Local limiting behavior of the zeros of approximating polynomials

Abstract

Let f be a piecewise analytic (but not analytic) function in Ck[a, b], k [ges] 0, and let p*n be the sequence of polynomials of best uniform approximation to f on [a, b]. It is well known that every point of [a, b] is a limit point of the zeros of the p*n. Let xgE [a, b], and suppose that f is analytic at x and f(x) [not equal to] 0. The main purpose of this paper is to show that there exists a constant [gamma] (which depends only on x) such that there is no zero of p*n within the circle of radius ([gamma]/n) log n centered at x, for all sufficiently large values of n.