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

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_M¹(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_M¹(D). Lastly, we solve an extremal problem for the dual of NH_M¹(D).