Bochner's tube theorem
# Bochner's tube theorem

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

Theorem Let $\omega \subset \mathbb {R} ^{n}$ be a connected open set. Then every function $f(z)$ holomorphic on the tube domain $\Omega =\omega +i\mathbb {R} ^{n}$ can be extended to a function holomorphic on the convex hull $\operatorname {ch} (\Omega )$.

A classic reference is (Theorem 9). See also for other proofs.

## Generalizations

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

Theorem Let $\omega$ be a connected submanifold of $\mathbb {R} ^{n}$ of class-$C^{2}$. Then every continuous CR function on the tube domain $\Omega (\omega )$ can be continuously extended to a CR function on $\Omega ({\text{ach}}(\omega )).\ \left(\Omega (\omega )=\omega +i\mathbb {R} ^{n}\subset \mathbb {C} ^{n}\ \left(n\geq 2\right),{\text{ach}}(\omega ):=\omega \cup {\text{Int}}\ {\text{ch}}(\omega )\right)$. By "Int ch(S)" we will mean the interior taken in the smallest dimensional space which contains "ch(S)".

