Special Cases of Orthogonal Polynomials on the Semicircle and Applications in Numerical Analysis

Gradimir V. Milovanović

Orthogonal polynomials on the semicircle was introduced by Gautschi and Milovanović in $[$Rend. Sem. Mat. Univ. Politec. Torino, Special Issue $($July $1985)$, $179-185]$ and $[$J. Approx. Theory {\bf46} $(1986)$, $230-250]$. In this paper we give an account of this kind of orthogonality, weighted generalizations mainly oriented to Chebyshev weights of the first and second kind, as well as the corresponding applications in numerical analysis. Moreover, we also present a number of new results including those for Laurent polynomials orthogonal to a semicircle and applications to quasi-singular integrals.