Representation of Holomorphic Functions by Boundary Integrals

Abstract

Let $K$ be a compact locally connected set in the plane and let $f$ be a function holomorphic in the extended complement of $K$ with $f(\infty ) = 0$. We prove that there exists a sequence of measures $\{ {\mu _n}\}$ on $K$ satisfying ${\lim _{n \to \infty }}||{\mu _n}|{|^{1/n}} = 0$ such that $f(z) = \sum \nolimits _{n = 0}^\infty {\int _K {{{(w - z)}^{ - n - 1}}d{\mu _n}(w)(z \in K)} }$. It follows from the proof that two topologies for the space of functions holomorphic on $K$ 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 $K$. 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 $K$ is a compact subset of the unit circle which is not locally connected.