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Journal of Convex Analysis 16 (2009), No. 2, 605--616
Copyright Heldermann Verlag 2009

Cauchy Transforms of Arens Bounded Measures for a Vitushkin Amendment. I

Norbert Trautmann
Department of Mathematics, University of Hamburg, Bundesallee 55, 20146 Hamburg, Germany

We amend the theorem of A. G. Vitushkin [J. Functional Analysis 20 (1975) 149 - 157], who solves the problem of the rational approximation by constructive methods and a quasi-geometric notation -- the so called analytic continuous capacity -- , by a new sufficient condition. The proof is functional-analytic abstract and at the same time we can almost see the effects of the condition. We start with measures -- a structure bearing level beneath the AC-capacity -- and go by Cauchy transforms directly to the level of functions. Moreover we confirm a conjecture of J. Garnett [Duke Math. J. 37(1970) 689 - 699] for functions, which are continuous on the complex numbers C including infinity and analytic off a Cantor-set.

Keywords: Cauchy-Transform, new norm for measures, Vitushkin, rational approximation, Garnett's conjecture.

MSC: 46J10, 46J15

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