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 Tr. Petrozavodsk. Gos. Univ. Ser. Mat., 1997, Issue 4, Pages 105–124 (Mi pa128)

Invariant subspaces in functional spaces of polynomial growth on $\mathbb{R}^{N}$

S. S. Platonov

Abstract: Let $G$ be a transitive group of transformations of a set $M, \mathcal{F}$ be some locally convex space consisting of complex-valued functions on $M, \pi(g): f(x)\to f(g^{-1}x), f(x)\in \mathcal{F}$ be the quasiregular representation of $G$. A linear subspace $H\subseteq \mathcal{F}$ we call an invariant subspace if $H$ is closed and invariant with respect to the representation $\pi$. In the paper we consider the case when $M$ is $n$-dimensional Euclidean space $R^{n}, G$ is the group of all orientation-preserving isometries. The function spaces are spaces of polynomial growth, for example $\mathcal{F}=S'$ is the space of tempered distributions on $R^{n}$. The main result of the paper is the complele description of invariant subspaces of this function spaces. In particular we obtain the description of irreductible and indecomposable subspaces.

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Bibliographic databases:
UDC: 517.518

Citation: S. S. Platonov, “Invariant subspaces in functional spaces of polynomial growth on $\mathbb{R}^{N}$”, Tr. Petrozavodsk. Gos. Univ. Ser. Mat., 1997, no. 4, 105–124

Citation in format AMSBIB
\Bibitem{Pla97} \by S.~S.~Platonov \paper Invariant subspaces in functional spaces of polynomial growth on $\mathbb{R}^{N}$ \jour Tr. Petrozavodsk. Gos. Univ. Ser. Mat. \yr 1997 \issue 4 \pages 105--124 \mathnet{http://mi.mathnet.ru/pa128} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1635433} \zmath{https://zbmath.org/?q=an:0916.43004} 

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This publication is cited in the following articles:
1. S. S. Platonov, “Invariantnye podprostranstva v funktsionalnykh prostranstvakh medlennogo rosta na svetovom konuse v $R^{3}$”, Trudy PGU. Matematika, 2011, no. 18, 21–60
2. S. S. Platonov, “Invariant subspaces in some function spaces on the light cone in $\mathbb R^3$”, Sb. Math., 203:6 (2012), 864–892
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