Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:
Journal of Computational and Applied Mathematics 99 (1998) 47–53
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, Logrono,
˜ Spain
b
Dpto. 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
c 1998 Elsevier Science
of xed poles represents a xed proportion with respect to the order of the rational approximant.
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 f
(x)|x|−
= 1:
(1)
|x|→∞
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:
(Within this class of measures, of particular interest are the so-called Freud weights
dw
(x) = exp{−|x|
}dx
∗
Corresponding author.
This research was partially supported by DGES under grant PB96-0120-C03-02.
2
This research was partially supported by DGES under grant PB96-0120-C03-01.
1
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 4 4 - 7
(2)
48
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
and their orthogonal polynomials.) He proved that
lim
n→∞
log |hn (d
; z)|
= D(
)|Im z|;
n1−
−1
(3)
where this limit is uniform on compact subsets of C\R,
D(
) =
−1
and
((
+ 1)=2)
(
=2)
(1=
)
;
(:) 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
log |ˆ
(z) − n (z)|
6−2D(
)|Im z|;
n→∞
n1−
−1
lim
(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 l2n be a polynomial of degree m(n) and 0 6 m(n) 6 n. We de ne the nth Pade-type approximants of ˆ
with xed poles at the zeros of l2n as the unique rational function
rn =
pn
;
qn l2n
where pn and qn are polynomials which satisfy
• deg pn 6 n − 1; deg qn 6 n − m(n); qn 6≡ 0,
• (qn l2n ˆ
− pn )(z) = O(1=(z n−m(n)+1 )); as z = ix → ∞; x¿0.
It is easy to prove (see, e.g., [4]) that
Z
0=
x qn (x)l2n (x)d
(x);
(ˆ
− rn )(z) =
1
(qn ln )2 (z)
Z
= 0; : : : ; n − m(n) − 1;
(5)
(qn ln )2 (x)d
(x)
:
z−x
(6)
If m(n) = 0, then rn is the nth diagonal Pade approximant with respect to ˆ
. If m(n) = n, all the
poles of the rational approximant are xed.
In recent years (see, e.g., [1–7]), the rate of convergence of Pade-type and multipoint Pade-type
approximants has been studied when the measure de ning the function has compact support. We will
show that results of type (4) take place for Pade-type approximants when the support of the measure
is unbounded. To this end, we will restrict the type of polynomials which carry as their zeros the
xed poles of the Pade-type approximants. In the sequel, ln denotes the orthonormal polynomial of
degree m(n)=2 with respect to the Freud measure dw (x) introduced above. Unless otherwise stated,
we take
¿ ¿1.
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
49
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 l2n and assume that
m(n)
= ∈ [0; 1):
n→∞
n
lim
Then
lim sup
n→∞
log |ˆ
(z) − rn (z)|
−1
6−2(1 − )1−
D(
)|Im z|;
−1
1−
n
(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 in nite number of points in its support
and nite moments. Denote
Kj (d; z) =
sup
p∈j ; p6≡0
R
|p2 (z)|
;
|p2 (x)|d(x)
(8)
where j is the set of all polynomials of degree 6 j.
If d = l2n d
we denote
Kn; j (z) = Kj (d; z):
Lemma 2.1. There exist constants D¿0 and ∈ R such that
Dn Kj (d ˜
; z) 6 Kn; j (z) 6 Kj (l2n d
|(−n1=
; n1=
) ; z);
(9)
where d ˜
(x) = exp{−(f
(x) − |x| )} dx; 1¡ ¡
; and l2n d
|(−n1=
; n1=
) is the restriction of the measure l2n d
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| } 6 D1 (m(n) + 1) 1 ;
Since 0 6 m(n) 6 n; we obtain
l2n (x) exp(−|x| ) 6 D2 n 1 :
x ∈ R:
50
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
Thus, if p ∈ j and p 6≡ 0, we get
R
|p2 (z)|
|p2 (z)|
R
¿
|p2 (x)|l2n (x) d
(x)
D2 n 1 |p2 (x)| d ˜
(x)
and the proof is concluded.
