Maximization of the size of monic orthogonal polynomials on the unit circle corresponding to the measures in the Steklov class (Q2352918)

The authors consider measures in the Steklov class of order \(\delta\), denoted here by \(S_{\delta}\), and the unique set of monic orthogonal polynomials defined by each \(d\mu \in S_{\delta}\), \(\left\{ \phi_{n} \left( z; d\mu \right) \right\}\). Then the correspondent orthonormal set \(\left\{ \varphi_{n}\left( z; d\mu \right) \right\}\) is defined in order to recall the Steklov conjecture which indicates that the sequence \[ M_{n,\delta}=\sup_{d\mu \in S_{\delta}} \; \max_{\{z \in \mathbb{C}: |z|=1 \}} |\varphi_{n}\left( z; d\mu \right)| \] is bounded in \(n\). This assertion was proved false by \textit{E. A. Rakhmanov} [Math. USSR, Sb. 36, 549--575 (1980); translation from Mat. Sb., Nov. Ser. 108(150), 581--608 (1979; Zbl 0452.33012)]. The techniques used by Rakhmanov are reviewed and applied by the authors in the pursuit of further boundaries for orthogonal polynomials within that class. This paper ends with an appendix with some illustrative numerical data and their discussion in view of the results previously proved.