Abstract

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\mathbb{C} including infinity and analytic off a Cantor-set.

Author information

Contact details are reproduced from the original publication and may be historical.

Norbert Trautmann

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

post@trautmann-hamburg.de

Keywords

  • Cauchy-Transform
  • new norm for measures
  • Vitushkin
  • rational approximation
  • Garnett's conjecture

Mathematics Subject Classification

46J10, 46J15

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Suggested citation

N. Trautmann. “Cauchy Transforms of Arens Bounded Measures for a Vitushkin Amendment. I.” Journal of Convex Analysis 16 (2009), No. 2, 605–616. https://doi.org/10.68381/jca16033

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.