Lemma 2.2. Let K be a compact subset of C\R; 1¡ ¡
; and lim(m(n)=n) = . Then
lim inf
n→∞
log |qn |(z)
¿ (1 − )1−1=
D(
)|Im z|
n1−1=
uniformly on K; where qn is the (n − m(n))th orthonormal polynomial with respect to l2n d
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
Kj (d; z) =
j
X
|tk (z)|2 ¿ |tj (z)|2 ;
z ∈ C;
(10)
k=0
Kj−1 (d; z) =
j−1 tj (z)tj−1 (z) − tj (z)tj−1 (z)
;
j
z−z
where z ∈ C\R and j is the leading coecient of tj . Thus, with the aid of (10), we obtain
Kj−1 (d; z) =
j−1 Im(tj (z)tj−1 (z))
j
Im z
6
j−1 |tj tj−1 (z)|
j
|Im z|
6
j−1 |tj (z)|Kj−1 (z)
:
j
|Im z|
1=2
This inequality yields
2
j−1
|tj (z)|2
;
Kj−1 (d; z) 6 2
j |Im z|2
(11)
therefore,
"
#
2
j−1
Kj (d; z) = Kj−1 + |tj (z)| 6 2
+ 1 |tj (z)|2 :
j |Im(z)|2
2
(12)
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
51
On the other hand,
1
= inf
P=z j +···
j2
Z
|P 2 (x)| d(x)
2
Z
tj−1 (x)
6 x
d(x);
j−1
or what is the same
2
j−1
6
j2
Z
|xtj−1 |2 d(x):
If d(x) satis es (2), there exist constants D1 , D2 , D3 ¿0 such that for all k ∈ N and p ∈ k , we
have (see Theorem 2.6 in [8])
Z
|p2 (x)| d(x) 6 D2
Z
D1 k 1=
|p(x)|2 d(x);
−D1 k 1=
in particular,
2
j−1
6 D2
j2
Z
D1 j 1=
|xtj−1 (x)|2 d(x);
−D1 j 1=
6 D3 j 2=
:
(13)
Take d(x) = d ˜
(x) = exp{−(f
(x) − |x| )} dx. Since 1¡ ¡
, the function f
(x) − |x| satis es (1).
Using (3), (10), (12), and (13) one obtains
log Kn (d ˜
; z)
= 2D(
)|Im(z)|:
n1−1=
This result appears in [9], Lemma 4.
For d(x) = l2n (x) d
(x) and j = n − m(n), (12) gives
lim
n→∞
"
#
2n; n−m(n)−1
Kn; n−m(n) (z) 6
+ 1 |qn (z)|2 ;
2n; n−m(n)
(14)
(15)
where qn is the (n−m(n))th orthonormal polynomial with respect to |ln |2 d
and n; n−m(n) its leading
coecient. Notice that, in nite– nite range L2 estimates give as above
2n−m(n)−1
6 D4 n2=
:
2n−m(n)
(16)
From the rst inequality in (9), (14)–(16), we obtain
lim inf
n→∞
n − m(n)
log |qn (z)|
¿ lim
1−1=
n
n
n
1−1=
log |Kn−m(n) (d ˜
; z)|
2(n − m(n))1−1=
= (1 − )1−1=
D(
)|Im(z)|
and the proof is nished.
52
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
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
Z
1
(qn ln )2 (x)
|(ˆ
− rn )(z)| =
d
(qn ln )2 (z)
z−x
6
1
:
D1 |(qn ln )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
[1] A. Ambroladze, H. Wallin, Pade type approximants of Markov and meromorphic functions, J. Approx. Theory 88
(1997) 354 –369.
[2] A. Ambroladze, H. Wallin, Convergence of rational interpolants with preassingned poles, J. Approx. Theory 89
(1997) 238 –256.
[3] F. Cala, G. Lopez, From Pade to Pade-type approximants. Exact rate of convergence, in: M. Florenzano et al. (Eds.),
Proc. 2nd Internat. Conf. on Approx. and Opt. in the Caribbean, Peter Series in Approx. and Optimization, vol. 8,
1995, pp. 155–163.
[4] F. Cala, G. Lopez, Multipoint Pade-type approximants. Exact rate of convergence, Constr. Approx. 14 (2) (1998)
259 –272.
[5] G. Freud, Orthogonal Polynomials, Pergamon Press, Budapest, 1966.
