Szegő's theorem for Jordan arcs

Abstract

The $n$-th Christoffel function for a point $z_0\in \mathbb{C}$ and a finite measure $\hat{I}¼$ supported on a Jordan arc $\hat{I}“$ is \[ λ_n(μ,z_0)=\inf\left\{\int_Γ|P|^2dμ\mid P\text{ is a polynomial of degree at most }n\text{ and } P(z_0)=1\right\}. \] It is natural to extend this notion to $z_0=\infty$ and define $\hat{I}{»}_n(\hat{I}¼,\infty )$ to be the infimum of the squared $L^2(\hat{I}¼)$-norm over monic polynomials of degree $n$. The classical Szegő theorem provides an asymptotic description of $\hat{I}{»}_n(\hat{I}¼,z_0)$ for $|z_0|>1$ and $z_0=\infty$ and arbitrary finite measures supported on the unit circle. Widom has proved a version of Szegő's theorem for measures supported on $C^{2+}$-Jordan arcs for the point $z_0=\infty$ and purely absolutely continuous measures belonging to the Szegő class. We extend this result in two directions. We prove explicit asymptotics of $\hat{I}{»}_n(\hat{I}¼,z_0)$ for any finite measure $\hat{I}¼$ supported on a $C^{1+}$-Jordan arc $\hat{I}“$, and for all points $z_0\in \mathbb{C}\cup \{\infty \}\setminus \hat{I}“$. Moreover, if the measure is in the Szegő class, we provide explicit asymptotics for the extremal and orthogonal polynomials.