Weighted Maximum over Minimum Modulus of Polynomials, Applied to Ray Sequences of Padé Approximants
چکیده
Let a 0; " > 0. We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set ( r 2 [0; 1] : r maxjtj=1 jP (t)j minjtj=r jP (t)j " ) : Here P is an arbitrary polynomial of degree n. We then apply this to diagonal and ray Padé sequences for functions analytic (or meromorphic) in the unit ball. For example, we show that the diagonal f[n=n]gn=1 sequence provides good approximation on almost 1 8 of the circles centre 0, and the f[2n=n]gn=1 sequence on almost 1 4 of such circles. 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. For an introduction to the subject, see [1]. The convergence theory of Padé approximation is rich and complex. It is known that if f is analytic at 0, and meromorphic in the whole plane, then f[n=n]gn=1 converges in measure, and in capacity the Nuttall-Pommerenke Theorem [13], [14]. More generally, given sequences of positive integers fmkgk=1 ; fnkg 1 k=1 that tend to 1 in such a way that for some xed 1; 1 mk nk ; k 1; and given r; " > 0; m2 fz : jzj r and jf [mk=nk]j (z) > "kg ! 0; k !1: Here m2 denotes planar measure, and it may be replaced by capacity. There are deeper analogues for functions with branchpoints [20], [21]. One unfortunate feature of the theorem is that it really requires f to be meromorphic in C. There are functions analytic in the unit ball for which f[n=n]gn=1 does Date : May 18, 2001 1991 Mathematics Subject Classi cation : 30E10, 30C15, 31A15, 41A21. 1