Representation of synthesizable differentiation-invariant subspaces of the Schwartz space

We consider a differentiation-invariant subspace $W$ in the Schwartz space $C^\infty(a;b)$ which admits weak spectral synthesis. We obtain the conditions under which W can be represented as the direct (algebraic and topological) sum of its residual subspace and the closed subspace spanned by the set of exponential monomials contained in $W$.