Distribution of Zeros of Polynomial Sequences, Especially Best Approximations

Abstract

In [2] the asymptotic behavior of the zeros of polynomials of near best approximation to functions f on a compact set E was studied in the case when f is not everywhere analytic on E. For example, suppose E is a finite union of compact intervals of the real line and f is continuous, but not analytic on E; then we have shown that every point of E is a limit point of zeros of the polynomials of best uniform approximation to f on E. Moreover, if the complement K of E is simply connected and the boundary of E consists of a finite number of analytic Jordan arcs, then the distribution of the zeros of the polynomials of best uniform approximation was analyzed. The purpose of this paper is to give a new interpretation of this distribution, namely to show that these zeros are uniformly distributed with respect to the normal derivative of G(x,y) on ∂E, where G(x,y) is Green’s function of K; furthermore results are obtained for the general case when K is connected.