CR functions and tube manifolds

Various generalizations of Bochner's theorem on the extension of holomorphic functions over tube domains are considered. It is shown that CR functions on tubes over connected, locally closed, locally starlike subsets of ${\textbf{R}^n}$ uniquely extend to CR functions on almost all of the convex hull of the tube set. A CR extension theorem on maximally stratified real submanifolds of ${\textbf{C}^n}$ is proven. The above two theorems are used to show that the CR functions (resp. CR distributions) on tubes over a fairly general class of submanifolds of ${\textbf{R}^n}$ uniquely extend to CR functions (CR distributions) on almost all of the convex hull.