Behavior of the lagrange interpolants in the roots of unity

Abstract

Let A ₀ be the class of functions f analytic in the open unit disk |z| < 1, continuous on |z| ≤ 1, but not analytic on |z| ≤ 1. We investigate the behavior of the Lagrange polynomial interpolants L _n−1(f, z) to f in the n-th roots of unity. In contrast with the properties of the partial sums of the Maclaurin expansion, we show that for any w, with |w| > 1, there exists a g ≠ A ₀ such that L _n−1(g, w) = 0 for all n. We also analyze the size of the coefficients of L _n−1(f, z) and the asymptotic behavior of the zeros of the L _n−1(f, z).

The research of this author was conducted while visiting the University of South Florida.

The research of this author was supported, in part, by the National Science Foundation under grant DMS-881-4026.