On the leading coefficient of polynomials orthogonal over domains with corners

TL;DR

This paper investigates the asymptotic behavior of the leading coefficients of orthogonal polynomials over domains with corners, showing that the known error estimate is generally optimal by providing a specific example.

Contribution

It proves that the O(1/n) estimate for the error term in the asymptotic behavior of leading coefficients is best possible, and introduces a conjecture related to Faber polynomials.

Findings

The O(1/n) error estimate cannot be improved in general.

An example is provided where the error term's lower limit is positive.

The proof involves Faber polynomials and formulates a related conjecture.

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 $\lambda_n>0$. N. Stylianopoulos has recently proven that the asymptotic behavior of $\lambda_n$ as $n\to\infty$ is given by \[ \frac{n+1}{\pi}\frac{\gamma^{2n+2}}{ \lambda_n^{2}}=1-\alpha_n, \] where $\alpha_n=O(1/n)$ as $n\to\infty$ and $\gamma$ is the reciprocal of the logarithmic capacity of the boundary $\partial G$. In this paper, we prove that the $O(1/n)$ estimate for the error term $\alpha_n$ 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.