NEW TOPOLOGICAL ℂ-ALGEBRAS WITH APPLICATIONS IN LINEAR SYSTEMS THEORY

Abstract

Motivated by the Schwartz space of tempered distributions [Formula: see text] and the Kondratiev space of stochastic distributions [Formula: see text] we define a wide family of nuclear spaces which are increasing unions of (duals of) Hilbert spaces [Formula: see text], 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 [Formula: see text], but not in [Formula: see text]. We give an example of such a ring which contains [Formula: see text]. We characterize invertible elements in these rings and present applications to linear system theory.