A New Family of Topological Rings with Applications in Linear System Theory

Details A New Family of Topological Rings with Applications in Linear System Theory A New Family of Topological Rings with Applications in Linear System Theory

2011-06-28

arxiv.org/abs/1106.5746v1

Mathematics

Functional Analysis

Scientific paper

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 $\|f \diamond g\|_{p} \leq A(p-q) \|f\|_{q}\|g\|_{p}$ for all $p\ge q+d$, where $\diamond$ 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 rings). 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.

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