We show that a CR function of class $C^k$, 0 ≤ k < ∞, on a tube submanifold of ${ℂ}^n$ holomorphically extends to the convex hull of the submanifold. The extension and all its derivatives through order k are shown to have nontangential pointwise boundary values on the original tube submanifold. The $C^k$-norm of the extension is shown to be no bigger than the $C^k$-norm of the original CR function.

It is proved that the Levi problem for certain locally convex Hausdorff spaces $E$ over $\mathbf{C}$ with a finite dimensional Schauder decomposition (for example for Fréchet or Silva spaces with a Schauder basis) the Levi problem has a solution, i.e. every pseudoconvex domain spread over $E$ is a domain of existence of an analytic function. It is also shown that a pseudoconvex domain spread over a Fréchet space or a Silva space with a finite dimensional Schauder decomposition is holomorphically convex and satisfies...

It is shown that a sequentially complete topological vector space X with a compact Schauder basis has WSPAP (see Definition 2) if and only if X has a pseudo-homogeneous norm bounded on every compact subset of X.

We prove that three pairwise disjoint, convex sets can be found, all congruent to a set of the form $(z_1,z_2,z_3)\in {ℂ}^3:|z_1{|}^2+|z_2{|}^2+{\left|z_3\right|}^{2m}\le 1$, such that their union has a non-trivial polynomial convex hull. This shows that not all holomorphic functions on the interior of the union can be approximated by polynomials in the open-closed topology.

This is a summary of recent work where we introduced a class of D-modules adapted to study ideals generated by exponential polynomials.