Spectral synthesis in certain spaces of entire functions of exponential type and its applications (Q2710698)

The author considers certain spaces \(P_{\Omega}\) of entire functions of exponential type in \({\mathbb C}^{n}\) associated with a domain \(\Omega \subset {\mathbb R}^{n}\). These functions are in fact Laplace transforms of distributions in \(\Omega\). It is shown that an arbitrary shift-invariant subspace of the functions under consideration admits spectral synthesis, i. e. coincides with the closure of the linear span of the exponential polynomials contained in it. As an application of this result, the author describes the solution space in \(P_{\Omega}\) of a system of homogeneous equations of infinite order for differential operators with characteristic functions infinitely differentiable in \(\Omega\).