On a class of infinitely differentiable functions on unbounded convex set in $\mathbb R^n$ admitting holomorphic continuation in $\mathbb C^n$

A subspace of the Schwartz space of rapidly decreasing functions on closed convex unbounded set in $\mathbb R^n$, admitting holomorphic extension in $\mathbb C^n$, is studied. The problem of description of the dual space for this space in terms of the Fourier-Laplace transform is considered.