An integral representation of holomorphic functions
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- by R. W. Hilger and J. F. Michaliček PDF
- Proc. Amer. Math. Soc. 74 (1979), 44-48 Request permission
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 ${\lim _{k \to \infty }}{({\Sigma _{m + n = k}}\left \|{\mu _{m,n}}\right \|)^{1/k}} = 0$ so that \[ f(z) = \sum \limits _{m,n} {\int _{{K^2}} {{{(z - {w_1})}^{ - m}}{{(z - {w_2})}^{ - n}}d{\mu _{m,n}}({w_1},{w_2})\quad (z \in \Omega ).} } \]References
- Albert Baernstein II, Representation of holomorphic functions by boundary integrals, Trans. Amer. Math. Soc. 160 (1971), 27–37. MR 283182, DOI 10.1090/S0002-9947-1971-0283182-0
- Gottfried Köthe, Dualität in der Funktionentheorie, J. Reine Angew. Math. 191 (1953), 30–49 (German). MR 56824, DOI 10.1515/crll.1953.191.30 —, Topological vector spaces. I, Springer-Verlag, Berlin, 1969.
Additional Information
- © Copyright 1979 American Mathematical Society
- 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