Representation of holomorphic functions by boundary integrals

Abstract

Let K K be a compact locally connected set in the plane and let f f be a function holomorphic in the extended complement of K K with f ( ∞ ) = 0 f(\infty ) = 0 . We prove that there exists a sequence of measures { μ n } \{ {\mu _n}\} on K K satisfying lim n → ∞ | | μ n | | 1 / n = 0 {\lim _{n \to \infty }}||{\mu _n}|{|^{1/n}} = 0 such that f ( z ) = ∑ n = 0 ∞ ∫ K ( w − z ) − n − 1 d μ n ( w ) ( z ∈ K ) f(z) = \sum \nolimits _{n = 0}^\infty {\int _K {{{(w - z)}^{ - n - 1}}d{\mu _n}(w)(z \in K)} } . It follows from the proof that two topologies for the space of functions holomorphic on K K are the same. One of these is the inductive limit topology introduced by Köthe, and the other is defined by a family of seminorms which involve only the values of the functions and their derivatives on K K . A key lemma is an open mapping theorem for certain locally convex spaces. The representation theorem and the identity of the two topologies is false when K K is a compact subset of the unit circle which is not locally connected.