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 $x\mapsto P(a_1(x), \dots, a_m(x))$ on $G$ is said to be polynomial function if $P(z_1,\dots, z_m)$ is a complex polynomial in $m$ variables and $a_1,\dots, a_m$ 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, $v_1, \dots, v_n$ be a system of generators of $G$. Any element $x\in G$ can be representd in the form $x=t_1 v_1+\dots +t_n v_n$, where $t_j\in{\mathbb Z}$ (this representation can be not unique). For $x\in G$, we define the number $|x|\in{\mathbb Z}_+=\{0,1,2,\dots\}$ by $|x|:=\min\{ |t_1|+\dots+|t_n| : x=t_1 v_1+\dots +t_n v_n, \,\, t_j\in{\mathbb Z}, \, j=1,\dots,n\}$. 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 $|f(x)| (1+|x|)^{-k} \to 0 \quad\text{ as }\quad |x|\to \infty$. 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.

UDC: 517.986.62
BBK: 22.152

DOI: 10.15688/jvolsu1.2016.5.5