V.A. Steklov's problem of estimating the growth of orthogonal polynomials

The well-known problem of V.A. Steklov is closely related to the following extremal problem. For a fixed $n\in \mathbb N$, find $M_{n,\delta }=\sup _{\sigma \in S_\delta } \mathopen \|\phi _n\|_{L^\infty (\mathbb T)}$, where $\phi _n(z)$ is an orthonormal polynomial with respect to a measure $\sigma \in S_\delta$ and $S_\delta$ is the Steklov class of probability measures $\sigma$ on the unit circle such that $\sigma '(\theta )\geq \delta /(2\pi )>0$ at every Lebesgue point of $\sigma$. There is an elementary estimate $M_n\lesssim \sqrt n$. E.A. Rakhmanov proved in 1981 that $M_n \gtrsim \sqrt n/ (\ln n)^{3/2}$. Our main result is that $M_n \gtrsim \sqrt n$, i.e., that the elementary estimate is sharp. The paper gives a survey of the results on the solution of this extremal problem and on the general problem of Steklov in the theory of orthogonal polynomials. The paper also analyzes the asymptotics of some trigonometric polynomials defined by Fejér convolutions. These polynomials can be used to construct asymptotic solutions to the extremal problem under consideration.