Let us say we have an analytic function $f(z)$ where $z \in \mathbb{C}$ defined in a tube $T$
$$T = \{ z \in \mathbb{C}, z = a+ib, b \in \mathcal{C}, a \in \mathbb{R}^n \}$$
where $\mathcal{C}$ is a convex cone at the origin. Given this, we can prove that some $f'(z)$ analytic continuation of $f(z)$ is defined in a similar tube. For this, we say that there exists a connected domain $G \subset \mathcal{R}^n$, which coincides with the boundary values of $f(z)$ and $f'(z)$. Then it implies that $f(z)$ and $f'(z)$ are the analytical continuations of each other and are analytic around the domain $G$. This is also known as the edge of the wedge problem.
Now we state the classical tube theorem. We also would make use of Malgrange–Zerner theorem.
Theorem 1. For every connected domain $G \subset \mathbb{R}^n$, there exists a holomorphic envelope $H(G)$ which contains $G$ as its subdomain.
Theorem 2 (Tube Theorem). For every connected domain $G \subset \mathbb{R}^n$, there is a tube given by
$$T(G) \{ z \in \mathbb{C}^n, Im(z) \in G \}$$
then the holomorphic envelope of the tube $T(G)$ is given by
$$H(T(G)) = T ( Co\ G)$$
where $Co\ G$ is the convex hull of $G$.
This tube theorem (and generalizations like the double cone theorem and Dyson's theorem) provide more insightful results in QFT. See Borchers1961. Timelike tube theorem can also be seen as a quantum generalization of Holmgren's uniqueness theorem, which also deals with analyticity. For more details on the timelike tube theorem, see Witten and Strohmaier & Witten.
Nice, is that what you were talking about when you said Witten and timelike tubes? After work you’re teaching me this stuff. Wish this post had more.