Motivated by the Schwartz space of tempered distributions S′ 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,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 S′. We give an example of such a ring which contains S′. We characterize invertible elements in these rings and present applications to linear system theory
D. Alpay and G. Salomon. New topological C-Algebras with Applications in Linear Systems Theory. Infinite Dimensional Analysis and Quantum Probability, vol. 12 (2012), pp. .