ASYMPTOTICS FOR ORTHOGONAL POLYNOMIALS OFF THE CIRCLE

We study the strong asymptotics of orthogonal polynomials with respect to a measure of the type dμ/2π + ∑∞ j=1Ajδ(z− zk), where μ is a positive measure on the unit circle Γ satisfying the Szegö condition and {zj}j=1 are fixed points outside Γ. The masses {Aj}j=1 are positive numbers such that ∑∞ j=1Aj < +∞. Our main result is the explicit strong asymptotic formulas for the corresponding orthogonal polynomials.


Introduction
Let ν be a finite positive measure defined on the Borelian σ-algebra of C and concentrated on the set Γ ∪ {z k } ∞ k=1 , where Γ = {z ∈ C : |z| = 1} and z k are such that |z k | > 1.The measure ν is defined as follows: where µ is concentrated on Γ and is absolutely continuous with respect to the Lebesgue measure dθ on [−π,+π], that is, The masses {A k } ∞ k=1 satisfy and δ(z − z k ) is the Dirac measure supported at the point z k .We denote by {Φ n (z)} and {ϕ n (z)} the systems of orthonormal polynomials associated to the measures ν and µ/2π, respectively.
A similar study has been done by Benzine in [2] and Khaldi and Benzine in [6], in the case of a curve instead of a circle.To prove their results, Benzine and Khaldi imposed some conditions on the measure ν which are difficult to check.In [2,6] k=1 , E being a curve.In the present work, we prove the same result in the case of a circle instead of a curve with some assumptions on ν which are not difficult to check.
To get the asymptotic formula of Φ n (z) (Theorem 5.4), we prove two intermediate results, Theorems 5.1 and 5.2.Theorem 5.1 establishes that the coefficients γ n of z n of the polynomials Φ n (z) tend (when → ∞) to the coefficient γ n of z n of the polynomials Φ n (z).
The new conditions on the measure ν, Theorems 5.1 and 5.2, as well as their consequences on the proof of Theorem 5.3 constitute our main contribution with respect to previous works by Benzine and Khaldi [2,6].
The asymptotic behavior of the polynomials {Φ n (z)} has been established by Li and Pan [9] in the case where the measure µ is not absolutely continuous, and by Kaliaguine and Benzine [5] in the case of a curve with µ an absolutely continuous measure.R. Khaldi and R. Benzine 39

The space
In what follows, we suppose that the weight function ρ (which defines the measure µ) belongs to the Szegö class, that is, This allows us to construct the so-called Szegö function D associated with the domain G and weight function ρ: such that where D is the angular limit of D. We say that Finally, the space L 2 (Γ,ρ|dξ|) is the space of functions f defined on the unit circle Γ, with values in C and for which +π Let f and g be in L 2 (Γ,ρ|dξ|), we define then (L 2 (Γ,ρ|dξ|), • L 2 (Γ,ρ|dξ|) ) is a Hilbert space.We summarize the basic properties of the space H 2 (G,ρ) in the following theorem and lemma.

The set of measures A
3.1.Definitions.We will give in this part the new conditions on the discrete part of the measure ν which allows us later to get the asymptotic formula of the orthonormal polynomials {Φ n (z)}.
We need to introduce some notation.The * -transform (P n ) * (z) of a polynomial P n (z) of degree n is defined as where As an example of families of points and masses satisfying condition (3.3), we can take the subsequences of the points {z k } ∞ k=1 and the masses {A k } ∞ k=1 of the following form: R. Khaldi and R. Benzine 41 On the other hand, the sequence (see [9, pages 66-67]).

Extremal properties of the orthogonal polynomials
In this section and the following one, we assume that the measure ν belongs to the class A.
We denote by ᏼ n the vector space of all polynomials of degree at most n, and by ᏼ n1 the set of monic polynomials of degree exactly equal to n.It is easy to see that the polynomial (1/γ n )Φ n is the optimal solution of the following extremal problem: with We define µ n (ν), µ n ( ), µ n (ρ), µ(ρ), µ(ν), m n (ρ), and m n ( ) to be the optimal values of the extremal problems ) ψ n e iθ 2 ρ(θ)dθ ) with Denote by ψ ∞ n , ψ n , ψ ρ n , ψ, and ψ ∞ the optimal solutions of the extremal problems (4.3), (4.4), (4.5), (4.6), and (4.7), respectively.We are now ready to state the lemmas which we need in the sequel.

Main results
Theorem 5.1.
(B1) First proof.For extremal functions ψ ∞ n , the relation (5.39) leads to For every compact set K ⊂ G, put then for some subsequence ρ < +∞.From (5.41), we deduce that {ψ ∞ n } n∈Λ is a normal (or Montel) family in G. Therefore, we can find a function ϕ that is the uniform limit (on the compact subsets of G) of some subsequence {ψ ∞ n } n∈Λ1 of {ψ ∞ n } n∈Λ (see [10]).From Lemma 2.2, we get that On the other hand, it is obvious that ϕ(∞) = 1, and (5.39) implies that (5.49) We also have (see [5,Lemma 4.2]) uniformly for every compact subset of {z ∈ C : |z| > 1}.ψ ∞ n and ψ ∞ are, respectively, the optimal solutions of the extremal problems (4.3)There remains the asymptotic formula (5.55).Recall that (