Some Polynomial Problems Arising from Padé Approximation

In the convergence theory of Padé approximation, one needs to estimate the size of a set on which a suitably normalized polynomial q is small. For example, one needs to estimate the size of the set of r 2 [0; 1] for which max jtj=1 jq (t)j =min jtj=r jq (t)j is not “too large”. We discuss some old and new problems of this type, and the methods used to solve them. 1. Introduction Let f be a function analytic at 0, and hence possessing a Maclaurin series there. Recall that if m;n 0, the (m;n) Padé approximant to f is a rational function [m=n] (z) = (p=q) (z) ; where p; q are polynomials of degree m;n respectively, with q not identically zero, and (fq p) (z) = O z : The order relation indicates that the coe¢ cients of 1; z; z; :::; z in the Maclaurin series of the left-hand side vanish. It may be reformulated as a system of homogeneous linear equations in the coe¢ cients of q and p, with more unknowns than equations, and hence has a nontrivial solution. After the division by q, the solution becomes unique. For an introduction to the subject, see [2], [3]. An essential tool in studying the convergence of Padé approximants as m and or n ! 1, is the contour integral error formula. Let us assume for simplicity that f is analytic in fz : jzj 1g. Then if [m=n] = p=q, Cauchy’s integral formula gives for jzj < 1; (fq p) (z) zm+n+1 = 1 2 i Z jtj=1 (fq p) (t) tm+n+1 dt t z ; jzj < 1: Date: September 19, 2000 1991 Mathematics Subject Classi…cation: 30E10, 30C15, 31A15, 41A21. 1