STRONG ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT IN THE PRESENCE OF DENUMERABLE SET OF MASS POINTS

. The strong asymptotics of the monic extremal polynomials with respect to a L p ( σ ) norm are studied. The measure σ is concentrated on the segment [ − 1 , 1] plus a denumerable set of mass points which accumulate at the boundary points of [ − 1 , 1] only. Under the assumptions that the mass points satisfy Blaschke’s condition and that the absolutely continuous part of σ satisﬁes Szeg˝o’s condition.


INTRODUCTION
Let 0 < p < ∞ and σ be a positive Borel measure supported on an infinite compact set E of the complex plane. We can then define for n = 1, 2, 3, .... where P n−1 denotes the class of complex polynomials of degree at most n − 1. It is easily seen that there is at least one monic polynomial T n,p (σ, z) = z n + ... ∈ P n such that (1) T n,p (σ, z) Lp(σ) = m n,p (σ) .
A special area of research in this subject has been the study of the asymptotic behavior of T n,p (z) when n tends to infinity. There exists an extensive literature on orthogonal polynomials, but not enough on extremal polynomials. Beginning by Geronimus results in 1952 [1], who considered the case where the support E of the measure is a rectifiable Jordan curve, in particular, Widom [11] investigated the case p = ∞. Then, in 1987, Lubinsky and Saff [7] proved the asymptotic of m n,p (σ) and T n,p outside the segment [−1, 1] under a general condition on the weight function. Another result on the zero distributions of the extremal polynomials on the unit circle, was presented by X. Li and K. Pan in [6]. In 1992, Kaliaguine [2], obtained the power asymptotic for extremal polynomials when E is a rectifiable Jordan curve plus a finite set of mass points and in 2004, Khaldi presented in [4] an extension of Kaliaguine's results, where he studied the case of a measure supported on a rectifiable Jordan curve plus an infinite set of mass points. Recently, Khaldi [5], solved this problem for a measure supported on the segment [−1, 1] plus a finite set of mass points.
We mentioned that in the special case p = 2 of orthogonal polynomials, Peherstorfer and Yudiskii in [8] established the asymptotic for such polynomial on a segment [−2, +2] plus a infinite set of mass points.
In this paper, we generalize the work of Peherstorfer and Yudiskii in [8] in the case where p ≥ 2, more precisely we establish the strong asymptotic of the L p extremal polynomials {T n,p (σ, z)} associated with the measure σ which has a decomposition of the form σ = α + γ, where α is a measure with supp(α) = [−1, 1], absolutely continuous with respect to the Lebesgue measure on the segment [−1, 1] i.e. (2) dα(x) = ρ(x)dx, ρ ≥ 0, satisfying Szegő's condition and γ is a discrete measure supported on the infinite set of points Let ρ be an integrable non negative weight function on E satisfying the Szegő's condition Then we can easily see that the weight function λ defined on the unit circle by Thus the Szegő function associated with the unit circle T = {t : |t| = 1} and the weight function λ is defined by satisfying the following properties: 2) D has boundary values, almost everywhere on the unit circle T such that λ(e iθ ) = ρ(cos θ) |sin θ| = |D(t)| −p a.e. for t = e iθ ∈ T . Any function f ∈ H p (Ω, ρ) has boundary values f + and f − on both sides of E, and f + , f − ∈ L p (α).
In the Hardy space H p (Ω, ρ) we will define 2.2. Notations and lemmas. Let 1 ≤ p < ∞. We denote by µ (ρ) and µ (σ) respectively the extremal values of the following problems: We denote by ϕ * and ψ * the extremal functions of the problems (6) and (7) respectively.
Lemma 2.2. The extremal functions ϕ * and ψ * are related by is a Blaschke product.
Proof. This lemma is proved for a curve in [4, p. 374]. This proof is valid in this case, too.
is said to belong to a class A, if the absolutely continuous part α satisfies the Szegö's condition (4) and the discrete part satisfies the the Blaschke's condition Proof. This lemma is proved for p = 2 by Peherstorfer and Yudiskii in [8]. We will prove this lemma for p > 2 following the same ideas as in [8].
Without loss of generality, we assume the boundness from below of the weight function 1 |D| , so 1 |D| ≥ 2. Let 1 |Dε| be a smooth function such that 1 |Dε| ≥ 1 and (10) for ε > 0. Let us choose η > 0 such that max 1 |Dε| ≤ 1 η and denote by E ± and E ± the vicinities of ±1 of the form
Now we give the main result of this paper: be a measure which belongs to a class A. Associate with the measure σ the functions D, B and the extremal values m n,p (σ) and µ(σ) given respectively by (5), (8), (1) and (7). Then the monic extremal polynomials T n,p (σ, z) have the following asymptotic behavior as n → ∞ Proof. We recall that by putting Then, applying the Hölder inequality to the first term of (13) for p ≥ 2 we get For the second term of (13) we transform it as the following sum Since the last term approaches 0 when n tends to ∞ then we have where α n → 0, as n → ∞.
In order to estimate the last integral of (13), we transform it as follows By applying the Hölder inequality to the first term of (16) we get For the last term of (16) by using the residue Theorem we get T t n Tn,p(Ψ(t)) the last term of (18) can be estimated as So, (18) becomes   On the other hand we have Since the last term in (23) approaches 0 as n → ∞, we conclude from (22), (23) and (24), the second statement of Theorem.