Müntz–Szász type approximation in direct products of spaces

We consider the problem of completeness of the system of exponentials $\exp\{-\lambda_nt\}$, $\operatorname{Re}\lambda_n>0$, in direct products $E=E_1\times E_2$ of the spaces $E_1=E_1(0,1)$ and $E_2=E_2(1,\infty)$ of functions defined on $(0,1)$ and $(1,\infty)$, respectively. We describe rather broad classes of spaces $E_1$ and $E_2$ such that the well-known condition of Szász is necessary for the completeness of the above system in $E$ and sufficient for this completeness.