Strong Asymptotic Analysis of OLPs on the Unit Circle by Riemann-Hilbert Approach

  • Yufeng Wang
  • Yifeng Lu
  • Jinyuan DuEmail author
Part of the Trends in Mathematics book series (TM)


In this article, we will deal with the asymptotics of the monic orthogonal Laurent polynomials (OLPs) on the unit circle with respect to a strictly-positive analytic weight by Riemann-Hilbert approach. We first construct a matrix Riemann-Hilbert problem (RHP) which is the Fokas-Its-Kitaev characterization. Then, the strong asymptotic formulas of OLPs are obtained by employing Deift-Zhou steepest descent analysis. Furthermore, the asymptotic formulas of the leading coefficient and the trailing coefficient are simultaneously obtained.


Orthogonal Laurent polynomial Riemann-Hilbert approach Riemann-Hilbert problem Strong asymptotics Cauchy-type integral operator 

Mathematics Subject Classification (2010)

42C05 41A35 30G30 



While the corresponding author visited Free University Berlin in summer 2005 on basis of State Scholarship Fund Award of China, our group began to explore the application of Riemann-Hilbert approach, and made a debut [3]. During that time Professor H. Begehr carefully reviewed this manuscript and offered a lot of suggestions. All the authors are very grateful to Professor H. Begehr for his long-term support and help.

This work was supported by NNSF for Young Scholars of China (No. 11001206) and NNSF (No. 11171260).


  1. 1.
    G. Szegő, Orthogonal Polynomials, 4th edn. AMS Colloquium Publications, vol. 23 (American Mathematical Society, Providence, 1975)Google Scholar
  2. 2.
    A. Martínez-Finkelshtein, K.T-R. McLaughlin, E.B. Saff, Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics. Constr. Approx. 24, 319–363 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Z.H. Du, J.Y. Du, Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on the unit circle. Chinese Ann. Math. 27A(5), 701–718 (2006)MathSciNetGoogle Scholar
  4. 4.
    Z.H. Du, J.Y. Du, Orthogonal trigonometric polynomials: Riemann-Hilbert analysis and relations with OPUC. Asymptot. Anal. 79, 87–132 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    T. Kriecherbauer, K.T-R. McLaughlin, Strong asymptotics of polynomials orthogonal with respect to Freud weights. Int. Math. Res. Not. 6, 299–333 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Deift, T. Kriecherbauer, K.T-R. McLaughlin, S. Venakides, X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Bo, R. Wong, A uniform asymptotic formula for orthogonal polynomials associated with \(\exp \left \{-x^4\right \}\). J. Approx. Theory 98, 146–166 (1999)Google Scholar
  8. 8.
    A.B.J. Kuijlaars, T.R. Mclaughlin, W.V. Assche, M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1]. Adv. Math. 188(2), 337–398 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A.S. Fokas, A.R. Its, A.V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147(2), 395–430 (1992)CrossRefGoogle Scholar
  10. 10.
    P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problem, asymptotics for the MKdV equation. Ann. Math. 137(2), 295–368 (1993)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Martínez-Finkelshtein, Szegő polynomials: a view from the Riemann-Hilbert window. Electron. Trans. Numer. Anal. 25, 369–392 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    A.B.J. Kuijlaars, Riemann-Hilbert analysis for polynomials, in Orthogonal Polynomials and Special Functions: Leuven 2002. Lecture Notes Mathematics, vol. 1817 (Springer, Berlin, 2003), pp. 167–210Google Scholar
  13. 13.
    P. Deift, Orthogonal polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 13 (Courant Institute of Mathematical Sciences, New York, 1999)Google Scholar
  14. 14.
    A.S. Fokas, A unified approach to boundary value problems, in CBMS-NSF Region Conference Series in Applied Mathematics, vol. 78 (Society for Industrial and Applied Mathematics, Philadelphia, 2008)CrossRefGoogle Scholar
  15. 15.
    H. Begehr, Complex Analytic Methods for Partial Differential Equation: An Introductory Text (World Scientific, Singapore, 1994)CrossRefGoogle Scholar
  16. 16.
    J.K. Lu, Boundary Value Problems For Analytic Functions (World Scientific, Singapore, 1993)zbMATHGoogle Scholar
  17. 17.
    Y.F. Wang, Y.J. Wang, On Riemann problems for single-periodic polyanalytic functions. Math. Nachr. 287, 1886C1915 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Y.F. Wang, P.J. Han, Y.J. Wang, On Riemann problem of automorphic polyanalytic functions connected with a rotation group. Complex Var. Elliptic Equ. 60(8), 1033–1057 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    F.D. Gakhov, Boundary Value Problems (Pergamon Press, Oxford, 1966)CrossRefGoogle Scholar
  20. 20.
    N.I. Muskhelishvili, Singular Integral Equations, 2nd edn. (Noordhoff, Groningen, 1968)zbMATHGoogle Scholar
  21. 21.
    J.A. Shohat, J.D. Tamarkin, The Problem of Moments. American Mathematical Society Surveys, vol. II (AMS, New York, 1943)Google Scholar
  22. 22.
    W.B. Jones, W.J. Thorn, H. Waadeland, A strong Stieltjes moment problem. Trans. Am. Math. Soc. 261, 503–528 (1980)MathSciNetCrossRefGoogle Scholar
  23. 23.
    A. Bultheel, P. González Vera, E. Hendriksen, O. Njåstad, Orthogonal Rational Functions. Cambridge Monographs on Applied & Computational Mathematics, vol. 5 (Cambridge University Press, Cambridge, 1999)Google Scholar
  24. 24.
    W.B. Jones, O. Njåstad, W.J. Thron, Orthogonal Laurent polynomials and the strong Hamburger moment problem. J. Math. Anal. Appl. 98, 528–554 (1984)MathSciNetCrossRefGoogle Scholar
  25. 25.
    W.B. Jones, O. Njåstad, Orthogonal Laurent polynomials and strong moment theory: a survey. J. Comput. Appl. Math. 105(1–2), 51–91 (1999)MathSciNetCrossRefGoogle Scholar
  26. 26.
    K.T-R. McLaughlin, A.H. Vartanian, X. Zhou, Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights. Int. Math. Res. Pap. 216, Art. ID 62815 (2006)Google Scholar
  27. 27.
    K.T-R. McLaughlin, A.H. Vartanian, X. Zhou, Asymptotics of Laurent polynomials of odd degree orthogonal with respect to varying exponential weights. Constr. Approx. 27(2), 149–202 (2008)MathSciNetCrossRefGoogle Scholar
  28. 28.
    R. Cruz-Barroso, C. Díaz Mendoza, R. Orive, Orthogonal Laurent polynomials. A new algebraic approach. J. Math. Anal. Appl. 408, 40–54 (2013)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina
  2. 2.School of ScienceLinyi UniversityShandongChina

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