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On the continuity of the Faber mapping. (English) Zbl 0598.30007

A compact connected plane set K with connected complement is called a Faber set if the Faber mapping \[ (Tp)(z)=\sum^{k}_{n=0}a_ nF_ n(z)\quad when\quad p(w)=\sum^{k}_{n=0}a_ nw^ n \] is a bounded mapping from the set of polynomials p on the unit disc to the set of polynomials on K, each equipped with the uniform norm. Here \(F_ n(z)\), \(n=0,1,..\). are the Faber polynomials. The main result of this paper is that K is a Faber set if the cyclic variation of K at z is a bounded function of \(z\in K\). This is obtained by combining Wazewski’s characterization of rectifiable continua with Young’s theory of length and applying earlier results of Anderson and Kövari and Pommerenke about Faber sets.
Reviewer: B.Øksendal

MSC:

30C10 Polynomials and rational functions of one complex variable
28A75 Length, area, volume, other geometric measure theory
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