Primary submodules dual to invariant subspaces of $\Omega$-ultradifferentiable functions

Abstract

In this work we consider locally convex spaces of entire functions dual to general spaces of $\Omega$-ultradifferentiable functions. These spaces are also topological modules over the polynomial ring, which makes the problem on local description of their closed subspaces invariant under multiplication by the independent variable, in short, submodules, meaningful. More precisely, we study primary submodules, that is, those generated by a single function. It is shown that among principal ones there are submodules that do not admit a local description
in the weak sense. Each result on submodules, in turn, leads to an equivalent dual statement on the (non)admissibility of spectral synthesis in the weak sense for subspaces of
$\Omega$-ultradifferentiable functions invariant under the differentiation operator.