On a Certain Difference Equation and Its Application to the Moment Problem

We study the difference equation
$$ \sum_{m_1,m_2}\Omega[\sigma_{m_1,m_2}(z)]=g(z), \qquad z\in D, $$
where $D$ is the unit square, $g(z)\in A(D)$, $\sigma_{m_1,m_2}(z)=z+m_1i+m_2$, $|m_1|+|m_2|=2$, and $\Omega(z)\in A(cD)$ is an unknown function.