[Paper Review] Strong asymptotics for Christoffel functions of planar measures

Tom Bloom, N. Levenberg|ArXiv.org|Sep 13, 2007

Mathematical functions and polynomials13 references4 citations

TL;DR

This paper establishes strong asymptotics for Christoffel functions with varying weights on compact sets in the complex plane using tools from weighted potential theory and pluripotential theory. It proves that the normalized Christoffel function converges weak-* to the weighted equilibrium measure, generalizing known results on the real line and extending them to planar measures with varying weights.

ABSTRACT

We prove a version of strong asymptotics of Christoffel functions with varying weights for a general class of sets E and measures in the complex plane. This class includes all regular measures in the sense of Stahl-Totik on regular compact sets E in the plane and even allows varying weights. Our main theorems cover some known results for subsets E of the real line R; in particular, we recover information in the case of E=R with Lebesgue measure dx and weight w(x) = exp(-Q(x)) where Q(x) is a nonnegative, even degree polynomial having positive leading coefficient.

Motivation & Objective

• To extend strong asymptotics results for Christoffel functions from the real line to compact sets in the complex plane.
• To establish convergence of the normalized Christoffel function to the weighted equilibrium measure for a general class of planar measures.
• To generalize known results on the real line, including those for $ E = \mathbb{R} $ with $ w(x) = \exp(-Q(x)) $, to non-polar compact sets in $ \mathbb{C} $.
• To provide a unified framework using pluripotential theory in $ \mathbb{C}^2 $ for analyzing varying weight orthogonal polynomials in the complex plane.
• To prove the existence of a 'free energy' and large deviation estimates via weighted Bernstein-Markov inequalities.

Proposed method

• Utilizes the correspondence between weighted potential theory in $ \mathbb{C} $ and pluripotential theory in $ \mathbb{C}^2 $, as developed in [3].
• Applies the weighted Bernstein-Markov inequality to ensure the existence of a limiting 'free energy' (Theorem 2.1).
• Employs the Laplacian of the logarithm of the Christoffel function to define a sequence of measures $ \mu_n = \Delta(\frac{1}{2n}\log K_n) $.
• Proves weak-* convergence of $ \mu_n $ to the weighted equilibrium measure $ \mu_{eq}^w $ on compact sets $ E \subset \mathbb{C} $.
• Uses a large deviation estimate (Proposition 4.2) derived from the existence of the limit in the free energy, relying on standard potential theory.
• Applies modifications of results from Deift's book [6] and Berman's work [1,2] to extend universality-type results to the complex plane.

Experimental results

Research questions

• RQ1 Can strong asymptotics for Christoffel functions with varying weights be extended from the real line to general compact sets in the complex plane?
• RQ2 What conditions on a compact set $ E \subset \mathbb{C} $, a weight $ w $, and a measure $ \mu $ ensure that the normalized Christoffel function converges to the weighted equilibrium measure?
• RQ3 How does the use of pluripotential theory in $ \mathbb{C}^2 $ facilitate the analysis of varying weight orthogonal polynomials in the complex plane?
• RQ4 Under what conditions does the free energy exist, and how does it relate to the large deviation principle for orthogonal polynomials?
• RQ5 To what extent do the results for $ E = \mathbb{R} $, $ w(x) = \exp(-Q(x)) $, and $ \mu = dx $ generalize to non-compact or non-real sets in $ \mathbb{C} $?

Key findings

• The sequence $ \frac{1}{2n}\log K_n(z) $ converges uniformly on $ \mathbb{C} $, implying weak-* convergence of $ \Delta(\frac{1}{2n}\log K_n) $ to the weighted equilibrium measure $ \mu_{eq}^w $.
• For any compact set $ E \subset \mathbb{C} $ with admissible weight $ w $ and measure $ \mu $ satisfying the weighted Bernstein-Markov inequality, the Christoffel function satisfies strong asymptotics: $ \frac{1}{n+1}K_n(x)w(x)^{2n}dx \to d\mu_{eq}^w(x) $ weak-*.
• The result generalizes Totik's strong asymptotics on the real line to the complex plane, including the case $ E = \mathbb{R} $, $ w(x) = \exp(-Q(x)) $, $ Q $ even degree polynomial with positive leading coefficient.
• The existence of the free energy (Theorem 2.1) is established via potential-theoretic consequences of the Bernstein-Markov inequality.
• The generalized $ m $-point correlation functions $ R_m^{(n)} $ converge weak-* to the product of $ m $ copies of $ \mu_{eq}^w $, extending universality results to the complex plane.
• For $ E = \mathbb{R} $, $ w(x) = \exp(-Q(x)) $, the limit $ \lim_{n\to\infty} \mathcal{Z}_n^{1/n^2} $ exists and equals $ \delta^w(\mathbb{R}) $, with the normalization on a large compact interval $ \tilde{E} $ preserving the limit due to exponential decay of $ w $.