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?$\{p_{n}\}_{n=0}^{\infty }$ be the sequence of polynomials that are orthonormal over G with respect to the area measure, with each pn having leading coefficient ?n>0. It has been proven in [9] that the asymptotic behavior of ?n as n??$n\to \infty $ is given by n+1??2n+2?n2=1??n,$$\frac{n+1}{\pi}\frac{\gamma^{2n+2}}{ {\lambda_{n}^{2}}}=1-\alpha_{n}, $$ where ?n=O(1/n) as n??$n\to \infty $ and ? 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 ?n is, in general, best possible, by exhibiting an example for which liminfn??n?n>0.$$\liminf_{n\to\infty}\,n\alpha_{n}>0. $$ The proof makes use of the Faber polynomials, about which a conjecture is formulated.