  We consider a space of infinitely smooth functions on an unbounded closed
convex set in ${\mathbb R}^n$. It is shown that each function of this space can
be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed
growth condition. Description of linear continuous functionals on this space in
terms of their Fourier-Laplace transform is obtained. Also a variant of the
Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.
