Akhiezer's orthogonal polynomials and Bernstein-Szegő method for a circular arc
An asymptotic result for polynomials orthogonal on a circular arc was stated without proof in a short note by \textit{N. I. Akhiezer} [Dokl. Akad. Nauk SSSR 130, 247-250 (1960); English transl. in Soviet Math. Dokl. 1, 31-34 (1960; Zbl 0129.28305)]. Although a rather general theory has been developed since then, the results and the methods applied there are still worth studying. The main goal of this paper is to present a rigorous and detailed exposition of Akhiezer's results, and to outline some possible extensions of the method. As a first step, a special class of weights on the arc is considered, for which both the orthogonal polynomials and the outer functions can be constructed explicitly, which yields the asymptotics. Then, a general weight on the arc is approximated by means of the Bernstein-Szegő's method, leading to the desired asymptotic formulas.
scientific article; zbMATH DE number 1230423
scientific article