On a theorem of Fuchs

We show that the method used by Fuchs to prove the completeness of the system of functions $\{e^{-t}t^\alpha\nu\}$, $a_\nu>0$, $a_{\nu+1}-a_\nu\ge c>0$, $\nu=1,2,\dots$ in $L^2(0,\infty)$ allows us to prove a more general statement.