On the leading coefficient of polynomials orthogonal over domains with corners

Abstract

Let G be the interior domain of a piecewise analytic Jordan curve without cusps. Let {pn} ∞ n=0 be the sequence of polynomials that are orthonormal over G with respect to the area measure, with each pn having leading coefficient λn > 0. It has been proven in [9] that the asymptotic behavior of λn as n → ∞ is given bywhere αn = O(1/n) as n → ∞ 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 αn is, in general, best possible, by exhibiting an example for which lim inf n→∞ nαn > 0.The proof makes use of the Faber polynomials, about which a conjecture is formulated.2010 Mathematics Subject Classification. 42C05, 30E10, 30E15, 30C10, 30C15.

Bergman Orthogonal Polynomials and the Grunsky Matrix

“…Finally, the case of piecewise analytic Γ without cusps has been studied recently by the second author, who obtained in [28,Theorem 2.4] the estimate ε n = O(1/n). An example of two overlapping disks considered by Miña-Díaz in [19] shows that this estimate cannot be improved, since lim inf n nε n > 0, for this particular Γ.…”