Generalized Fourier series by double trigonometric system

Abstract: Necessary and sufficient conditions are obtained on the function $M$ such that ${ M(x,y) e^{i kx}e^{i my}: (k,m)\in \Omega }$ is complete and minimal in $L^{p}(\mathbb{T}^{2})$ when $\Omega^{c}={(0,0)}$ and $\Omega^{c} = 0\times\mathbb{Z}$. If $\Omega^{c} = 0\times\mathbb{Z}{0},$ $\mathbb{Z}{0} = \mathbb{Z}\setminus{0}$ it is proved that the system ${ M(x,y) e^{i kx}e^{i my}: (k,m)\in \Omega }$ cannot be complete minimal in $L^{p}(\mathbb{T}^{2})$ for any $M\in L^{p}(\mathbb{T}^{2})$. In the case, $\Omega^{c}={(0,0)}$ necessary and conditions are found in terms of the one-dimensional case.