Fejer convolutions for an extremal problem in the Steklov class

The famous problem of V. A. Steklov is intimately related with the following extremal problem. Fix degree and find a maximum of the orthonormal polynomial with respect to measure from the Steklov class (i.e. class of probability measures on the unit circle, such that its density is bounded away from zero at every Lebesgue point. We study asymptotics of certain trigonometric polynomials defined by the Fejer convolutions. These polynomials can be used to construct asymptotical solutions of the above extremal problem.