Ranges of joint Laplace-Fourier transforms

Abstract:

If $f \in {L^1}([0,\infty [ \times {\mathbf {R}})$ and $\hat f(z,s) = \smallint _{ - \infty }^\infty \smallint _0^\infty f(u,v){e^{iuz}}{e^{isv}}dudv$, where $z \in H = \{ z \in {\mathbf {C}}:\operatorname {Im} \geqslant 0\} ,s \in {\mathbf {R}}$, then $\hat f(H \times {\mathbf {R}}) \cup \{ 0\} = \hat f({\mathbf {R}} \times {\mathbf {R}}) \cup \{ 0\}$.