OFF A COMPLEX CURVE

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L

p

OFF A COMPLEX CURVE

RABAH KHALDI

Received 29 September 2003 and in revised form 15 June 2004

We study the asymptotic behavior ofLp(σ) extremal polynomials with respect to a measure of the formσ=α+γ, whereαis a measure concentrated on a rectifiable Jordan curve in the complex plane andγis a discrete measure concentrated on an infinite number of mass points.

1. Introduction

LetFbe a compact subset of the complex planeCand letBbe a metric space of functions defined onF. We suppose thatBcontains the set of monic polynomials. Then the extremal or general Chebyshev polynomialT_n of degreen is a monic polynomial that minimizes the distance between zero and the set of all monic polynomials of degreen, that is,

distTn, 0=mindistQn, 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 ofB=L_p(F,σ), 1≤p≤ ∞, whereσis a Borel measure onF; see, for example, [3,7,8,12]. Whenp=2, we have the special case of orthogonal polynomials with respect to the measureσ. A lot of research work has been done on this subject; see, for example, [1,4,5,9,11,13]. The case of the spacesL_p(F,σ), 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 measureσhas a decomposition of the form

σ=α+γ, (1.2)

whereαis a measure supported on a closed rectifiable Jordan curveEas defined in [2]

andγis 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 measureσ is a rectifiable Jordan curveEplus an infinite discrete set of

Copyright©2004 Hindawi Publishing Corporation Journal of Applied Mathematics 2004:5 (2004) 371–378 2000 Mathematics Subject Classification: 42C05, 30E15, 30E10 URL:http://dx.doi.org/10.1155/S1110757X0430906X

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mass points which accumulate onE. More precisely,σ=α+γ, where the measureαand its supportEare defined as in [3], that is,

dα(ξ)=ρ(ξ)|dξ|, ρ≥0,ρ∈L¹E,|dξ|

; (1.3)

γ is a discrete measure concentrated on{zk}^∞_k₌1⊂Ext(E) (Ext(E) is the exterior ofE), that is,

γ=

+∞

k=1

Akδz−zk

, Ak>0,

+∞

k=1

Ak<∞. (1.4)

Note that the result of the special casep=2 is also a generalization of [4]. More precisely, in the proof ofTheorem 4.3, we show that condition [4, page 265, (17)] imposed on the points{zk}^∞_k₌1is redundant.

2. TheH^p(Ω,ρ)spaces (1≤p <∞)

LetEbe a rectifiable Jordan curve in the complex plane,Ω=Ext(E),G= {z∈C,|z|>1} (∞belongs toΩandG).

We denote byΦthe conformal mapping ofΩintoGwithΦ(∞)= ∞and 1/C(E)= lim_z_→∞(Φ(z)/z)>0, whereC(E) is the logarithmic capacity ofE. We denoteΨ=Φ⁻¹.

Letρbe an integrable nonnegative weight function onEsatisfying the Szeg¨o condition

E

logρ(ξ)Φ(ξ)|dξ|>−∞. (2.1) Condition (2.1) allows us to construct the so-called Szeg¨o functionDassociated with the curveEand the weight functionρ:

D(z)=exp

− 1 2pπ

+π

−π

w+e^it

w−e^itlog ρ(ξ) Φ(ξ)

dt

w=Φ(z),ξ=Ψe^it (2.2) such that

(i)Dis analytic inΩ,D(z)=0 inΩ, andD(∞)>0;

(ii)|D(ξ)|⁻^p|Φ(ξ)| =ρ(ξ) a.e. onE, whereD(ξ)=lim_z_→_ξD(z).

We say that f ∈H^p(Ω,ρ) if and only if f is analytic inΩand f0Ψ/D0Ψ∈H^p(G).

For 1≤p <∞,H^p(Ω,ρ) is a Banach space. Each functionf ∈H^p(Ω,ρ) has limit values a.e. onEand

f _H^pp(Ω,ρ)=

E

f(ξ)^pρ(ξ)|dξ| = lim

R→1⁺

1 R

ER

f(z)^p D(z)^p

Φ(z)dz, (2.3)

whereE_R= {z∈Ω:|Φ(z)| =R}.

