STRONG ASYMPTOTICS FOR Lp EXTREMAL POLYNOMIALS OFF A COMPLEX CURVE

Recently, a series of results concerning the asymptotic of the extremal polynomials was established for the case of B = Lp(F,σ), 1≤ p ≤∞, where σ 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 σ . 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,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 measure σ has a decomposition of the form σ = α+ γ, (1.2)


Introduction
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 T n 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, dist T n ,0 = min dist Q n ,0 : Q n (z) = z n + a n−1 z n−1 + ••• + a 0 = m n (B). (1.1) Recently, a series of results concerning the asymptotic of the extremal polynomials was established for the case of B = L p (F,σ), 1 ≤ p ≤ ∞, where σ 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 σ.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 L 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 measure σ has a decomposition of the form where α is a measure supported on a closed rectifiable Jordan curve E as defined in [2] and γ 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 σ is a rectifiable Jordan curve E plus an infinite discrete set of mass points which accumulate on E.More precisely, σ = α + γ, where the measure α and its support E are defined as in [3], that is, (1.4) 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 {z k } ∞ k=1 is redundant.

The
Let E be a rectifiable Jordan curve in the complex plane, We denote by Φ the conformal mapping of Let ρ be an integrable nonnegative weight function on E satisfying the Szegö condition where Rabah Khaldi 373
Let T l n,p (z) and T n,p (z) be the extremal polynomials with respect to the measures σ l and σ, respectively, that is, .., and let be the Blaschke product.Then Proof.This lemma is proved for p = 2 in [1].The proof is based on the fact that if f ∈ H 2 (U), where U = {z ∈ C, |z| < 1}, and B is the Blaschke product formed by the zeros of f , then f /B ∈ H 2 (U).It remains true in H p (U) for 1 ≤ p < ∞; see [6,10].

Lemma 3.2. An extremal function ψ ∞ of problem (3.3) is given by ψ
On the other hand, since the function (3.13) Finally, the lemma follows from (3.12) and (3.13).

The main results
Definition 4.1.A measure σ = α + γ is said to belong to a class A if the absolutely continuous part α and the discrete part γ satisfy conditions (1.3), (1.4), and (2.1) and Blaschke's condition, that is, We denote λ n = Φ n − Φ n , where Φ n is the polynomial part of the Laurent expansion of Φ n in the neighborhood of infinity.Definition 4.2 [2].A rectifiable curve E is said to be of class Γ if λ n (ξ) → 0 uniformly on E. Theorem 4.3.Let a measure σ = α + γ satisfy conditions (1.3), (1.4) On the other hand, from the extremal property of T l n,p (z k ), we can write Note that C n does not depend on l; so for all l = 1,2,3,..., This implies that there is a constant C n 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 Then the monic orthogonal polynomials T n,p (z) with respect to the measure σ have the following asymptotic behavior: , where ε n (z) → 0 uniformly on compact subsets of Ω and ψ ∞ is an extremal function of problem (3.3).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 γ n /γ l n ≤ l k=1 |z k |, where γ l n = 1/m n,2 (l) and γ n = 1/m n,2 (ρ) are, respectively, the leading coefficients of the orthonormal polynomials associated to the measures σ l and α.
Proof of Theorem 4.4.Taking the limit when l tends to infinity in (4.9) and using Theorem 4.3, we get  for all compact subsets K of Ω.This achieves the proof of the theorem.

. 1 )
Condition (2.1) allows us to construct the so-called Szegö function D associated with the curve E and the weight function ρ: