#### Strong asymptotics for Lp extremal polynomials off a complex curve

Hindawi Publishing Corporation
Journal of Applied Mathematics
We study the asymptotic behavior of Lp(?) extremal polynomials with respect to a measure of the form ? = ? + ?, where ? is a measure concentrated on a rectifiable Jordan curve in the complex plane and ? is a discrete measure concentrated on an infinite number of mass points. Let F be a compact subset of the complex plane C and let B be a metric space of functions defined on F. We suppose that B contains the set of monic polynomials. Then the extremal or general Chebyshev polynomial Tn of degree n is a monic polynomial that minimizes the distance between zero and the set of all monic polynomials of degree n, that is, where ? is a measure supported on a closed rectifiable Jordan curve E as defined in [2] and ? is a discrete measure with a finite number of mass points. In this paper, we generalize Kaliaguine's work [3] in the case where 1 ? p < ? and the support of the measure ? is a rectifiable Jordan curve E plus an infinite discrete set of
? = ? + ?
1. Introduction
dist Tn, 0 = min dist Qn, 0 : Qn(z) = zn + an?1zn?1 + ? ? ? + a0 = mn(B).
(1.1)
Recently, a series of results concerning the asymptotic of the extremal polynomials was
established for the case of B = Lp(F, ?), 1 ? p ? ?, where ? is a Borel measure on F; see,
for example, [
3, 7, 8, 12
]. When p = 2, we have the special case of orthogonal polynomials
with respect to the measure ?. A lot of research work has been done on this subject; see,
for example, [
1, 4, 5, 9, 11, 13
]. The case of the spaces Lp(F, ?), where 0 < p < ? and
F is a closed rectifiable Jordan curve with some smoothness conditions, was studied by
Geronimus [2]. An extension of Geronimus?s result has been given by Kaliaguine [
3
] who
found asymptotics when 0 < p < ? and the measure ? has a decomposition of the form
+?
k=1
Note that the result of the special case p = 2 is also a generalization of [
4
]. More
precisely, in the proof of Theorem 4.3, we show that condition [4, page 265, (17)] imposed
on the points {zk}k?=1 is redundant.
2. The H p(? , ?) spaces (1 ? p < ?)
Let E be a rectifiable Jordan curve in the complex plane, ? = Ext(E), G = {z ? C, |z| > 1}
(? belongs to ? and G).
We denote by ? the conformal mapping of ? into G with ?(?) = ? and 1/C(E) =
limz??(?(z)/z) > 0, where C(E) is the logarithmic capacity of E. We denote ? = ??1.
Let ? be an integrable nonnegative weight function on E satisfying the Szego? condition
mass points which accumulate on E. More precisely, ? = ? + ?, where the measure ? and
its support E are defined as in [
3
], that is,
sup
f (z) : z ? K
? CK f H p(? ,?).
E
log ?(?)
? (?) |d?| > ??.
Condition (2.1) allows us to construct the so-called Szego? function D associated with
the curve E and the weight function ?:
(1.3)
(1.4)
(2.1)
(2.3)
(2.4)
d?(?) = ?(?)|d?|,
? ? 0, ? ? L1 E, |d?| ;
? is a discrete measure concentrated on {zk}k?=1 ? Ext(E) (Ext(E) is the exterior of E),
that is,
+?
k=1
? =
Ak? z ? zk ,
Ak > 0,
Ak < ?.
D(z) = exp
1 +? w + eit
? 2p? ?? w ? eit log
?(?)
? (?)
dt
w = ?(z), ? = ? eit
(2.2)
such that
(i) D is analytic in ? , D(z) = 0 in ? , and D(?) > 0;
(ii) |D(?)|?p|? (?)| = ?(?) a.e. on E, where D(?) = limz?? D(z).
We say that f ? H p(? , ?) if and only if f is analytic in ? and f0?/D0? ? H p(G).