Lemma2.1 [3]. If f ∈H^p(Ω,ρ), then for every compact setK⊂Ω, there is a constantC_K such that

supf(z):z∈K≤CK f H^p(Ω,ρ). (2.4)

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3. The extremal problems

Let 1≤p <∞; we denoteσl=α+^l_k₌₁Akδ(z−zk) and byµ(ρ), µ(l), µ^∞(ρ),mn,p(ρ), m_n,p(l), andm_n,p(σ) the extremal values of the following problems, respectively:

µ(ρ)=inf ϕ _H^p^p(Ω,ρ):ϕ∈H^p(Ω,ρ), ϕ(∞)=1, (3.1) µ(l)=inf ϕ _H^p^p(Ω,ρ):ϕ∈H^p(Ω,ρ), ϕ(∞)=1,ϕzk

=0,k=1, 2,...,l, (3.2) µ^∞(ρ)=inf ϕ _H^p^p(Ω,ρ):ϕ∈H^p(Ω,ρ), ϕ(∞)=1,ϕzk

=0,k=1, 2,..., (3.3) mn,p(ρ)=minQn

Lp(α):Qn(z)=zⁿ+···

, (3.4)

m_n,p(l)=minQ_n_L_p₍_σ_l₎:Q_n(z)=zⁿ+···

, (3.5)

m_n,p(σ)=minQ_n_L_p₍_σ₎:Q_n(z)=zⁿ+···

. (3.6)

As usual,

f Lp(σ):=

E

f(ξ)^pdσ(ξ) 1/ p

. (3.7)

We denote byϕ^∗andψ^∞the extremal functions of problems (3.1) and (3.3), respectively.

LetT_n^l,p(z) andT_n,p(z) be the extremal polynomials with respect to the measuresσ_l andσ, respectively, that is,

T_n,p^l _L_p(σl)=mn,p(l), Tn,p

Lp(σ)=mn,p(σ). (3.8) Lemma3.1. Letϕ∈H^p(Ω,ρ)such thatϕ(∞)=1andϕ(zk)=0fork=1, 2,...,and let

B∞(z)=

+∞

k=1

Φ(z)−Φzk Φ(z)Φzk

−1

Φzk²

Φzk (3.9)

be the Blaschke product. Then

(i)B∞∈H^p(Ω,ρ),B∞(∞)=1,|B∞(ξ)| =+_∞

k=1|Φ(zk)|(ξ∈E);

(ii)ϕ/B∞∈H^p(Ω,ρ)and(ϕ/B∞)(∞)=1.

Proof. This lemma is proved forp=2 in [1]. The proof is based on the fact that if f ∈ H²(U), whereU= {z∈C,|z|<1}, andBis the Blaschke product formed by the zeros of f, thenf /B∈H²(U). It remains true inH^p(U) for 1≤p <∞; see [6,10].

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Lemma3.2. An extremal functionψ^∞of problem (3.3) is given byψ^∞=ϕ^∗B∞; in addition, µ^∞(ρ)=

+_∞

k=1

Φz_k^pµ(ρ). (3.10)

Proof. Ifϕ∈H^p(Ω,ρ), ϕ(∞)=1 and ϕ(z_k)=0 for k=1, 2,....Then by Lemma 2.1, we have f =ϕ/B∞∈H^p(Ω,ρ), f(∞)=1, and|B∞(ξ)| =+_∞

k=1|Φ(zk)|forξ∈E. These lead to

f ^p=

+∞

k=1

Φzk−p

ϕ ^p. (3.11)

Thus

µ(ρ)≤

+_∞

k=1

Φz_k −p

µ^∞(ρ). (3.12)

On the other hand, since the functionψ^∞=ϕ^∗B∞∈H^p(Ω,ρ),ϕ(∞)=1 andϕ(zk)= 0 fork=1, 2,..., we get

µ^∞(ρ)≤ψ^∞^p=

+∞

k=1

Φzkp

µ(ρ). (3.13)

Finally, the lemma follows from (3.12) and (3.13).

4. The main results

Definition 4.1. A measureσ=α+γis said to belong to a classAif the absolutely contin- uous partαand the discrete partγsatisfy conditions (1.3), (1.4), and (2.1) and Blaschke’s condition, that is,

+∞

k=1

Φzk−1<∞. (4.1)

We denoteλn=Φⁿ−Φn, whereΦnis the polynomial part of the Laurent expansion ofΦⁿin the neighborhood of infinity.

Definition 4.2[2]. A rectifiable curveEis said to be of classΓifλn(ξ)→0 uniformly onE.

Theorem4.3. Let a measureσ=α+γsatisfy conditions (1.3), (1.4) and Blaschke’s condi- tion (4.1); then

l→lim+∞m_n,p(l)=m_n,p(σ). (4.2)

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Proof. The extremal property ofT_n,p(z_k) gives m_n,p(σ)^p≤

E

T_n^l,p(ξ)^pρ(ξ)|dξ|+ l k=1

A_kT_n^l,p

z_k^p+

+_∞

k=l+1

A_kT_n^l,p

z_k^p

=

mn,p(l)^p+

+∞

k=l+1

AkT_n^l,p

zk^p.

