Spectral synthesis in certain spaces of entire functions of exponential type and its applications

We consider certain spaces $P_\Omega$ of entire functions of exponential type in $\mathbb C^n$ associated with a domain $\Omega\in\mathbb R^n$ that are in fact Laplace transforms of distributions in $\Omega$. It is shown that any shift-invariant subspace of these functions admits spectral synthesis, that is, coincides with the closure of the linear span of the exponential polynomials contained in it. As an application of this result, we describe the solution space in $P_\Omega$ of a system of homogeneous equations of infinite order for differential operators with characteristic functions infinitely differentiable in $\Omega$.