L p Christoffel functions, L p universality, and Paley-Wiener spaces

Abstract

Let ω be a regular measure on the unit circle in ℂ, and let p > 0. We establish asymptotic behavior, as n→∞, for the Lp Christoffel function $${\lambda _{n,p}}(\omega ,z) = \mathop {\inf }\limits_{\deg (P) \leqslant n - 1} \frac{{\int_{ - \pi }^\pi {{{\left| {P({e^{i\theta }})} \right|}^p}dw(\theta )} }}{{{{\left| {P(z)} \right|}^p}}}$$ at Lebesgue points z on the unit circle in ℂ, where ω′ is lower semi-continuous. While bounds for these are classical, asymptotics have never been established for p ≠ 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].