The topology of the class of functions representable by Carleman type formulae, duality and applications

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Abstract

We set D to be a simply connected domain and we consider exhaustion function spaces, X(D) with the projective topology (see §1). We show that the natural topology on the topological dual of X (D), (X(D))′, is the inductive topology. As a main application we assume that D has a Jordan rectifiable boundary ∂D, and M ⊂ ∂D to be an open analytic arc whose Lebesgue measure satisfies 0 < m(M) < m(∂D). We prove a result for the dual NHM1(D), which is the class of holomorphic functions in D which are represented by Carleman formulae on M ⊂ ∂D. Furthermore we show that the Cauchy Integral associated to f ∈ NHM 1(D) is an element of NHM1(D). Lastly, we solve an extremal problem for the dual of NHM1(D).

Original languageEnglish
Pages (from-to)629-639
Number of pages11
JournalBulletin of the Belgian Mathematical Society - Simon Stevin
Volume14
Issue number4
Publication statusPublished - Oct 2007

Keywords

  • Carleman formulas
  • Cauchy integrals
  • Extremal problems
  • Projective-inductive limit spaces

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