Rev. Anal. Num´er. Th´eor. Approx., vol. 36 (2007) no. 1, pp. 89–95 ictp.acad.ro/jnaat
ON THE ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS ON THE CURVE WITH A DENUMERABLE MASS POINTS
KHALDI RABAH∗and AGGOUNE FATEH∗
Abstract. We investigate the asymptotic behavior of orthogonal polynomials with respect to a measure of the typeσ=α+γ, whereαis a measure concen- trated on a rectifiable Jordan curve andγ is an infinite discrete measure.
MSC 2000. Primary 42C05; Secondary 30E15, 30E10.
Keywords. Orthogonal Polynomials, Asymptotic Behavior.
1. INTRODUCTION
Let σ be a finite positive Borel measure on a compact set of the complex plane whose support contains an infinite set of points. Denote by Tn(z) the monic polynomial of degreen with respect to the measureσ i.e.
Tn(z) = zn+....
Z
E
Tn(z)zmdσ = 0; m= 0,1,2, ..., n−1.
Let Pn be the set of polynomials of degree n, it is well known that Tn(z) satisfies the extremal properties:
(1) kTnk2L
2(σ):= min
Q∈Pn−1
kzn+Qk2L
2(σ)=mn(σ), where as usual
kfkL
2(σ) :=
Z
|f(ξ)|2dσ(ξ) 1/2
.
One of the major areas of research in the study of orthogonal polynomials is to investigate the asymptotics behavior of Tn(z) as n→ ∞. There exists different type of asymptotics behavior, in this context, we can mention the most frequent ones:
a)nths-root (or weak) asymptotic behavior of orthogonal polynomials is the asymptotic of pn |Tn(z)|,n∈N, it requires weakest assumptions and depends on the regularity properties of the measure σ. The main tool of study in this case is the logarithmic potential. Among the applications one can cite the
∗Department of Mathematics, University of Annaba, B.P. 12, 23000, Annaba, Algeria, e-mail: [email protected], [email protected].
location and asymptotics of zeros distribution. Extremely important results on this subject are given in the book of H. Stahl and V. Totik [8].
b) Ratio asymptotic is the asymptotic of TTn+1(z)
n(z) .The best condition to es- tablish the ratio asymptotics for orthogonal polynomials is the strict positivity of the density of the measure, that is known as Rakhmanov condition.
c) Szeg˝o (or strong) asymptotics is the uniform asymptotic ofTn(z) outside the support of the measure. The essential condition imposed on the measure to obtain the strong asymptotic is the so-called Szeg˝o condition, that permits to construct the associated Hardy spaces and to get the asymptotic from extremal properties of orthogonal polynomials.
It is easy to see that the strong asymptotic implies the two other types of asymptotics.
In the present work we establish the strong asymptotic of the orthogonal polynomials Tn(z) associated with a measure σ supported on a rectifiable Jordan curve and perturbed by an infinite Blaschke sequence of point masses outside the curve.
For the case that σ =α is an absolutely continuous measure with respect to the Lebesgue measure |dξ|on a rectifiable Jordan curve i.e.
(2) dα(ξ) =ρ(ξ)|dξ|, ρ:E →R+, Z
E
ρ(ξ)|dξ|<+∞,
Geronimus [1] has given such strong asymptotics. An extension of Geronimus results has been given by Kaliaguine [2] in the case when the measure σ = α+γl, whereαis the same as given in [1] andγlis a point measure supported on {zk}lk=1 (|zk|>1). In [3] Khaldi et al. presented an extension of Kaliaguine’s results, where they studied the case of a measure of the form
σ =α+γ,
where α is the same as given in [1] and γ is a point measure supported on a denumerable set of points{zk}∞k=1 in the region exterior to the curve E, i.e.
(3) γ =
∞
X
k=1
Akδ(z−zk);Ak>0,
∞
X
k=1
Ak <∞.
We note that, in this paper we generalize the results found in [3], more precisely in the proof of Theorem 6, we show that the condition (17) in [3, page 265] imposed on the points{zk}∞k=1 is redundant.
The structure of this paper is the following: In the next section we define Szeg˝o function, Hardy space H2(Ω, ρ) and the extremal problem. In section 3, we give the main results, first we find the limit of a sequence of extremal values, then we prove the asymptotic formulas.
