zbMATH — the first resource for mathematics

Equilibrium measure and the distribution of zeros of extremal polynomials. (English. Russian original) Zbl 0618.30008
Math. USSR, Sb. 53, 119-130 (1986); translation from Mat. Sb., Nov. Ser. 125(167), No. 1, 117-127 (1984).
Let \(F=[a,b]\) be an interval, s a fixed number and \(\phi_ n\), \(n=1,2,..\). a sequence of continuous nonnegative functions on F. Let \(P_ n\) be a polynomial that minimizes the norm \((\int^{b}_{a}| P|^ s\phi_ ndx)^{1/s}\) in the class of all polynomials of degree n with leading coefficient 1. It follows from a general theorem proved in the paper that if the sequence \(\phi_ n^{1/n}\) converges uniformly on the interval F to a function \(\phi\not\equiv 0\), then the zeros of the polynomials \(P_ n\), \(n=1,2,..\). have a limit distribution, which is characterized by the equilibrium measure \(\mu\) (f) corresponding to \(f=(1/s)\log (1/\phi)\) (here we put \(\log (1/0)=+\infty):\) for any interval \(I\subset F\), \[ \lim_{n\to \infty}(1/n)\kappa_ n(I)=\int_{I}d\mu \] where \(\kappa_ n(I)\) is the number of zeros of the polynomial \(P_ n\) on the interval I.
Reviewer: W.Pleśniak

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
Full Text: DOI