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Transactions of the American Mathematical Society

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A representation theorem for functions holomorphic off the real axis


Author: Albert Baernstein
Journal: Trans. Amer. Math. Soc. 165 (1972), 159-165
MSC: Primary 30A86
DOI: https://doi.org/10.1090/S0002-9947-1972-0293111-2
MathSciNet review: 0293111
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Abstract: Let f be holomorphic in the union of the upper and lower half planes, and let $ p \in [1,\infty )$. We prove that there exists an entire function $ \varphi $ and a sequence $ \{ {f_n}\} $ in $ {L^p}(R)$ satisfying $ \left\Vert {{f_n}} \right\Vert _p^{1/n} \to 0$ such that

$\displaystyle f(z) = \varphi (z) + \sum\limits_{n = 0}^\infty {\int_{ - \infty }^\infty {{{(t - z)}^{ - n - 1}}{f_n}(t)dt.} } $

This complements an earlier result of the author's on representation of function holomorphic outside a compact subset of the Riemann sphere. A principal tool in both proofs is the Köthe duality between the spaces of functions holomorphic on and off a subset of the sphere. A corollary of the present result is that each hyperfunction of one variable can be represented by a sum of Cauchy integrals over the real axis.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0293111-2
Keywords: Integral representation of holomorphic functions, Cauchy integrals, Köthe duality, hyperfunction
Article copyright: © Copyright 1972 American Mathematical Society

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