Generalized Fourier series by double trigonometric system

Abstract

Necessary and sufficient conditions are obtained on the function $M$ such that $\{M(x,y)e^{ikx}{e}^{imy}:(k,m)\in \Omega \}$ 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 $\{M(x,y)e^{ikx}{e}^{imy}:(k,m)\in \Omega \}$ 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.