Asymptotic properties of extremal polynomials corresponding to measures supported on analytic regions

Abstract

Let G be a bounded region with simply connected closure G and analytic boundary and let µ be a positive measure carried by G together with finitely many pure points outside G. We provide estimates on the norms of the monic polynomials of minimal norm in the space L q (µ) for q > 0. In case the norms converge to 0, we provide estimates on the rate of convergence, generalizing several previous results. Our most powerful result concerns measures µ that are perturbations of measures that are absolutely continuous with respect to the push-forward of a product measure near the boundary of the unit disk. Our results and methods also yield information about the strong asymptotics of the extremal polynomials and some information concerning Christoffel functions.

Ratio Asymptotics, Hessenberg Matrices, and Weak Asymptotic Measures

“…It is a separate matter to calculate the moments of the limiting measure. Some results on this subject can be found in [13,Chapter 9] as well as [9,10,11]. Our contribution to this effort is Theorem 1.8, which we now prove.…”

“…for appropriate values of z. Many examples of measures whose orthonormal polynomials exhibit ratio asymptotics can be found in [9,10,16]. One usually obtains root and ratio asymptotics for z outside the polynomial convex hull of the support of the measure.…”

“…It is very often the case that if ∂G ⊆ supp(µ), then the ratio ϕ n (z; µ)ϕ n−1 (z; µ) −1 approaches φ G (z) when |z| is sufficiently large (see [9,16] for examples). Furthermore, it is often the case that the probability measures {|ϕ n | 2 dµ} n∈N converge weakly to the equilibrium measure of G as n → ∞ (see [9] for examples). Therefore, in such cases one expects the integral in (4) to approach…”

Asymptotics of polynomials orthogonal over circular multiply connected domains

“…Logarithmic/zero asymptotics with applications to shape reconstruction have been given in [8,17] for polynomials orthogonal over an archipelago (a finite union of Jordan domains). The papers [16,18,21,22], although more general in scope, also carry important implications for planar orthogonality.…”

Asymptotics of Carleman Polynomials for Level Curves of the Inverse of a Shifted Zhukovsky Transformation

“…Recently, the subject has experienced a new surge, with many new interesting results in a variety of topics such as the asymptotic behavior and zero distribution of the orthogonal polynomials [4,6,7,8,10,11,12,14,16], universality and Christoffel functions [9,14,20], and the existence of recurrence relations [1,18,19]. In particular, [6] and [20] consider orthogonality over several domains, while [14] considers orthogonality with respect to certain potential theoretic varying weights.…”