On condition of representing a subspace in Schwartz space invariant with respect to differentiation as direct sum of its residual and exponential components

Abstract

In the work we consider the Schwartz space $\mathcal E$ of infinitely differentiable functions on the real line and its closed subspaces invariant with respect to the differentiation operator. It is known that each such space possesses, possibly trivial, exponential and residual components, which are defined by a multiple sequence of points $(-\mathrm{i}\Lambda)$ in the complex plane (spectrum $W$) and by a relatively closed in $\mathbb{R}$ segment $I_W$ (residual interval of the subspace $W$). Recent studies showed that under certain restrictions for the behavior of $\Lambda$ and $I_W$, the corresponding invariant subspace $W$ is uniquely recovered by these characteristics, that is, it admits a spectral synthesis in a weak sense. In the case when the spectrum $(-\mathrm{i}\Lambda)$ is a finite sequence, the exponential component of the subspace $W$ is finite-dimensional and the subspace $W$ is the algebraic sum of the residual subspace and a finite-dimensional span of the set of exponential monomials contained in $W$. In the case of an infinite discrete spectrum we obtained the conditions, under which the algebraic sum of the residual and exponential subspaces in $W$ is closed, and hence, it is a direct topological sum coinciding with $W$. These conditions were general but not convenient enough for a straightforward checking. Here we obtain transparent easily checked conditions for the infinite sequence $\Lambda,$ under which the invariant subspace $W$ with the spectrum $(-\mathrm{i}\Lambda)$ and the residual interval $I_W$ is a direct algebraic and topological sum of its exponential and residual components, that is, each element in $W$ is uniquely represented as a sum of two functions, one of which is the limit of a sequence of exponential monomials in $\mathcal E,$ while the other vanishes identically on $I_W.$