ON THE ASYMPTOTIC BEHAVIOR OF L p EXTREMAL POLYNOMIALS

. Let β denotes a positive Szeg¨o measure on the unit circle Γ and δ z k denotes an anatomic measure ( δ Dirac) centered on the point z k . We study, for all p > 0 , the asymptotic behavior of L p extremal polynomials with respect to a measure of the type α = β + ∞ X k =1 A k δ z k , where { z k } ∞ k =1 is an inﬁnite collection of points outside Γ. MSC 2000. 42C05, 30E15.


INTRODUCTION
Let α be a finite measure defined on the borel sets of C and of compact support F. We denote by m n,p (α, F ), n ∈ N, p > 0 the extremal constants m n,p (α, F ) = min Q n Lp(α,F ) : Q n = z n + a n−1 z n−1 + ... + a 0 , a 0 , ..., a n−1 ∈ C , and by T n,p (α, F ) the associated extremal polynomials (we suppose that z n ∈ L p (α, F ), n ∈ N).The case p = 2 is the special case of L 2 (α, F ) monic orthogonal polynomials.
There are many interesting problems about orthogonal or extremal polynomials.The most important and difficult ones are their asymptotic behavior and zero distributions.
In this paper we shall study the strong asymptotics of m n,p (α, F ) and T n,p (α, F ) in the case where 0 This work is a generalization of the one of Kaliaguine [9], as well as those of [15] and [12].In [9] Kaliaguine uses a measure concentrated on a rectifiable Jordan curve plus a finite number of points {z k } l k=1 .The passage from a finite number to an infinite number of points is a difficult problem and its resolution required, in the case p = 2, several years (see [2], [8], [10], [11] and [29]).In ( [15], [12]) the authors only consider the relatively simple case 1 ≤ p ≤ ∞.We will get in this paper the asymptotic behavior of the L p extremal polynomials T n,p,α (z), for 0 < p < ∞.
We give in section 2 some basic definitions and lemmas in the H p (G, ρ) spaces and define extremal problems on these spaces.Our main result, Theorem 3.1, is stated in section 3.
Let E be a Jordan closed rectifiable curve, The method used in this paper is applicable to the case of a contour instead of a circle by considering the functions Φ(z) and Ψ(w).We shall give a result in our future paper (see [15]).

EXTREMAL PROBLEMS IN THE H P (G, ρ) SPACES
In this section, we introduce some notations and definitions concerning the Let ρ (ξ) be an integrable non negative function on Γ.If the weight function ρ (ξ) satisfies the Szegö condition ( 1) then, one can construct the so-called Szegö function D ρ (z) associated with the domain G and the weight function ρ (ξ) with the following properties: where D ρ (ξ) = lim z→ξ D ρ (z) , (a.e. on Γ).Define the function D as follows One gets a construction and an explicit representation of D(w) in [26] (4) )) e iθ +w e iθ −w dθ .
Let f be an analytic function in G.For p > 0, we say that f belongs to H p (G, ρ) if f.D ρ is a function from the space H p (G).For a function F analytic in G, we say that F ∈ H p (G) if and only if F (1/w) ∈ H p (U ) .The space H p (U ) is well known (see [4,13,23,14]).
Each function f (z) from H p (G, ρ) has limit values on Γ and (5) The following lemmas summarize some properties of the H p (G, ρ) spaces.

MAIN RESULTS
We now study the asymptotic behavior of the extremal polynomials {T n,p,α (z)}.As previously let α = β + γ be a finite positive measure defined on the Borelian σ-algebra of C and concentrated on the set k=1 be an infinite set of points which lay at the exterior of Γ, β and γ are defined as follows: β is a measure concentrated on Γ and is absolutely continuous with respect to the Lebesgue measure |dξ| on the arc, i.e.: (14) dβ and γ is a discrete measure with masses A k at the points z k ∈ Ext(Γ), k = 1, 2, ..., i.e.: where δ z k denotes the (Dirac delta) unit measure supported at the point z k .By P n,1 we denote the set of monic polynomials of degree n.For 0 < p < ∞, define m n,p (α), m n,p (α ), m n,p (β), T n,p,α (z) = z n + ... ∈ P n,1 , T n,p,α (z) ∈ P n,1 and T n,p,β (z) ∈ P n,1 as follows: where we say that the measure α belongs to the class BA (denoted by α ∈ BA) if the absolute and discrete parts of α, satisfy in addition to the natural relations (1), ( 14) and (15), the following conditions Condition (18) was proven by Khaldi and Benzine [11] and Perherstorfer and Yuditskii ([29], pp.3217-3219 ) for the case p = 2.To arrive at their results, Khaldi and Benzine used properties of orthogonality of the polynomials T n,2,α , but the proof by Perherstorfer and Yuditskii ( [29]) is essentially based in many points on the extremal properties of the polynomials T n,2,α .
We conclude this section by formulating the main result of this paper.
Proof.(i) p > 0 and α ∈ L, then We will present two proofs of this inequality.First proof of (21).
The extremal property of T n,p,α (z) and T n,p,α (z) imply Using this result and theorem 2.2 of [9] , we obtain Now, using the fact that (see [9], formula (1.9)) we obtain when → ∞ Second proof of (21).Putting (26) φ * n,p = T n,p,α (z) /z n , and using (20) we get: , we have ( 28) . This result and lemma 2.1 imply that φ * n,p , n ∈ N 1 is a normal family in G.So that we can find a function ψ (z) that is the uniform limit (on the compact subsets of G) of some subsequence φ From Lemma 2.2, we get ψ ∈ H p (G, ρ) and ( 29) .
We use the extension of Keldysh theorem (see theorem 2 pp.430-431 of [1]).More precisely if one notes that in our case the singular part of the measure β is equal to zero and if one takes in consideration the transformation z → 1 z , we obtain the following version of theorem 2 of [1].This completes the proof of Theorem 3.1.