Proceedings of the 3rd Conference will be published in Aportaciones Matemáticas, a series edited by the Mexican Society of Mathematics, and also in the Eletronic Library of the European Mathematical Society, which is accessable under http://www.emis.de/ on the internet, edited by B. Bank, J. Bustamante, M. Florenzano, M. Gudat, D. Hinrichsen, M. A. Jimenez, H. Th. Jongen, G. López Lagomasino and F. Marcellán.

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Title of the paper

Author address

Abstract

Let {h_{n}(z)} be the sequence of polynomials that satisfies

∫_{0}^{+∞}h_{m}(x)h_{n}(x)x^{−λ_{n}}dρ(x) = δ_{mn}, 0 ≤ m ≤ n,

where {λ_{n}} is a sequence of reals. For a wide class of weights dρ(x), the logarithmic asymptotics of {h_{n}(z)} is obtained under the additional assumption

lim_{n→∞} \frac{λ_{n}}{2n} = θ ∈ R.

Some applications in approximation theory are presented.

Keywords and phrases: Laurent-type orthogonal polynomials, asymptotics, Christoffel function, potential theory.

1 Introduction

Let ρ be a positive Borel measure on IR_{+} = [0; +∞). Consider a sequence of reals {λ_{n}}, n ≥ 0, and set

Λ_{1} := sup_{n} λ_{n} Λ_{2} := sup_{n}(2n − λ_{n}).  (1)

Assume that

c_{ν} := ∫_{0}^{+∞}x^{ν}dρ(x) < +∞, −Λ_{1} ≤ ν ≤ Λ_{2}, ν ∈ Z.  (2)

This condition guarantees that all forthcoming integrals exist. If ([1]) holds, we

call the pair (ρ, {λ_{n}}) admissible . By h_{n}(z) = κ_{n}z^{n} + …, κ_{n} > 0, we denote the n-th orthonormal polynomial with respect to the measure x^{−λ_{n}}dρ(x), x > 0; hence,

∫_{0}^{+∞}h_{n}(x)h_{m}(x)\frac{dρ(x)}{x^{λ_{n}}} = δ_{n,m}, 0 ≤ m ≤ n.

These polynomials arise in several problems of rational approximation of Stieltjes functions

ρ̂(z) := ∫_{0}^{+∞}\frac{dρ(t)}{z − t}.

2 Title of the section

See [1], [2,4]

Theorem 1 This is Theorem 1.

Σf(x)  (3)

References

[1] A.A. Gonchar, E.A. Rakhmanov: Equilibrium measures and the distribution of zeros of extremal polynomials. Math. Sb. 125 (167), 1984; Engl. transl. in Math. USSR Sb. 53, 1984, 1, 119-130.

[2] G. López Lagomasino, A. Martínez Finkelshtein: Rate of convergence of twopoint Pade approximants and logarithmic asymptotics of Laurent-type orthogonal polynomials. Submitted to Constr. Approx., to appear.

Title of the paper 5

[3] G. López Lagomasino: Rational approximation and orthogonal polynomials. Proceedings of the 2nd International Seminar on Approximation and Optimization, Havana 1993.

[4] G.G. Lorentz: Approximation by incomplete polynomials (problem and results). In: ”Pade and Rational Approximation: Theory and Applications” (E.B. Saff and R.S. Varga, eds.), Academic Press, N.Y., 1977, 289-302.

[5] E.A. Rakhmanov: On asymptotic properties of polynomials, orthogonal on the real axis. Mat. Sb. 119 (161), 1982, 161-203; Engl. transl. in Math. USSR Sb. 47, 1984, 155-193.