L p CHRISTOFFEL FUNCTIONS , L p UNIVERSALITY , AND PALEY - WIENER SPACES

Let ω be a regular measure on the unit circle and let p > 0. We establish asymptotic behavior, as n→∞, for the Lp Christoffel function λn,p (ω, z) = inf deg(P )≤n−1 ∫ π −π ∣∣P (eiθ)∣∣p dω (θ) |P (z)| at Lebesgue points z on the unit circle, where ω′ is lower semi-continuous. While bounds for these are classical, asymptotics have never been established for p 6= 2. The limit involves an extremal problem in Paley-Wiener space. As a consequence, we deduce universality type limits for the extremal polynomials, which reduce to random-matrix limits involving the sinc kernel in the case p = 2. We also present analogous results for Lp Christoffel functions on [−1, 1] . Lp Christoffel functions, Universality Limits, Paley-Wiener Spaces 42C05