New Topological C-Algebras with Applications in Linear Systems Theory

TL;DR

This paper introduces a new class of nuclear topological algebras based on spaces of functions on monoids, characterizes their properties, and explores applications in linear systems theory, especially focusing on convolution inequalities and invertibility.

Contribution

It defines a broad family of nuclear spaces with algebraic structures satisfying specific convolution inequalities, extending known spaces like $ ext{Schwartz space}$ and $ ext{Kondratiev space}$, and applies these to linear systems.

Findings

Identified conditions under which these spaces form topological algebras.

Demonstrated the inequality holds in $ ext{Kondratiev space}$ but not in $ ext{Schwartz space}$.

Provided examples of rings containing $ ext{Schwartz space}$ and characterized invertible elements.

Abstract

Motivated by the Schwartz space of tempered distributions $\mathscr S^\prime$ 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 $\mathscr H_p^\prime,p\in\mathbb N$, 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 $|f * g|_{p} \leq A(p-q) |f|_{q}|g|_{p}$ 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 $\mathscr S^\prime$. We give an example of such a ring which contains $\mathscr S^\prime$. We characterize invertible elements in these rings and present applications to linear system theory