A conjecture was given in [3] about the possibility of decomposition of an arbitrary $f$ in $L^2(R)$ in terms of the family of functions $$ \f_{mn}(x) =\pi^{-1/4} \exp\{-(1/2) imnab+imxa-(1/2) (x-nb)^2\}, \qquad a,b>0; ab<2\pi. $$ We prove this conjecture for $ab<2\pi$ and $b$ sufficiently large. Also, we give some applications for the Bargman space of entire functions.