Numerical Algorithms

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

On the leading coefficient of polynomials orthogonal over domains with corners

  • Erwin Miña-Díaz
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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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MississippiUniversityUSA

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