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).
|Number of pages||11|
|Journal||Bulletin of the Belgian Mathematical Society - Simon Stevin|
|Publication status||Published - Oct 2007|
- Carleman formulas
- Cauchy integrals
- Extremal problems
- Projective-inductive limit spaces