Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle (Q1069086): Difference between revisions

Consider a system \(\{\phi_ n\}\) of polynomials orthonormal on the unit circle with respect to a measure \(d\mu\), with \(\mu '>0\) almost everywhere. Denoting by \(\kappa_ n\) the leading coefficient of \(\phi_ n\), a simple new proof is given for \textit{E. A. Rakhmanov}'s important result that \(\lim_{n\to \infty}\kappa_ n/\kappa_{n+1}=1;\) this result plays a crucial role in extending Szegö's theory about polynomials orthogonal with respect to measures \(d\mu\) with log \(\mu\) '\(\in L^ 1\) to a wider class of orthogonal polynomials.