Restrictions on the zeroes of lagrange interpolants to analytic functions

Abstract

We consider restrictions on the zeroes of Lagrange interpolants to analytic functions in bounded convex regions G. First we prove that if all linear interpolants (with distinct nodes in [Gbar]) vanish only in G, and if all quadratic Taylor interpolants to f have both zeroes in [Gbar] then f must be linear. The key to our proof is the fact that compact convex domains in the plane have the fixed point property. Our second result says that if all interpolants with nodes in G have all their zeroes in G, f is either linear or non-continuable past the boundary of G. Our proof relies on Jentsch's Theorem and the Gauss-Lucas Theorem (which depends on the convexity of G). Finally we consider the class TUG of analytic function in G such that every Lagrange interpolant to f is univalent in G. Such functions turn out to be analytic in a region (E (G)), that properly contains G, where E(G) is the envelope of G (defined in Section 3). We also show that there are non-polynomials in TUG by investigating an appropriate Ba...