## Abstract

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} ∈ N, 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 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.

Original language | English |
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Article number | 1250011 |

Journal | Infinite Dimensional Analysis, Quantum Probability and Related Topics |

Volume | 15 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jun 2012 |

## Keywords

- Kondratiev spaces
- Nuclear spaces
- Schwartz space of tempered distributions
- Våge inequality
- Wick product
- convolution
- linear systems on commutative rings
- topological rings
- white noise space

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Mathematical Physics
- Applied Mathematics