[6] A.A. Gonchar, On convergence of Pade approximants for some classes of meromorphic functions, Mat. Sb. 97 (139)
(1975) 4; English Transl. in: Math. USSR Sb. 26 (1975) 555–575.
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
53
[7] A.L. Levin, D.S. Lubinsky, Christo el functions, Orthogonal Polynomials, and Nevai’s Conjecture for Freud Weights,
Constr. Approx. 8 (1992) 463–535.
[8] H.N. Mhaskar, E.B. Sa , Where does the Lp norm of a weighted polynomial live?, Trans. Amer. Math. Soc. 303
(1987) 109 –124.
[9] E.A. Rakhmanov, On the asymptotic properties of polynomials orthogonal on the real axis, Mat. Sb. 119 (161)
(1982) 161–203; English Transl. in: Math. USSR Sb. 47 (1984) 155–193.
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, Logrono,
˜ Spain
b
Dpto. 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
c 1998 Elsevier Science
of xed poles represents a xed proportion with respect to the order of the rational approximant.
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 f
(x)|x|−
= 1:
(1)
|x|→∞
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:
(Within this class of measures, of particular interest are the so-called Freud weights
dw
(x) = exp{−|x|
}dx
∗
Corresponding author.
This research was partially supported by DGES under grant PB96-0120-C03-02.
2
This research was partially supported by DGES under grant PB96-0120-C03-01.
1
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 4 4 - 7
(2)
48
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
and their orthogonal polynomials.) He proved that
lim
n→∞
log |hn (d
; z)|
= D(
)|Im z|;
n1−
−1
(3)
where this limit is uniform on compact subsets of C\R,
D(
) =
−1
and
((
+ 1)=2)
(
=2)
(1=
)
;
(:) 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
log |ˆ
(z) − n (z)|
6−2D(
)|Im z|;
n→∞
n1−
−1
lim
(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 l2n be a polynomial of degree m(n) and 0 6 m(n) 6 n. We de ne the nth Pade-type approximants of ˆ
with xed poles at the zeros of l2n as the unique rational function
rn =
pn
;
qn l2n
where pn and qn are polynomials which satisfy
• deg pn 6 n − 1; deg qn 6 n − m(n); qn 6≡ 0,
• (qn l2n ˆ
− pn )(z) = O(1=(z n−m(n)+1 )); as z = ix → ∞; x¿0.
It is easy to prove (see, e.g., [4]) that
Z
0=
x qn (x)l2n (x)d
(x);
(ˆ
− rn )(z) =
1
(qn ln )2 (z)
Z
= 0; : : : ; n − m(n) − 1;
(5)
(qn ln )2 (x)d
(x)
:
z−x
(6)
If m(n) = 0, then rn is the nth diagonal Pade approximant with respect to ˆ
. If m(n) = n, all the
poles of the rational approximant are xed.
In recent years (see, e.g., [1–7]), the rate of convergence of Pade-type and multipoint Pade-type
approximants has been studied when the measure de ning the function has compact support. We will
show that results of type (4) take place for Pade-type approximants when the support of the measure
is unbounded. To this end, we will restrict the type of polynomials which carry as their zeros the
xed poles of the Pade-type approximants. In the sequel, ln denotes the orthonormal polynomial of
degree m(n)=2 with respect to the Freud measure dw (x) introduced above. Unless otherwise stated,
we take
¿ ¿1.
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
49
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 l2n and assume that
m(n)
= ∈ [0; 1):
n→∞
n
lim
Then
lim sup
n→∞
log |ˆ
(z) − rn (z)|
−1
6−2(1 − )1−
D(
)|Im z|;
−1
1−
n
(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 in nite number of points in its support
and nite moments. Denote
Kj (d; z) =
sup
p∈j ; p6≡0
R
|p2 (z)|
;
|p2 (x)|d(x)
(8)
where j is the set of all polynomials of degree 6 j.
