Generalized Fourier series by double trigonometric system

Necessary and sufficient conditions are obtained on the function $M$ such that $\{M(x,y)e^{ikx}{e}^{imy}:(k,m)\in \mathrm{\Omega }\}$ is complete and minimal in $L^p({\mathbb{T}}^2)$ when ${\mathrm{\Omega }}^c=\{(0,0)\}$ and $\Omega^{c} = 0\times$\mathbb{Z}$$\mathrm{.}If$\Omega^{c} = 0\times$\mathbb{Z}${0},$ $$\mathbb{Z}${0} = $\mathbb{Z}$\setminus{0}$itisprovedthatthesystem${ M(x,y) e^{i kx}e^{i my}: (k,m)\in \Omega }$cannotbecompleteminimalin$L^{p}($\mathbb{T}$^{2})$forany$M\in L^{p}($\mathbb{T}$^{2})$\mathrm{.}Inthecase,$\Omega^{c}={(0,0)}$ necessary and conditions are found in terms of the one-dimensional case.