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 Mathematical Physics and Computer Simulation: Year: Volume: Issue: Page: Find

 Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, Issue 5(36), Pages 42–59 (Mi vvgum130)

Mathematics

On spectral synthesis in the space of tempered functions on finitely generated Abelian groups

S. S. Platonov

Petrozavodsk State University

Abstract: Let $G$ be an arbitrary locally compact Abelian group (LCA-group) and let ${\mathscr F}$ be a topological vector space (TVS) consisting of complex-valued functions on $G$. The space ${\mathscr F}$ is said to be translation invariant if ${\mathscr F}$ is invariant with respect to the transformations $\tau_y: f(x)\mapsto f(x-y), \quad f(x)\in{\mathscr F}, y\in G$, and all operators $\tau_y$ are continuous on ${\mathscr F}$.
A closed linear subspace ${\mathscr H}\subseteq{\mathscr F}$ is referred to as an invariant subspace if $\tau_y({\mathscr H})\subseteq {\mathscr H}$ for every $y\in G$.
By an exponential function or generalized character we mean an arbitrary continuous homomorphism from a group $G$ to the multiplicative group ${\mathbb C}_*:={\mathbb C}\setminus \{0\}$ of nonzero complex numbers. Continuous homomorphisms of $G$ to the additive group of complex numbers are referred to as additive functions. A function $x\mapsto P(a_1(x), …, a_m(x))$ on $G$ is said to be polynomial function if $P(z_1,…, z_m)$ is a complex polynomial in $m$ variables and $a_1,…, a_m$ are additive functions. A product of a polynomial and an exponential function is referred to as an exponential monomial, and a sum of exponential monomials is referred to as an exponential polynomial on $G$.
Let ${\mathscr F}$ be a translation-invariant function space on the group $G$ and let be ${\mathscr H}$ an invariant subspace of ${\mathscr F}$. An invariant subspace ${\mathscr H}$ admits spectral synthesis if it coincides with the closure in ${\mathscr F}$ of the linear span of all exponential monomials belonging to ${\mathscr H}$. We say that spectral synthesis holds in ${\mathscr F}$ if every invariant subspace ${\mathscr H}\subseteq{\mathscr F}$ admits spectral synthesis.
One of the natural function space is the space ${\mathscr S}'(G)$ of all tempered distributions on a LCA-group $G$. In the present paper we study spectral synthesis in the space ${\mathscr S}'(G)$ for the case when $G$ is a discrete Abelian group. In this case the distributions from ${\mathscr S}'(G)$ coincide with usual functions, thus we will refer to ${\mathscr S}'(G)$ as the space of tempered functions. Let us consider a convenient definition of the space ${\mathscr S}'(G)$ on a discrete finite generated Abelian group $G$.
Let $G$ be a finitely generated Abelian group, $v_1, …, v_n$ be a system of generators of $G$. Any element $x\in G$ can be representd in the form $x=t_1 v_1+…+t_n v_n$, where $t_j\in{\mathbb Z}$ (this representation can be not unique). For $x\in G$, we define the number $|x|\in{\mathbb Z}_+=\{0,1,2,…\}$ by $|x|:=\min\{ |t_1|+…+|t_n| : x=t_1 v_1+…+t_n v_n, t_j\in{\mathbb Z}, j=1,…,n\}$. The function $|x|$ is a special example of a quasinorm on $G$.
For every $k>0$, we denote by ${\mathscr S}'_k(G)$ the set of all compex-valued functions $f(x)$ on $G$ that satisfy $|f(x)| (1+|x|)^{-k} \to 0 \quad as \quad |x|\to \infty$. The set ${\mathscr S}'(G)$ is a Banach space with respect to the norm $\|f\|_{G,k}=\|f\|_k:=\sup_{x\in G} |f(x)| (1+|x|)^{-k}$. Clearly, ${\mathscr S}'_{k_1}(G)\subseteq {\mathscr S}'_{k_2}(G)$ if $k_1<k_2$, and this embedding is continuous. We equip the space ${\mathscr S}'(G):=\bigcup_{k>0} {\mathscr S}'_k(G)$ with the topology of the inductive limit of the Banach spaces ${\mathscr S}'_k(G)$. Thus ${\mathscr S}'(G)$ is a translation invariant locally convex space.
The main results of the paper is the theorem, that spectral synthesis holds in the space ${\mathscr S}'(G)$ for any finitely generated Abelian group $G$.

Keywords: spectral synthesis, locally compact Abelian groups, finitely generated Abelian groups, tempered functions, Bruhat–Schwartz functions.

DOI: https://doi.org/10.15688/jvolsu1.2016.5.5

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Citation: S. S. Platonov, “On spectral synthesis in the space of tempered functions on finitely generated Abelian groups”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, no. 5(36), 42–59

Citation in format AMSBIB
\Bibitem{Pla16} \by S.~S.~Platonov \paper On spectral synthesis in the space of tempered functions on finitely generated Abelian groups \jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica \yr 2016 \issue 5(36) \pages 42--59 \mathnet{http://mi.mathnet.ru/vvgum130} \crossref{https://doi.org/10.15688/jvolsu1.2016.5.5}