Szegő's theorem for Jordan arcs

Abstract

The $n$-th Christoffel function for a point $z_0\in\mathbb C$ and a finite measure $μ$ supported on a Jordan arc $Γ$ 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 $λ_n(μ,\infty)$ to be the infimum of the squared $L^2(μ)$-norm over monic polynomials of degree $n$. The classical Szegő theorem provides an asymptotic description of $λ_n(μ,z_0)$ 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 $λ_n(μ,z_0)$ for any finite measure $μ$ supported on a $C^{1+}$-Jordan arc $Γ$, and for all points $z_0\in\mathbb{C}\cup\{\infty\}\setminusΓ$. Moreover, if the measure is in the Szegő class, we provide explicit asymptotics for the extremal and orthogonal polynomials.