
Journal of Convex Analysis 16 (2009), No. 2, 605616 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 post@trautmannhamburg.de 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 quasigeometric notation  the so called analytic continuous capacity  , by a new sufficient condition. The proof is functionalanalytic 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 ACcapacity  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 Cantorset. Keywords: CauchyTransform, new norm for measures, Vitushkin, rational approximation, Garnett's conjecture. MSC: 46J10, 46J15 [ Fulltextpdf (143 KB)] for subscribers only. 