Bochner's tube theorem

In mathematics, Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in can be extended to the convex hull of this domain.

Theorem Let be a connected open set. Then every function holomorphic on the tube domain can be extended to a function holomorphic on the convex hull .

A classic reference is [1] (Theorem 9). See also [2][3] for other proofs.

Generalizations

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The generalized version of this theorem was first proved by Kazlow (1979),[4] also proved by Boivin and Dwilewicz (1998)[5] under more less complicated hypothese.

Theorem Let   be a connected submanifold of   of class- . Then every continuous CR function on the tube domain   can be continuously extended to a CR function on  . By "Int ch(S)" we will mean the interior taken in the smallest dimensional space which contains "ch(S)".

References

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  1. ^ Bochner, S.; Martin, W.T. (1948). Several Complex Variables. Princeton mathematical series. Princeton University Press. ISBN 978-0-598-34865-4.
  2. ^ Hounie, J. (2009). "A Proof of Bochner's Tube Theorem". Proceedings of the American Mathematical Society. 137 (12). American Mathematical Society: 4203–4207. doi:10.1090/S0002-9939-09-10057-6. JSTOR 40590656.
  3. ^ Noguchi, Junjiro (2020). "A brief proof of Bochner's tube theorem and a generalized tube". arXiv:2007.04597 [math.CV].
  4. ^ Kazlow, M. (1979). "CR functions and tube manifolds". Transactions of the American Mathematical Society. 255: 153. doi:10.1090/S0002-9947-1979-0542875-5.
  5. ^ Boivin, André; Dwilewicz, Roman (1998). "Extension and Approximation of CR Functions on Tube Manifolds". Transactions of the American Mathematical Society. 350 (5): 1945–1956. doi:10.1090/S0002-9947-98-02019-4. JSTOR 117646..