Local limiting behavior of the zeros of approximating polynomials

Abstract

Let f be a piecewise analytic (but not analytic) function in @[a, b], k > 0, and let p,* 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,*. Let x E [a, b], and suppose that f is analytic at x and f(x) # 0. The main purpose of this paper is to show that there exists a constant y (which depends only on x) such that there is no zero of p,* within the circle of radius (y/n) log n centered at X, for all sufficiently large values of n.