We develope a theory of generalized function spaces $LG'_\alpha$ and its generalizations $LG'_{ea}$, $\alpha>-1$, which elements have orthonormal expansions with respect to the Laguerre orthonormal systems $l_{n,\alpha}$, $n\to N_0$, $\alpha>- 1$. We define the $(\alpha,\beta)$-convolution product and find conditions of solvability of a convolution equations in these spaces. Finally, we give some applications of it in solving integral equations.