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 $\{p_{n}\}_{n=0}^{\infty }$ 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.