Generalized Fourier Series by the Double Trigonometric System

Abstract

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