### 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 NH_{M}^{1}(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 ∈ NH_{M} ^{1}(D) is an element of NH_{M}^{1}(D). Lastly, we solve an extremal problem for the dual of NH_{M}^{1}(D).

Original language | English |
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Pages (from-to) | 629-639 |

Number of pages | 11 |

Journal | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Volume | 14 |

Issue number | 4 |

Publication status | Published - Oct 2007 |

### Keywords

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