Convergence rate of Pade-type approximants for Stieltjes
functions
M. Bello Hernandeza;∗;1, F. Cala Rodrguezb, J.J. Guadalupea;1, G. Lopez Lagomasinoc;2
a
Dpto. de Matematicas y Computacion, Univ. de La Rioja, Logron˜o, Spain
bDpto. de Analisis Matematico, Univ. de La Laguna, Tenerife, Spain c
Dpto. de Matematicas, Univ. Carlos III de Madrid, Madrid, Spain
Received 30 October 1997; received in revised form 20 March 1998
Abstract
For a wide class of Stieltjes functions we estimate the rate of convergence of Pade-type approximants when the number of xed poles represents a xed proportion with respect to the order of the rational approximant. c1998 Elsevier Science B.V. All rights reserved.
Keywords:Orthogonal polynomials; Pade-type approximation
1. Introduction
Let ¿1, by f we denote a continuous almost everywhere positive function on the real line such that
lim
|x|→∞f(x)|x|
−= 1: (1)
In [9], Rakhmanov studied the asymptotic behavior of the sequence hn(d;:) of orthonormal polynomials with respect to
d(x) = exp{−f(x)}dx; x ∈R: (2)
(Within this class of measures, of particular interest are the so-called Freud weights
dw(x) = exp{−|x|}dx
∗Corresponding author.
1
and their orthogonal polynomials.) He proved that
lim n→∞
log|hn(d;z)|
n1−−1 =D()|Imz|; (3)
where this limit is uniform on compact subsets of C\R,
D() =
and (:) denotes the Gamma function. Set
ˆ (z) =
Z d
(x) z−x :
Let n denote the nth diagonal Pade approximant with respect to ˆ. From Rakhmanov’s result it is not hard to deduce that
lim n→∞
log|ˆ(z)−n(z)|
n1−−1 6−2D()|Imz|; (4)
uniformly on compact subsets of C\R. We aim to obtain similar results when instead of Pade approximants, Pade-type approximants are used.
Let l2
n be a polynomial of degree m(n) and 06m(n)6n. We dene the nth Pade-type approxi-mants of ˆ with xed poles at the zeros of l2
n as the unique rational function
rn= pn qnl2n
;
where pn and qn are polynomials which satisfy
• degpn6n−1; degqn6n−m(n); qn6≡0, poles of the rational approximant are xed.
We prove
Theorem 1. Let ln denote the orthonormal polynomial of degree m(n)=2 with respect to the Freud
measure dw(x) where 1¡¡. Let rn denote the nth Pade-type approximant of ˆ with xed
poles at the zeros of l2
n and assume that
lim n→∞
m(n)
n = ∈[0;1):
Then
lim sup n→∞
log|ˆ(z)−rn(z)|
n1−−1 6−2(1−)
1−−1
D()|Imz|; (7)
where convergence takes place uniformly on each compact subset of C\R.
The paper is divided as follows. In Section 2, we give some auxiliary results. Section 3 is devoted to the proof of the theorem stated above and some comments.
2. Auxiliary results
Let d be a nite positive Borel measure on R, with an innite number of points in its support and nite moments. Denote
Kj(d; z) = sup p∈j; p6≡0
|p2(z)|
R
|p2(x)|d(x); (8)
where j is the set of all polynomials of degree 6j. If d=l2
nd we denote
Kn; j(z) =Kj(d; z):
Lemma 2.1. There exist constants D¿0 and ∈R such that
DnKj(d ˜; z)6Kn; j(z)6Kj(l2nd|(−n1=; n1=); z); (9)
where d ˜(x) = exp{−(f(x)− |x|)}dx; 1¡¡; andl2
nd|(−n1=; n1=) is the restriction of the mea-sure l2
nd to (−n1=; n1=).
Proof. From (8) the inequality on the right side of (9) follows directly. On the other hand from Corollary 1.4 in [7] there exist constants D1¿0 and 1∈R such that
l2n(x) exp{−|x|}6D
1(m(n) + 1)1; x∈R:
Since 06m(n)6n; we obtain
l2
n(x) exp(−|x| )6D
Thus, if p∈j and p6≡0, we get
and the proof is concluded.
Lemma 2.2. Let K be a compact subset of C\R; 1¡¡; and lim(m(n)=n) =. Then
and ln denotes the orthonormal polynomial of degree m(n)=2 with respect to the Freud measure dw(x).
Proof. Let tk be the kth orthonormal polynomial with respect to d. From the general theory of orthogonal polynomials, we know (see [5]) that
On the other hand,
or what is the same
2 have (see Theorem 2.6 in [8])
Z Using (3), (10), (12), and (13) one obtains
lim n→∞
logKn(d ˜; z)
n1−1= = 2D()|Im(z)|: (14) This result appears in [9], Lemma 4.
For d(x) =l2
coecient. Notice that, innite–nite range L2 estimates give as above
2
n−m(n)−1
2
n−m(n)
6D4n2=: (16)
From the rst inequality in (9), (14)–(16), we obtain
lim inf
3. Proof of Theorem 1
Let K be a compact subset of C\R, then there exists D1=D1(K)¿0 such that
|z−x|¿D1; z∈K; x∈R:
Using (6) and the orthonormality of qn, we get
|( ˆ−rn)(z)|=
1 (qnln)2(z)
Z (q
nln)2(x) z−x d
6 1
D1|(qnln)2(z)| :
Now, from Lemma 2.2 and (3) as applied to the sequence {ln}, we obtain (7).
Corollary 1. Under the assumptions of Theorem 1
rn →ˆ;
uniformly on each compact set of C\R.
Proof. It is immediate from the fact that the right-hand side of (7) is continuous and negative on
C\R.
Remark 1. In the case when = 1 and 1¡= it is possible to construct examples where there is divergence. For example, taking m(n) =n and f(x) =|x|− |x|
′
with ′¡ suciently close to .
For this reason we do not discuss this limiting situation.
Remark 2. When 1¡¡ and m(n) =n there is always divergence.
References
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[5] G. Freud, Orthogonal Polynomials, Pergamon Press, Budapest, 1966.
[7] A.L. Levin, D.S. Lubinsky, Christoel functions, Orthogonal Polynomials, and Nevai’s Conjecture for Freud Weights, Constr. Approx. 8 (1992) 463–535.
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