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Constructive Approximation

, Volume 33, Issue 2, pp 219–263 | Cite as

Asymptotics of Orthogonal Polynomials for a Weight with a Jump on [−1,1]

  • A. Foulquié Moreno
  • A. Martínez-Finkelshtein
  • V. L. SousaEmail author
Article

Abstract

We consider the orthogonal polynomials on [−1,1] with respect to the weight
$$w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \varXi _{c}(x),\quad\alpha,\beta>-1,$$
where h is real analytic and strictly positive on [−1,1] and Ξ c is a step-like function: Ξ c (x)=1 for x∈[−1,0) and Ξ c (x)=c 2, c>0, for x∈[0,1]. We obtain strong uniform asymptotics of the monic orthogonal polynomials in ℂ, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n→∞. In particular, we prove for w c a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel–Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior.

For the asymptotic analysis we use the steepest descent method of Deift and Zhou applied to the noncommutative Riemann–Hilbert problems characterizing the orthogonal polynomials. The local analysis at x=0 is carried out in terms of confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.

Keywords

Orthogonal polynomials Asymptotics Riemann–Hilbert analysis Zeros Local behavior Confluent hypergeometric functions Reproducing kernel Universality de Branges space 

Mathematics Subject Classification (2000)

42C05 30C15 33C15 41A60 46E22 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • A. Foulquié Moreno
    • 1
  • A. Martínez-Finkelshtein
    • 2
    • 3
  • V. L. Sousa
    • 4
    Email author
  1. 1.Department of Mathematics and Center of Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal
  2. 2.Department of Statistics and Applied MathematicsUniversity of AlmeríaAlmeríaSpain
  3. 3.Instituto Carlos I de Física Teórica y ComputacionalGranada UniversityGranadaSpain
  4. 4.Escola Secundária João da Silva Correia, and Center of Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal

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