Rational Approximation with Varying Weights, II

Abstract

We consider two problems concerning uniform approximation by weighted rational functions {wnrn}∞n=1, wherern=pn/qnhas real coefficients, degpn⩽[αn] and degqn⩽[βn], for givenα>0 andβ⩾0. Forw(x):=exwe show that on any interval [0, a] witha∈(0, â(α, β)), every real-valued functionf∈C([0, a]) is the uniform limit of some sequence {wnrn}. An implicit formula forâ(α, β) was given in the first part of this series of papers; in particular,â(1, 1)=2π. Forw(x):=xθwithθ>1 we show that uniform approximation of real-valuedf∈C([b, 1]) on [b, 1] by weighted rationalswnrnis possible for anyb∈(b(θ; α, β),1), whereb(θ; α, β) was also found in Part I; in particular,b(θ; 1, 1)=tan4((π/4)((θ−1)/θ)). Both of the mentioned results are sharp in the sense that approximation is no longer possible ifâis replaced byâ+εorbis replaced byb−εwithε>0. We use potential theoretic methods to prove our theorems.