On a characeterization of integrability for the reciprocal weight of orthogonal polynomials on the circle (Q1591299)

In this paper, the authors consider extremal polynomials on the circle with respeet to a prescribed finite and positive Borel measure \(\mu\), with a constrain in the \(r\)-th coefficient. The authors state and prove a necessary and sufficient condition for the density \(\mu'\) to satisfy the integrability condition \[ \int_0^{2\pi} {d\theta \over\mu'} <\infty. \] More precisely, if \(_rm_n\) denotes the square of the \(L^2(\mu)\) norm of an extremal polynomial among all the polynomials of degree \(\leq n\), it can be proved that there exists \(\lim_{r\to\infty} \lim_{n\to\infty} {_rm_n}\). If this limit is denoted by \(m\), then \(\int_0^{2\pi} {d\theta\over \mu'}< \infty\) if and only if \(m>0\). The proof of this result involves interesting calculations using the Szegö's kernel, which is defined in terms of extremal polynomials for \(r=n\).