For 1 ? p < ?, H p(? , ?) is a Banach space. Each function f ? H p(? , ?) has limit
values a.e. on E and
f Hp p(? ,?) =
E
1 f (z) p
f (?) p?(?)|d?| = Rl?im1+ R ER D(z) p ? (z)dz ,
where ER = {z ? ? : |?(z)| = R}.
Lemma 2.1 [
3
]. If f ? H p(? , ?), then for every compact set K ? ? , there is a constant CK
such that
3. The extremal problems
Let 1 ? p < ?; we denote ?l = ? + lk=1 Ak?(z ? zk) and by ?(?), ?(l), ??(?), mn,p(?),
mn,p(l), and mn,p(?) the extremal values of the following problems, respectively:
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
?(?) = inf
?(l) = inf
??(?) = inf
mn,p(?) = min
mn,p(l) = min
mn,p(?) = min
As usual,
? Hp p(? ,?) : ? ? H p(? , ?), ?(?) = 1 ,
? Hp p(? ,?) : ? ? H p(? , ?), ?(?) = 1, ? zk = 0, k = 1, 2, . . . , l ,
? Hp p(? ,?) : ? ? H p(? , ?), ?(?) = 1, ? zk = 0, k = 1, 2, . . . ,
Qn Lp(?) : Qn(z) = zn + ? ? ? ,
Qn Lp(?l) : Qn(z) = zn + ? ? ? ,
Qn Lp(?) : Qn(z) = zn + ? ? ? .
f Lp(?) :=
E
f (?) pd?(?)
1/p
.
We denote by ?? and ?? the extremal functions of problems (3.1) and (3.3),
respectively.
Let Tnl,p(z) and Tn,p(z) be the extremal polynomials with respect to the measures ?l
and ?, respectively, that is,
Tnl,p Lp(?l) = mn,p(l),
Tn,p Lp(?) = mn,p(?).
Lemma 3.1. Let ? ? H p(? , ?) such that ?(?) = 1 and ?(zk) = 0 for k = 1, 2, . . . , and let
B?(z) =
+? ?(z) ? ? zk
k=1 ?(z)? zk ? 1
? zk
? zk
2
be the Blaschke product. Then
(i) B? ? H p(? , ?), B?(?) = 1, |B?(?)| =
(ii) ?/B? ? H p(? , ?) and (?/B?)(?) = 1.
k+=?1 |?(zk)| (? ? E);
Proof. This lemma is proved for p = 2 in [
1
]. The proof is based on the fact that if f ?
H2(U), where U = {z ? C, |z| < 1}, and B is the Blaschke product formed by the zeros of
f , then f /B ? H2(U). It remains true in H p(U) for 1 ? p < ?; see [
6, 10
].
Lemma 3.2. An extremal function ?? of problem (3.3) is given by ?? = ??B?; in addition,
??(?) =
On the other hand, since the function ?? = ??B? ? H p(? , ?), ?(?) = 1 and ?(zk) =
0 for k = 1, 2, . . . , we get
??(?) ? ?? p =
+?
k=1
? zk
p
?(?).
Finally, the lemma follows from (3.12) and (3.13).
4. The main results
Thus
(3.10)
(3.11)
(3.12)
(3.13)
(4.1)
(4.2)
Definition 4.1. A measure ? = ? + ? is said to belong to a class A if the absolutely
continuous part ? and the discrete part ? satisfy conditions (1.3), (1.4), and (2.1) and Blaschke?s
condition, that is,
We denote ?n = ?n ? ?n, where ?n is the polynomial part of the Laurent expansion
of ?n in the neighborhood of infinity.
Definition 4.2 [
2
]. A rectifiable curve E is said to be of class ? if ?n(?) ? 0 uniformly on E.
Theorem 4.3. Let a measure ? = ? + ? satisfy conditions (1.3), (1.4) and Blaschke?s
condition (4.1); then
+?
k=1
lim mn,p(l) = mn,p(?).
l?+?
Proof. The extremal property of Tn,p(zk) gives
E
mn,p(?) p ?
