Spectral synthesis and analytic continuation

A closed subspace of functions holomorphic in a domain of the $n$-dimensional complex space is considered. It is assumed that the subspace is invariant with respect to the partial differentiation operators and admits spectral synthesis, that is, coincides with the closure of the linear span of the common root elements in it of the partial differentiation operators. Conditions under which the elements of the invariant subspace admit analytic continuation to a larger domain are studied. The geometry of this domain depends both on the original domain and on the existence of functions admitting special lower bounds in the annihilator submodule of the invariant subspace. The same problem is also considered for topological products of invariant subspaces. The results are applied to the analytic continuation of solutions of homogeneous convolution equations.