We consider Chebyshev polynomials, $T_n(z)$, for infinite, compact sets $\frak{ e} \subset \mathbb{ R}$ (that is, the monic polynomials minimizing the sup-norm, $\Vert T_n \Vert_{ \frak{ e}}$, on $\frak{ e}$). We resolve a $45+$ year old conjecture of Widom that for finite gap subsets of $\mathbb{ R}$, his conjectured asymptotics (which we call Szeg\H{ o}-Widom asymptotics) holds. We also prove the first upper bounds of the form $\Vert T_n \Vert_{ \frak{ e}} \leq Q C({ \frak{ e}})^n$ (where $C(\frak{ e})$ is the logarithmic capacity of $\frak{ e}$) for a class of $\frak{ e}$'s with an infinite number of components, explicitly for those $\frak{ e} \subset \mathbb{ R}$ that obey a Parreau-Widom condition.
Categories: math.CA