(4.3)

On the other hand, from the extremal property ofT_n^l,p(zk), we can write mn,p(l)≤

E

Tn,p(ξ)^pρ(ξ)|dξ|+ l k=1

AkTn,p

zk^p1/ p

≤mn,p(σ)=Cn<∞.

(4.4)

Note thatC_ndoes not depend onl; so for alll=1, 2, 3,...,

E

T_n^l,p(ξ)^pρ(ξ)|dξ| 1/ p

< Cn. (4.5)

This implies that there is a constantC_nindependent oflsuch that for alll=1, 2, 3,..., maxT_n^l,p(z)^p:|z| ≤2< C_n. (4.6) Using (4.6) in (4.3) for large enoughlwith (4.4), we get

m_n,p(l)^p≤

m_n,p(σ)^p≤

m_n,p(l)^p+C_n

+_∞

k=l+1

A_k. (4.7)

Lettingl→ ∞, we obtain

llim→∞mn,p(l)=mn,p(σ). (4.8) Theorem4.4. Let1≤p <∞,E∈Γ, and letσ=α+γbe a measure which belongs toA. In addition, for allnandl,

m_n,p(l)≤ ^l

k=1

Φz_k

m_n,p(ρ). (4.9)

Then the monic orthogonal polynomialsTn,p(z)with respect to the measureσ have the following asymptotic behavior:

(i) lim_n_→∞(mn,p(σ)/(C(E))ⁿ)=(µ^∞(ρ))¹^{/ p}; (ii) lim_n→∞ T_n,p/[C(E)Φ]ⁿ−ψ^∞ _H^p(Ω,ρ)=0;

(iii)Tn,p(z)=[C(E)Φ(z)]ⁿ[ψ^∞(z) +εn(z)],

where εn(z)→0 uniformly on compact subsets of Ωand ψ^∞ is an extremal function of problem (3.3).

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Remark 4.5. For p=2 andEthe unit circle, condition (4.9) is proved (see [5, Theorem 5.2]). In this case, this condition can be written asγn/γ^l_n≤_l

k=1|zk|, whereγ^l_n=1/mn,2(l) andγ_n=1/m_n,2(ρ) are, respectively, the leading coeﬃcients of the orthonormal polynomials associated to the measuresσ_landα.

Proof ofTheorem 4.4. Taking the limit whenltends to infinity in (4.9) and usingTheorem 4.3, we get

m_n,p(σ) C(E)ⁿ ^≤

+_∞

k=1

Φz_k

m_n,p(ρ)

C(E)ⁿ. (4.10)

On the other hand, it is proved in [2] that

nlim→∞

mn,p(ρ) C(E)ⁿ⁼

µ(ρ)¹^{/ p}. (4.11)

Using (4.10), (4.11), andLemma 3.2, we obtain

lim sup

n→∞

mn,p(σ) C(E)ⁿ ^≤

+∞

k=1

Φzk

µ(ρ)¹^{/ p}=

µ^∞(ρ)¹^{/ p}. (4.12)

It is well known that (see [3, page 231])

∀l >0, µ(l)=µ(ρ) l k=1

Φz_k p

. (4.13)

We also have (see [3, Theorem 2.2])

nlim→∞

mn,p(l) C(E)ⁿ⁼

µ(l)¹^{/ p}. (4.14)

From (4.4), we deduce that

∀l >0, m_n,p(σ) C(E)ⁿ ^≥

m_n,p(l)

C(E)ⁿ. (4.15)

By passing to the limit whenntends to infinity in (4.15) and taking into account (4.13) and (4.14), we get

∀l >0, lim inf

n→∞

m_n,p(σ) C(E)ⁿ^≥

l k=1

Φz_k

µ(ρ)¹^{/ p}. (4.16)

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Finally, by usingLemma 3.2, we obtain

lim inf

n→∞

mn,p(σ) C(E)ⁿ ^≥

+∞

k=1

Φzk

µ(ρ)^{1/ p}=

µ^∞(σ)^{1/ p}. (4.17)

Inequalities (4.12) and (4.17) proveTheorem 4.4(i).

We obtain (ii) by proceeding as in [3, pages 234, 235].

To prove (iii), we consider the function εn= Tn,p

C(E)Φⁿ⁻ψ^∞ (4.18)

which belongs to the spaceH^p(Ω,ρ). Then by applyingLemma 2.1, we obtain

sup Tn,p(z)

C(E)Φ(z)ⁿ⁻ψ^∞(z):z∈K

=supε_n(z):z∈K≤C_Kε_n_Hp(Ω,ρ)−→0

(4.19)

for all compact subsetsKofΩ. This achieves the proof of the theorem.

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Rabah Khaldi: Department of Mathematics, University of Annaba, P.O. Box 12, 23000 Annaba, Algeria

E-mail address:[email protected]