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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
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?)

  • [1] A. Baernstein, Representation of holomorphic functions by boundary integrals, Trans. Amer. Math. Soc. 160 (1971), 27-37. MR 0283182 (44:415)
  • [2] G. Köthe, Dualität in der Funktionentheorie, J. Reine Angew. Math. 191 (1953), 30-49. MR 15, 132. MR 0056824 (15:132g)
  • [3] C. Roumieu, Sur quelques extensions de la notion de distribution, Ann. Sci. École Norm. Sup. (3) 77 (1960), 41-121. MR 22 #12377. MR 0121643 (22:12377)
  • [4] M. Sato, Theory of hyperfunctions. I, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1959), 139-193. MR 22 #4951. MR 0114124 (22:4951)
  • [5] H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 33 #1689. MR 0193469 (33:1689)
  • [6] A. Wilansky, Functional analysis, Blaisdell, New York, 1964. MR 30 #425. MR 0170186 (30:425)

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Keywords: Integral representation of holomorphic functions, Cauchy integrals, Köthe duality, hyperfunction
Article copyright: © Copyright 1972 American Mathematical Society

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