Numerical Algorithms

, Volume 70, Issue 1, pp 1–8 | Cite as

On the leading coefficient of polynomials orthogonal over domains with corners

Original Paper


Let G be the interior domain of a piecewise analytic Jordan curve without cusps. Let \(\{p_{n}\}_{n=0}^{\infty }\) be the sequence of polynomials that are orthonormal over G with respect to the area measure, with each p n having leading coefficient λ n >0. It has been proven in [9] that the asymptotic behavior of λ n as \(n\to \infty \) is given by
$$\frac{n+1}{\pi}\frac{\gamma^{2n+2}}{ {\lambda_{n}^{2}}}=1-\alpha_{n}, $$
where α n =O(1/n) as \(n\to \infty \) and γ is the reciprocal of the logarithmic capacity of the boundary G. In this paper, we prove that the O(1/n) estimate for the error term α n is, in general, best possible, by exhibiting an example for which
$$\liminf_{n\to\infty}\,n\alpha_{n}>0. $$
The proof makes use of the Faber polynomials, about which a conjecture is formulated.


Orthogonal polynomials Bergman polynomials Faber polynomials Asymptotic behavior Piecewise analytic curve 


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  1. 1.
    Carleman, T.: Über die approximation analytischer funktionen durch lineare aggregate von vorgegebenen potenzen. Archiv. för Math. Atron. och Fysik 17, 1–30 (1922)Google Scholar
  2. 2.
    Dragnev, P., Miña-Díaz, E., Michael Northington, V.: Asymptotics of Carleman polynomials for level curves of a shifted Zhukovsky transformation. Comput. Methods Funct. Theory 13, 75–89 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dragnev, P., Miña-Díaz, E.: On a series representation for Carleman orthogonal polynomials. Proc. Amer. Math. Soc. 138, 4271–4279 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dragnev, P., Miña-Díaz, E.: Asymptotic behavior and zero distribution of Carleman orthogonal polynomials. J. Approx. Theory 162, 1982–2003 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Levin, A.L., Saff, E.B., Stylianopoulos, N.S.: Zero distribution of Bergman orthogonal polynomials for certain planar domains. Constr. Approx. 19, 411–435 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Maymeskul, V., Saff, E.B.: Zeros of polynomials orthogonal over regular N-gons. J. Approx. Theory 122, 129–140 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Miña-Díaz, E.: An asymptotic integral representation for Carleman orthogonal polynomials, Int. Math. Res. Notices, 2008 (2008), article ID rnn065, 38 pagesGoogle Scholar
  8. 8.
    Miña-Díaz, E., Saff, E.B., Stylianopoulos, N.S.: Zero distributions for polynomials orthogonal with weights over certain planar regions. Comput. Methods Funct. Theory 5, 185–221 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Stylianopoulos, N.: Strong asymptotics of Bergman polynomials over domains with corners and applications. Constr. Approx. 38, 59–100 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Suetin, P.K.: Polynomials orthogonal over a region and Bieberbach polynomials, vol. 100 of Proc. Steklov Inst. Math. Amer. Math. Soc. Translations (1974)Google Scholar
  11. 11.
    Suetin, P.K.: Series of Faber Polynomials. Gordon and Breach Science Publications, Amsterdam (1998)MATHGoogle Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MississippiUniversityUSA

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