Strong asymptotic behavior for extremal polynomials with respect to varying measures on the unit circle. (Q1418953)

Given a finite Borel measure \(\mu\) on the unit circle and a sequence \(\{W_n\}\) of polynomials of degree \(n\) such that all zeros \(w_{n1},\ldots,w_{nn}\) lie in the open unit disk with \[ \lim\limits_{n\to\infty} \; \sum\limits^n_{j=1} \; (1-| w_{nj}| ) \, = \, \infty \; , \] the authors prove a result on strong asymptotics (a so-called Szegö-type theorem) for \(L^p\)-extremal polynomials of degree \(\leq n\) with respect to varying measures of the form \(d\mu_n := d\mu / | W_n| ^p\) on the unit circle. Similar results are given for \(L^p\)-extremal polynomials with respect to varying measures of the above kind on closed rectifiable Jordan curves in the complex plane and for measures of the form \(\beta_n = \mu_n + \eta\), where \(\eta\) is a discrete measure with positive mass at a finite set of points \(z_1,\ldots,z_N\) in the open unit disk.