New topological C-algebras with applications in linear systems theory

Motivated by the Schwartz space of tempered distributions ℓ′ and the Kondratiev space of stochastic distributions S ^-1 we define a wide family of nuclear spaces which are increasing unions of (duals of) Hilbert spaces H′ _p ∈ N, with decreasing norms ||·|| _p. The elements of these spaces are functions on a free commutative monoid. We characterize those rings in this family which satisfy an inequality of the form ||f * g|| _p ≤ A(p-q)||f|| _q||g|| _p for all p < q + d, where _* denotes the convolution in the monoid, A(p-q) is a strictly positive number and d is a fixed natural number (in this case we obtain commutative topological C-algebras). Such an inequality holds in S _-1, but not in ℓ′. We give an example of such a ring which contains ℓ′. We characterize invertible elements in these rings and present applications to linear system theory.