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Christoffel functions on curves and domains. (English) Zbl 1189.26028
Many results concerning the asymptotics of Christoffel functions on the real line and on the unit circle have been well understood. The present work extends these studies of asymptotic behaviour on general curves: measures on smooth Jordan curves and area type measures on domains. In particular, a new feature in the case of asymptotics on arc-like or area-like measures is that on the boundary the order of the Christoffel function will be $1/{n}^{2}$ (opposite to the order $1/n$ above). The author begins by proving some properties of lemniscates and lemniscate domains, studies Green’s functions and equilibrium measures, constructs some fast decreasing polynomials and then he proves two main theorems on the asymptotics in question. In addition he establishes some Markov and Bernstein inequalities.
##### MSC:
 26C05 Polynomials: analytic properties (real variables) 31A99 Two-dimensional potential theory 41A10 Approximation by polynomials 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)