If d = l2n d
we denote
Kn; j (z) = Kj (d; z):
Lemma 2.1. There exist constants D¿0 and ∈ R such that
Dn Kj (d ˜
; z) 6 Kn; j (z) 6 Kj (l2n d
|(−n1=
; n1=
) ; z);
(9)
where d ˜
(x) = exp{−(f
(x) − |x| )} dx; 1¡ ¡
; and l2n d
|(−n1=
; n1=
) is the restriction of the measure l2n d
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| } 6 D1 (m(n) + 1) 1 ;
Since 0 6 m(n) 6 n; we obtain
l2n (x) exp(−|x| ) 6 D2 n 1 :
x ∈ R:
50
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
Thus, if p ∈ j and p 6≡ 0, we get
R
|p2 (z)|
|p2 (z)|
R
¿
|p2 (x)|l2n (x) d
(x)
D2 n 1 |p2 (x)| d ˜
(x)
and the proof is concluded.
Lemma 2.2. Let K be a compact subset of C\R; 1¡ ¡
; and lim(m(n)=n) = . Then
lim inf
n→∞
log |qn |(z)
¿ (1 − )1−1=
D(
)|Im z|
n1−1=
uniformly on K; where qn is the (n − m(n))th orthonormal polynomial with respect to l2n d
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
Kj (d; z) =
j
X
|tk (z)|2 ¿ |tj (z)|2 ;
z ∈ C;
(10)
k=0
Kj−1 (d; z) =
j−1 tj (z)tj−1 (z) − tj (z)tj−1 (z)
;
j
z−z
where z ∈ C\R and j is the leading coecient of tj . Thus, with the aid of (10), we obtain
Kj−1 (d; z) =
j−1 Im(tj (z)tj−1 (z))
j
Im z
6
j−1 |tj tj−1 (z)|
j
|Im z|
6
j−1 |tj (z)|Kj−1 (z)
:
j
|Im z|
1=2
This inequality yields
2
j−1
|tj (z)|2
;
Kj−1 (d; z) 6 2
j |Im z|2
(11)
therefore,
"
#
2
j−1
Kj (d; z) = Kj−1 + |tj (z)| 6 2
+ 1 |tj (z)|2 :
j |Im(z)|2
2
(12)
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
51
On the other hand,
1
= inf
P=z j +···
j2
Z
|P 2 (x)| d(x)
2
Z
tj−1 (x)
6 x
d(x);
j−1
or what is the same
2
j−1
6
j2
Z
|xtj−1 |2 d(x):
If d(x) satis es (2), there exist constants D1 , D2 , D3 ¿0 such that for all k ∈ N and p ∈ k , we
have (see Theorem 2.6 in [8])
Z
|p2 (x)| d(x) 6 D2
Z
D1 k 1=
|p(x)|2 d(x);
−D1 k 1=
in particular,
2
j−1
6 D2
j2
Z
D1 j 1=
|xtj−1 (x)|2 d(x);
−D1 j 1=
6 D3 j 2=
:
(13)
Take d(x) = d ˜
(x) = exp{−(f
(x) − |x| )} dx. Since 1¡ ¡
, the function f
(x) − |x| satis es (1).
Using (3), (10), (12), and (13) one obtains
log Kn (d ˜
; z)
= 2D(
)|Im(z)|:
n1−1=
This result appears in [9], Lemma 4.
For d(x) = l2n (x) d
(x) and j = n − m(n), (12) gives
lim
n→∞
"
#
2n; n−m(n)−1
Kn; n−m(n) (z) 6
+ 1 |qn (z)|2 ;
2n; n−m(n)
(14)
(15)
where qn is the (n−m(n))th orthonormal polynomial with respect to |ln |2 d
and n; n−m(n) its leading
coecient. Notice that, in nite– nite range L2 estimates give as above
2n−m(n)−1
6 D4 n2=
:
2n−m(n)
(16)
From the rst inequality in (9), (14)–(16), we obtain
lim inf
n→∞
n − m(n)
log |qn (z)|
¿ lim
1−1=
n
n
n
1−1=
log |Kn−m(n) (d ˜
; z)|
2(n − m(n))1−1=
= (1 − )1−1=
D(
)|Im(z)|
and the proof is nished.
52
M. Bello Hernandez et al. / Journal of Computational and Applied Mathematics 99 (1998) 47–53
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
Z
1
(qn ln )2 (x)
|(ˆ
− rn )(z)| =
d
(qn ln )2 (z)
z−x
6
1
:
D1 |(qn ln )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.
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