L p extremal polynomials (0 < p < ∞ ) in the presence of a denumerable set of mass points

. We study, for all p > 0 the asymptotic behavior of L p extremal polynomials with respect to the measure α = β + γ,α denotes a positive measure whose support is the unit circle Γ plus a denumerable set of mass points, which accumulate at Γ and satisfy Blaschke’s condition and β = β a + β s , β s the absolutely continuous part of the measure satisfies Szeg¨o condition and β s the singular part. Our main result is the explicit strong asymptotic formulas for the L p extremal polynomials.


Introduction
Let α be a finite positive Borel measure with an infinite compact support in the complex plane. We denote T n,p,α (z) = z n + a n−1 z n−1 + ... + a 0 , a n−1 , ..., a 0 ∈ C, the monic polynomial of degree n with respect to measure α. Then the extremal or general Chebyshev polynomial T n,p,α is a monic polynomial that minimize the L p (α) norm in the set of monic polynomials of degree n m n,p (α) = T n,p,α L p (α) = min ∥Q n ∥ L p (α) : Q n (z) = z n + a n−1 z n−1 + ... + a 0 .
For p = 2, we have the special case of orthogonal polynomials with respect to the measure α. A large number of works have been done on this subject; see, for example [2], [3].
A series of results concerning the asymptotics of the L p extremal polynomials was established. In [1] Geronimus has given such asymptotics in the case where the support of the measure α is a rectifiable Jordan curve with some smoothness condition. An extension of the Geronimus's result has been given by Kaliaguine [5], where the measure is supported by a rectifiable Jordan curve plus a finit number of mass points. In [8] Laskri, Benzine have obtained the asymptotics of L p extremal polynomials on a complete curve plus an infinite number of mass points with some conditions of smoothness. Recently, X. Li and K. Pan [4] investigated the zero distributions of L p extremal polynomials on the unit circle (1 < p < ∞).
In this note we shall study the power asymptotic of the L p (α) extremal polynomials T n,p,α outside the unit circle Γ. We are also inspired by the work of Peherstorfer and Yuditskii [6] and Bello Hernandez, Marcellan and Minguez [7] to reach the desired asymptotic formula for L p extremal polynomials T n,p,α . However, we can use a new technic due to Peherstorfer and Yuditskii [6]. In which the authors showed the asymptotic formula for the orthogonal polynomials, their method based on the use of a measure concentred on a segment plus an infinite points. we try to carry over some of the main ideas of [6] for L p (α) extremal polynomials with the necessary modification imposed by the nature of our problem, we can show the asymptotic formula for the case of L p (α) extremal polynomials. First of all, we set some notations.
Let Γ = {z ∈ C : |z| = 1} , G = {z ∈ C : |z| > 1} the exterior of the unit circle and let α be a measure which has a decomposition of the form where β a is the absolutely continuous part of β with supp(β a ) = Γ, respect to the Lebesgue measure |dζ| on [−π, +π], that is and supp(β s ) ⊂ Γ ( β s the singular part of β) and γ is a point measure supported on where each δ z k is the Dirac measure at the point z k . Suppose that the absolutely continuous part β a of β, satisfies the following Szegö condition : Condition (4) allows us to construct the so-colled Szegö function D G,ρ associated with G and the weight function ρ(ζ) with the following properties [5]: The following function is the Szegö function for the domain G : One can find in the literature several technics to solve the problem of the asymptotic behavior of L p extremal polynomials.
The technic that we use consists to generate and to study some sequences of extremal problems in Hardy spaces.

Asymptotic behavior
Let Γ be a unit circle, and the support of the measure α is Γ plus an infinite discrete set of mass points which accumulate on Γ. We associated to the measures β a and α the extremal constants m n,p (β a ), m n,p (α) and the L p extremal polynomial T n,p,β a , T n,p,α , as follows : m n,p (β a ) = T n,p,β a L p (Γ) = min Q n (z)=z n +...
m n,p (α) = T n,p,α L p (α) = min . ∥Q n ∥ L p (α) , Q n (z) = z n + ... , with We pose 0 < p < 1. The optimal solution φ * of the following extremal problem : is given by i.e., the infinimum (9) denote µ(β a ) is reached for (9): where D Γ,ρ , Szegö function associated with the unit disk and the weight function ρ and the analytic function φ * belongs to H p (G, ρ) if and only if φ * (z)D G,ρ (z) ∈ H p (G), where H p (G) is the usual Hardy space associated with the exterior G of the unit circle.
We denote by µ(α) the extremal value of the problem: We denote by the Blaschke product and we denote by ψ * (z) = φ * (z)B(z) is an extremal function of problem (12). The optimal values of the problems (9) and (12) are connected by: Lemma 2.1. [5] If f (z) ∈ H p (G, ρ), then for every compact set K ⊂ G there is a constant C(K) (C(K) depending only on K ) such that :

Definition 2.3.
If the measure α = β a + β s + γ is such that a verifies the condition (4) and its discrete part verifies then we say that α ∈ E.
Theorem 2.4. Let Γ be the unit circle and α = β a + β s + γ, such that α ∈ E, and D G,ρ , the Szegö function associated with G, then ϵ n (z) → 0 uniformly on the compact sets of G.