2. EXTREMAL PROBLEM IN THE HARDY SPACEH2(Ω, ρ)
Let E be a rectifiable Jordan curve in the complex plane, Ω = Ext (E), G = {z∈C,|z|>1} (∞ ∈Ω, ∞ ∈G), and Φ : Ω → G is the conformal mapping with Φ (∞) =∞.We denote Ψ = Φ−1.
If the weight function ρ (which defines the absolutely part of the measure σ) satisfies the Szeg˝o condition:
Z
E
(logρ(ξ))Φ0(ξ)|dξ|>−∞,
then, the Szeg˝o functionDassociated with the curveEand the weight function ρ defined by
D(z) = exp
−4π1 Z2π
0
Φ(z)+eiθ
Φ(z)−eiθ log ρ(ξ)
|Φ0(ξ)|dθ
, ξ= Ψ(eiθ), satisfies the following properties:
(i) D(z) is analytic in Ω,D(z) 6= 0 in Ω, andD(∞)>0.
(ii) |D(ξ)|−2Φ0(ξ)=ρ(ξ),ξ ∈E , whereD(ξ) = lim
z→ξD(z) (a.e. on E).
One says that a functionf analytic in Ω is fromH2 (Ω, ρ) space if the function f◦Ψ/D◦Ψ is from the usual Hardy spaceH2(G). (Let’s recall that a function f analytic in Ω is from H2(G) space if lim
R→1+ 1 R
R
CR
|f(w)|2|dw| < ∞, where CR={w∈G:|w|=R}).
Each function f from H2(Ω, ρ) has limit values on E and kfk2H2(Ω,ρ)= lim
R→1+ 1 R
Z
ER
|f(z)|2
|D(z)|2
Φ0(z) dz= Z
E
|f(ξ)|2ρ(ξ)|dξ|, whereER={z∈Ω :|Φ (z)|=R}.
Lemma 1. [2] If f ∈H2(Ω, ρ) then for every compact set K ⊂Ω there is a constant CK such that:
sup{|f(z)|:z∈K} ≤CKkfkH2(Ω,ρ).
Now we define µ(ρ) as the extremal value of the following problem:
(4) µ(ρ) = infnkϕk2H2(Ω,ρ):ϕ∈H2(Ω, ρ), ϕ(∞) = 1o.
It is proved in [2] that the extremal function of the problem (4) is exactly the functionϕ∗ =D/D(∞).
Lemma 2. [3] The extremal function of the problem
µ∞(ρ) = infnkϕk2H2(Ω,ρ), ϕ∈H2(Ω, ρ), ϕ(∞) = 1, ϕ(zk) = 0, k= 1,2, ...o
is given by ψ∞=ϕ∗B∞, in addition µ∞(ρ) =µ(ρ)
+∞
Y
k=1
|Φ (zk)|2,
where the constantµ(ρ) and the functionϕ∗are defined by the problem(4)and B∞ is the Blaschke product:
(5) B∞(z) =
+∞
Y
k=1
Φ(z)−Φ(zk) Φ(z)Φ(zk)−1
|Φ(zk)|2 Φ(zk) .
3. MAIN RESULTS
Definition 3. A measure σ =α+γ is said to belong to a class A, if the absolutely continuous partα and the discrete partγ satisfy the conditions(2), (3) and the Blaschke’s condition, i.e.
(6)
+∞
X
k=1
(|Φ (zk)| −1)<∞.
Remark4. The condition (6) is natural and it guarantees the convergence
of the Blaschke product (5).
We denote byλn= Φn−Φn, where Φnis the polynomial part of the Laurent expansion of Φnin the neighborhood of infinity.
Definition 5. [1]A rectifiable curveE is said to be of classΓifλn(ξ)→0 (n→ ∞) uniformly onE.
Theorem 6. Let a measure σ=α+
∞
P
k=1
Akδ(z−zk) satisfy the conditions (2) and (3), then
l→∞limmn(σl) =mn(σ), where the measure σl = α+ Pl
k=1
Akδ(z−zk) and mn(.) are defined as in (1) i.e.
mn(σ) = Z
E
|Tn(ξ)|2ρ(ξ)|dξ|+
∞
X
k=1
Ak|Tn(zk)|2,
mn(σl) = Z
E
Tnl(ξ)2ρ(ξ)|dξ|+
l
X
k=1
AkTnl(zk)2. (7)
Proof. It is easy to see that the extremal property ofTnl(z) (see (7)) implies that the sequences {mn(σl)}∞l=1 is increasing and mn(σl) ≤mn(σ) for every l≥1, and so Theorem 6 tells us what the limit is.
According to the reproducing property of the kernel polynomial Kn(ξ, z) (see [9]), we have:
Tnl(zj) = Z
E
Tnl(ξ)Kn+1(ξ, zj)ρ(ξ)|dξ|. The Schwarz inequality implies
Tnl(zj)2 ≤ Z
E
Tnl(ξ)2ρ(ξ)|dξ|
Z
E
|Kn+1(ξ, zj)|2ρ(ξ)|dξ|
≤ mn(σl) sup
ξ∈E
|Kn+1(ξ, zj)|2, (8)
the extremal property of Tn(z) implies that
mn(σ) ≤ mn(σl) +
∞
X
k=l+1
Ak
Tnl(zk)2
≤ mn(σl)
1 + sup
ξ∈E,k≥l+1
|Kn+1(ξ, zk)|2
∞
X
k=l+1
Ak
.
This gives
mn(σ)≤lim inf
l→+∞mn(σl)≤lim sup
l→+∞
mn(σl)≤mn(σ).
The proof of the Theorem is complete.
Theorem 7. Let E be a curve from the class Γ and the measure σ∈A. If
(9) mn(σl)≤
l
Y
k=1
|Φ (zk)|
mn(α),∀n,∀l,
then the orthogonal polynomialsTn(z) and the extremal valuemn(σ) have the following asymptotic behavior (n→ ∞):
(i) lim mn(σ)
(C(E))2n =µ∞(σ);
(ii) lim[C(E)Φ]Tn n −ψ∞
H2(Ω,ρ) = 0;
(iii) Tn(z) = [C(E) Φ (z)]n[ψ∞(z) +εn(z)], εn(z) → 0 uniformly on the compact subsets ofΩ.
Where C(E)1 = lim
z→∞
Φ(z)
z >0, the constantµ∞(σ) and the functionψ∞ are defined in Lemma 2.
Remark 8. Note that in the case of the circle the condition (9) is satisfied
(see [4, th. 5.2]).
Proof. By passing to the limit when l tends to infinity in (9) and using Theorem 6, we obtain
(10) mn(σ)
(C(E))2n ≤ +∞
Y
k=1
|Φ (zk)|
mn(α) (C(E))2n.
It is proved in [2] that
(11) lim
n→∞
mn(α)
(C(E))2n =µ(α). Using (10), (11) and Lemma 2, we get
(12) lim sup
n→∞
mn(σ)
(C(E))2n ≤µ∞(σ).
Putting ϕ∗n = [C(E)Φ]Tn n, then from (7) and (12) we deduce that the prod- ucts |ϕ∗n(zk)|2Φ (zk)2n are bounded for all k ≥ 1, so ϕ∗n(zk) → 0, n →
∞,(|Φ (zk)|>1).
If we setψn= 12[ϕ∗n+ψ∞], then we can see thatψn(∞) = 1 andψn(zk)→ 0, n→ ∞,therefore (see [2, page 234]),
(13) lim inf
n→∞ kψnkH2(Ω,ρ) ≥µ∞(σ).
From the parallelogram identity, Lemma 2 and (7), it yields
kϕ∗n−ψ∞k2H2(Ω,ρ) = 2kϕ∗nk2H2(Ω,ρ)+ 2kψ∞k2H2(Ω,ρ)−412[ϕ∗n+ψ∞]2
H2(Ω,ρ)
≤ 2 mn(σ)
(C(E))2n + 2µ∞(σ)−4kψnk2H2(Ω,ρ). (14)
Using (13) and the fact that the norm is non negative we obtain lim inf
n→∞
mn(σ)
(C(E))2n ≥µ∞(σ). This, with (12), proves (i) of the Theorem.
The inequalities (12), (13) and (14) imply (15) lim sup
n→∞ kϕ∗n−ψ∞k2H2(Ω,ρ)≤2µ∞(σ) + 2µ∞(σ)−4µ∞(σ) = 0.
(ii) of the Theorem follows immediately from (15).
Now, to prove (iii) of the Theorem, we apply Lemma 1 for the function ϕ∗n−ψ∞ which belongs to H2(Ω, ρ), then, for all compactK ⊂Ω, we have
sup
z∈K
Tn(z)
[C(E)Φ(z)]n −ψ∞(z) = sup
z∈K
|(ϕ∗n−ψ∞) (z)|
≤ CKkϕ∗n−ψ∞k2H2(Ω,ρ) →
n→∞0.
This achieves the proof of Theorem 7.
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Received by the editors: April 26, 2006.
