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', p is an element of N, with decreasing norms parallel to.parallel to(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 parallel to f * g parallel to (p) <= A(p - q) parallel to f parallel to(q) parallel to g parallel to(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.