Fine Asymptotics of Christoffel Functions for General Measures

Let μ be a measure on the unit circle satisfying Szegő’s condition. In 1991, A. Máté calculated precisely the first-order asymptotic behavior of the sequence of Christoffel functions associated with μ when the point of evaluation lies on the circle, resolving a long-standing open problem. We extend his results to measures supported on smooth curves in the plane. In the process, we derive new asymptotic estimates for the Cesáro means of the corresponding 1-Faber polynomials and investigate some applications to orthogonal polynomials, linear ill-posed problems and the mean ergodic theorem. Introduction Let μ be a Borel measure of the complex plane with compact support, Γ. The so-called Christoffel functions associated with μ are defined by (1) λn(μ, z) := inf 1 |P (z)|2 1 2π ∫ |P | dμ, where the infimum is evaluated over all complex polynomials of degree at most n−1 which do not vanish at z. We want to understand how the asymptotic behavior of this sequence depends on the geometry of the support, Γ, when the point of evaluation, z ∈ Γ. It is easy to see, for instance, that λn(μ, z) → μ({z}) as n → ∞ if Γ is sufficiently regular and z lies on the outer boundary of Γ. What is the rate of this convergence? For certain classes of measures supported on a circle or a union of intervals, the answer is already known to first order ([6], [12]): If μ is absolutely continuous with weight, μ′, in Szegő’s class (defined below), then (2) lim n→∞ nλn(μ, e ) = 1 2π μ′(eiθ) or lim n→∞ nλn(μ, x) = π √ 1− x2μ′(x) (almost everywhere on the supports) when Γ is, respectively, the unit circle or the interval, [−1, 1]. The geometric significance of these equations is elucidated by elementary potential theory, which we briefly review. Every measure, μ, with compact support, Γ, has associated logarithmic energy defined as follows: I(μ) := ∫ ∫ log 1 |z − w| Received by the editors October 7, 2008 and, in revised form, March 2, 2009, April 20, 2009 and May 21, 2009. 2010 Mathematics Subject Classification. Primary 30E10. This research was partially supported by NSF grant NSF DMS 0700471. c ©2011 American Mathematical Society Reverts to public domain 28 years from publication 4553 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use