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


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.


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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  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972) zbMATHGoogle Scholar
  2. 2.
    Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the second row of a Young diagram under Plancherel measure. Geom. Funct. Anal. 10(4), 702–731 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Borodin, A., Olshanski, G.: Infinite random matrices and ergodic measures. Commun. Math. Phys. 223(1), 87–123 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cassels, J.W.S.: An Introduction to Diophantine Approximation. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 45. Cambridge University Press, New York (1957) zbMATHGoogle Scholar
  5. 5.
    Deift, P.A.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. New York University Courant Institute of Mathematical Sciences, New York (1999) Google Scholar
  6. 6.
    Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137(2), 295–368 (1993) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Deift, P.A., Zhou, X.: Asymptotics for the Painlevé II equation. Commun. Pure Appl. Math. 48(3), 277–337 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Deift, P., Venakides, S., Zhou, X.: New results in small dispersion KdV by an extension of the steepest descent method for Riemann–Hilbert problems. Int. Math. Res. Not. 6, 286–299 (1997) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52(12), 1491–1552 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fokas, A., Its, A., Kitaev, A.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147, 395–430 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Foulquié Moreno, A., Martínez-Finkelshtein, A., Sousa, V.L.: On a Conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials. J. Approx. Theory 162(4), 807–831 (2010) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gakhov, F.D.: Boundary Value Problems. Dover, New York (1990). Translated from the Russian, Reprint of the 1966 translation zbMATHGoogle Scholar
  13. 13.
    Its, A., Krasovsky, I.: Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump. Contemp. Math. 458, 215–247 (2008) MathSciNetGoogle Scholar
  14. 14.
    Kuijlaars, A.B.J., McLaughlin, K.T.-R., Van Assche, W., Vanlessen, M.: The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]. Adv. Math. 188(2), 337–398 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lubinsky, D.S.: Universality limits for random matrices and de Branges spaces of entire functions. J. Funct. Anal. 256, 3688–3729 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Levin, E., Lubinsky, D.S.: Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials. J. Approx. Theory 150, 69–95 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Magnus, A.P.: Asymptotics for the simplest generalized Jacobi polynomials recurrence coefficients from Freud’s equations: numerical explorations. Ann. Numer. Math. 2, 311–325 (1995) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Simon, B.: The Christoffel–Darboux kernel. In: “Perspectives in PDE, Harmonic Analysis and Applications”, a volume in honor of V.G. Maz’ya’s 70th birthday. Proceedings of Symposia in Pure Mathematics, vol. 79, pp. 295–335 (2008) Google Scholar
  19. 19.
    Simon, B.: Fine structure of the zeros of orthogonal polynomials: a progress report. In: “Recent Trends in Orthogonal Polynomials and Approximation Theory”, a volume in honor of Guillermo López’s 60th birthday, Contemporary Math., vol. 507, pp. 241–254 (2010) Google Scholar
  20. 20.
    Slater, L.J.: Confluent Hypergeometric Functions. Cambridge University Press, Cambridge (1960) zbMATHGoogle Scholar
  21. 21.
    Szegő, G.: Orthogonal Polynomials, 4th edn. Am. Math. Soc. Colloq. Publ., vol. 23. Am. Math. Soc., Providence (1975) Google Scholar

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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