Szegő's theorem for Jordan arcs

The $n$-th Christoffel function for a point $z_0\in \mathbb{C}$ and a finite measure $\mu$ supported on a Jordan arc $\mathrm{\Gamma }$ is $${\lambda }_n(\mu ,z_0)=\inf \{{\int }_{\mathrm{\Gamma }}\mathrm{\mid }P{\mathrm{\mid }}^{2}d\mu \mid P\text{ is a polynomial of degree at most }n\text{ and }P(z_0)=1\}\mathrm{.}$$ It is natural to extend this notion to $z_0=\mathrm{\infty }$ and define ${\lambda }_n(\mu ,\mathrm{\infty })$ to be the infimum of the squared $L^2(\mu )$-norm over monic polynomials of degree $n$. The classical Szegő theorem provides an asymptotic description of ${\lambda }_n(\mu ,z_0)$ for $\mathrm{\mid }{z}_0\mathrm{\mid }>1$ and $z_0=\mathrm{\infty }$ 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 $z_0=\mathrm{\infty }$ and purely absolutely continuous measures belonging to the Szegő 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 $\mathrm{\Gamma }$, and for all points $z_0\in \mathbb{C}\cup \{\mathrm{\infty }\}\setminus \mathrm{\Gamma }$. Moreover, if the measure is in the Szegő class, we provide explicit asymptotics for the extremal and orthogonal polynomials.