New Topological C-Algebras with Applications in Linear Systems Theory

Abstract

Motivated by the Schwartz space of tempered distributions ${\mathcal{S}}^{\prime }$ and the Kondratiev space of stochastic distributions ${\mathcal{S}}_{-1}$ we define a wide family of nuclear spaces which are increasing unions of (duals of) Hilbert spaces ${\mathcal{H}}_p^{\prime },p\in \mathbb{N}$, with decreasing norms $|\cdot {|}_p$. 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 $|f\ast g{|}_p\le A(p-q)|f{|}_q|g{|}_p$ for all $p\ge q+d$, 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 $\mathbb{C}$-algebras). Such an inequality holds in ${\mathcal{S}}_{-1}$, but not in ${\mathcal{S}}^{\prime }$. We give an example of such a ring which contains ${\mathcal{S}}^{\prime }$. We characterize invertible elements in these rings and present applications to linear system theory

Version accepted and to appear in Infinite Dimensional Analysis, Quantum Probability and Related Topics (Vol. 15, number 2, 2012)