Tnl,p(?) p?(?)|d?| +
Ak Tnl,p zk
= mn,p(l) p +
Ak Tnl,p zk
+?
k=l+1
l
k=1
p +
+?
k=l+1
Ak Tnl,p zk
p
On the other hand, from the extremal property of Tnl,p(zk), we can write
mn,p(l) ?
E
Tn,p(?) p?(?)|d?| +
Ak Tn,p zk
l
k=1
p
1/p
? mn,p(?) = Cn < ?.
Note that Cn does not depend on l; so for all l = 1, 2, 3, . . . ,
This implies that there is a constant Cn independent of l such that for all l = 1, 2, 3, . . . ,
Using (4.6) in (4.3) for large enough l with (4.4), we get
mn,p(l) p ? mn,p(?) p ? mn,p(l) p + Cn
+?
k=l+1
Ak.
Letting l ? ?, we obtain
E
max
Tnl,p(?) p?(?)|d?|
1/p
< Cn.
Tnl,p(z) p : |z| ? 2 < Cn.
lim mn,p(l) = mn,p(?).
l??
l
k=1
mn,p(l) ?
? zk
mn,p(?).
Theorem 4.4. Let 1 ? p < ?, E ? ?, and let ? = ? + ? be a measure which belongs to A. In
addition, for all n and l,
Then the monic orthogonal polynomials Tn,p(z) with respect to the measure ? have the
following asymptotic behavior:
(i) limn??(mn,p(?)/(C(E))n) = (??(?))1/p;
(ii) limn?? Tn,p/[C(E)?]n ? ?? H p(? ,?) = 0;
(iii) Tn,p(z) = [C(E)?(z)]n[??(z) + ?n(z)],
where ?n(z) ? 0 uniformly on compact subsets of ? and ?? is an extremal function of
problem (3.3).
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
Remark 4.5. For p = 2 and E the unit circle, condition (4.9) is proved (see [5, Theorem
5.2]). In this case, this condition can be written as ?n/?nl ? lk=1 |zk|, where ?nl = 1/mn,2(l)
and ?n = 1/mn,2(?) are, respectively, the leading coefficients of the orthonormal
polynomials associated to the measures ?l and ?.
Proof of Theorem 4.4. Taking the limit when l tends to infinity in (4.9) and using Theorem
4.3, we get
On the other hand, it is proved in [
2
] that
Using (4.10), (4.11), and Lemma 3.2, we obtain
mn,p(?)
C(E) n ?
+?
k=1
lim mn,p(?)
n?? C(E) n = ?(?) 1/p.
lim sup mn,p(?)
n?? C(E) n ?
+?
k=1
? zk
?(?) 1/p = ??(?) 1/p.
It is well known that (see [3, page 231])
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
We also have (see [3, Theorem 2.2])
From (4.4), we deduce that
?l > 0, ?(l) = ?(?)
lim mn,p(l)
n?? C(E) n = ?(l) 1/p.
?l > 0,
mn,p(?)
C(E) n ?
mn,p(l)
C(E) n .
l
k=1
l
k=1
By passing to the limit when n tends to infinity in (4.15) and taking into account (4.13)
and (4.14), we get
?l > 0, lim inf mn,p(?)
n?? C(E) n ?
(4.17)
(4.18)
(4.19)
Finally, by using Lemma 3.2, we obtain
lim inf
n??
mn,p(? )
C(E) n ?
+?
k=1
.
Inequalities (4.12) and (4.17) prove Theorem 4.4(i).
We obtain (ii) by proceeding as in [3, pages 234, 235].
To prove (iii), we consider the function
?n =
Tn,p
C(E)?
n ? ??
which belongs to the space H p(? , ?). Then by applying Lemma 2.1, we obtain
sup
Tn,p(z)
C(E)?(z)
n ? ??(z) : z ? K
= sup
?n(z) : z ? K
? CK ?n H p(? ,?) ?? 0
for all compact subsets K of ? . This achieves the proof of the theorem.
Rabah Khaldi: Department of Mathematics, University of Annaba, P.O. Box 12, 23000 Annaba,
Algeria
E-mail address:
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