ASYMPTOTIC BEHAVIOR OF ORTHOGONAL POLYNOMIALS CORRESPONDING TO A MEASURE WITH INFINITE DISCRETE PART OFF AN ARC

We study the asymptotic behavior of orthogonal polynomials. The measure is concentrated on a complex rectifiable arc and has an infinity of masses in the region exterior to the arc. 2000 Mathematics Subject Classification. 42C05, 30E15, 30E10.


Introduction. Kaliaguine has studied in
the asymptotic behavior of orthogonal polynomials associated to a measure of the type σ l = α + γ l , where α is concentrated on a complex rectifiable arc E and is absolutely continuous with respect to the Lebesgue measure |dξ| on the arc, and γ l is a finite discrete measure with masses A k at the points z k ∈ Ext(E), k = 1, 2,...,l, that is, γ l = l k=1 A k δ z k , A k > 0, where δ z k being the Dirac measure at the points z k .In this paper, we generalize the previous study, when σ = α + γ, where α possess the same properties as in [3] and γ is concentrated on an infinite discrete part We note that the cases of a closed curve and a circle studied in [4,5] are different from the case of an arc with respect to the asymptotics of orthogonal polynomials.

The space H 2 (Ω,ρ).
Suppose that E is a rectifiable arc in the complex plane, where Φ : Ω → G is the conformal mapping.We denote by Ψ the inverse of Φ.
Let ρ(ξ) be an integrable nonnegative function on E. If the weight function ρ(ξ) satisfies the Szegö condition Let f (z) be an analytic function in Ω, we say that f (z) ∈ H 2 (Ω,ρ) if and only if f (Ψ (w))/D(Ψ (w)) ∈ H 2 (G), and for a function F analytic in G, F ∈ H 2 (G) if and only if The space H 2 (D) is well known (see [6]).Any function from H 2 (Ω,ρ) has boundary values f + , f − on both sides of E, f + ,f − ∈ L 2 (ρ).We define the norm in Hardy space by Here, we take the integral on both sides of E.
3. Extremal properties of the orthogonal polynomials.We denote by P n the set of polynomials of degree almost n.Define µ(ρ), µ * (ρ), m n (ρ), m n (σ l ), and m n (σ ) as the extremal values of the following problems: ) ) We denote, respectively, by ϕ * and ψ * the extremal functions of the problems (3.1) and (3.2).We denote by {T l n (z)} and {T n (z)} the systems of the monic orthogonal polynomials, respectively, associated to the measures σ l and σ , that is, It is easy to see that the polynomials {T l n (z)} and {T n (z)} are, respectively, the optimal solutions of the extremal problems (3.4) and (3.5).

, and let
be the Blashke product, then The proof is the same as that of Lemma 3.1 given in [1].Lemma 3.2.The extremal functions ϕ * and ψ * are connected by The proof is the same as that of a closed curve given in [2, Lemma 4.2].We replace the finite Blashke product by the infinite product B and using its properties announced by Lemma 3.1.

Main results
Definition 4.1.The measure σ = α+γ belongs to the class A (and we write σ ∈ A), if the absolutely continuous part α and the discrete part of σ satisfy (in addition to conditions (1.1) and ( 2 An arc E is from C α+ class if E is rectifiable and its coordinates are α-times differentiable, with αth derivatives satisfying a Lipschitz condition positive exponent.Proof.The extremal property of T n (z) implies that then On the other hand, the extremal property of T n (z) implies that (4.5) According to the reproducing property of the kernel function K n (ξ, z) (see [7]), and T l n (z) ∈ P n , we have The Scharwz inequality and the fact that |Φ(ξ)| = 1 for ξ ∈ E and K n (z, z k ) ∈ P n imply (4.7) The inequalities (1.1), (4.5), and (4.7) imply , ∀n, ∀l.(4.12) Suppose that E ∈ C 2+ .Then we have where , n → 0 uniformly on the compact subsets of Ω.
Proof.By passing to the limit when l tends to infinity and using Theorem 4.2 and (4.12), we obtain The extremal property of the polynomials T n (z) and the fact that |Φ(ξ)| = 1, for ξ ∈ E imply (see [2] for details) Now we take the integral by the triangular inequality, we have Then we deduce that

.21)
By using the parallelogram rule in H 2 (Ω,ρ), we have where we have used the fact that lim inf n→∞ g n 2 For the asymptotics in the region exterior to the arc E we need the Szegö reproducing kernel function K(ξ, z) (see [8, page 173]) and the fact that

.26)
The first integral approaches 0 as n → ∞ (part 2 of Theorem 4.3), the second one may be transformed into the form where λ n → 0 (coefficients of an integrable function).This proves part 3.
Remark 4.4.It is not difficult to find families of points {A k } ∞ k=1 and {z k } ∞ k=1 satisfying condition (4.12).For example if E = [−1, +1], then We can take z k such that