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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Representation of holomorphic functions by boundary integrals

Author: Albert Baernstein
Journal: Trans. Amer. Math. Soc. 160 (1971), 27-37
MSC: Primary 30.30; Secondary 46.00
MathSciNet review: 0283182
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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 }}\vert\vert{\mu _n}\vert{\vert^{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.

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Keywords: Representation of holomorphic functions, Cauchy integrals, boundary integrals, generalization of Laurent series, duality in function theory, spaces of holomorphic functions, Roumieu's generalized functions, open mapping theorem, (DF) space, Montel space
Article copyright: © Copyright 1971 American Mathematical Society