1?k Vk (x)(1?xt)?1 > (ek +ek+1t+ ); that is
h
i
? < c; Vk (x)(1 ? xt)?1 >= gk (t) c0vk + t < c; x1?k Vk (x)(1 ? xt)?1 > : But c0vk =< c; x?k Vk (x) > and we have
c0vk + t < c; x1?k Vk (x)(1 ? xt)?1 > = < c; x?k Vk (x) > +t < c; xx?k Vk (x)(1 ? xt)?1 > = < c; x?k Vk (x)(1 ? xt)?1 > : It follows
? xt)?1 > ; e(t) = 1 ? tk ; f (t) ? (k ? 1=k)f (t) = tk f (t)
