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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An integral representation of holomorphic functions


Authors: R. W. Hilger and J. F. Michaliček
Journal: Proc. Amer. Math. Soc. 74 (1979), 44-48
MSC: Primary 30E20
DOI: https://doi.org/10.1090/S0002-9939-1979-0521871-3
MathSciNet review: 521871
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Abstract: Let K be a compact set in the complex plane and let f be a function holomorphic on the complement $ \Omega $ of K and vanishing at infinity. We prove that there are finite complex-valued Borel measures $ {\mu _{m,n}}(m,n = 0,1,2, \ldots ;m + n \geqslant 1)$ on $ {K^2}$ satisfying $ {\lim _{k \to \infty }}{({\Sigma _{m + n = k}}\left\Vert{\mu _{m,n}}\right\Vert)^{1/k}} = 0$ so that

$\displaystyle f(z) = \sum\limits_{m,n} {\int_{{K^2}} {{{(z - {w_1})}^{ - m}}{{(z - {w_2})}^{ - n}}d{\mu _{m,n}}({w_1},{w_2})\quad (z \in \Omega ).} } $


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DOI: https://doi.org/10.1090/S0002-9939-1979-0521871-3
Article copyright: © Copyright 1979 American Mathematical Society