# Generalized Fourier series by double trigonometric system

## TL;DR

This paper establishes conditions for the completeness and minimality of a generalized Fourier system on the torus, depending on the function M and the set Omega, extending classical results to a two-dimensional setting.

## Contribution

It provides necessary and sufficient conditions for the completeness and minimality of a double trigonometric system in L^p spaces, generalizing previous one-dimensional results.

## Key findings

- Conditions for completeness and minimality depending on Omega and M.
- Proof that certain systems cannot be complete minimal for specific Omega.
- Extension of one-dimensional results to two-dimensional Fourier systems.

## 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.