Szeg\H{o}'s theorem for Jordan arcs

Abstract

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