Remark on a theorem of Aharonov and Walsh

Abstract

The following theorem is proved: there is a functionf(z) analytic in |z|<1 and having the natural boundary |z|=1 such that for an infinite sequence of rational functions of degreen, r _n(z)=P_n(z)/q_n(z), the inequality

$$\left| {f(z) - r_n (z)} \right|< \varepsilon _n $$

(1)

holds in the closed unit circle |z|≦1. Hereɛ ₁,ɛ ₂,...,ɛ _n is any sequence of positive numbers, tending to zero asn approaches infinity. This theorem is a refinement of a theorem of Aharonov and Walsh, who showed the existence of anf(z) satisfying (*) in |z|≦1 (with an infinite sequence {r _n(z)}) but having the natural boundary |z|=3.