Spectral synthesis in a complex domain for a differential operator with constant coefficients. I: A duality theorem

The problem of spectral synthesis in a complex domain for a differential operator with symbol $\pi(z)=z^q+a_1z^{q-1}+\dots+a_q$, $a_i\in\mathbf C$, is reduced to the problem of a local description of the closed submodules of a module (of entire functions of exponential type) over the ring $\mathbf C[\pi]$ of polynomials of the form $c_0+c_1\pi+\dots+c_n\pi^n$, $c_i\in\mathbf C$.