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.