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

Let be an arbitrary locally compact Abelian group (LCAgroup) and let F be a topological vector space (TVS) consisting of complexvalued functions on 𝐺. The space F is said to be translation invariant if F is invariant with respect to the transformations : 𝑓(𝑥) ↦→ 𝑓(𝑥 - 𝑦), 𝑓(𝑥) ∈ ∈ F, ∈ 𝐺, and all operators are continuous on F. A closed linear subspace H ⊆ F is referred to as an invariant subspace if 𝑦(H ) ⊆ H for every ∈ 𝐺. By an exponential function or generalized character we mean an arbitrary continuous homomorphism from a group to the multiplicative group C* := := C ∖ {0} of nonzero complex numbers. Continuous homomorphisms of to the additive group of complex numbers are referred to as additive functions. A function ↦→ 𝑃(𝑎1(𝑥),..., 𝑎𝑚(𝑥)) on is said to be polynomial function if 𝑃(𝑧1,..., 𝑧𝑚) is a complex polynomial in variables and 𝑎1,..., 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 F be a translation-invariant function space on the group and let be H an invariant subspace of F. An invariant subspace H admits spectral synthesis if it coincides with the closure in F of the linear span of all exponential monomials belonging to H. We say that spectral synthesis holds in F if every invariant subspace H ⊆ F admits spectral synthesis. One of the natural function space is the space S′(𝐺) of all tempered distributions on a LCA-group 𝐺. In the present paper we study spectral synthesis in the space S′(𝐺) for the case when is a discrete Abelian group. In this case the distributions from S′(𝐺) coincide with usual functions, thus we will refer to S′(𝐺) as the space of tempered functions. Let us consider a convenient definition of the space S′(𝐺) on a discrete finite generated Abelian group 𝐺. Let be a finitely generated Abelian group, 𝑣1,..., be a system of generators of 𝐺. Any element ∈ can be representd in the form = 𝑡1𝑣1 + +· · ·+𝑡𝑛𝑣𝑛, where ∈ Z (this representation can be not unique). For ∈ 𝐺, we define the number |𝑥| ∈ Z+ = {0, 1, 2,... } by |𝑥| := min{|𝑡1| + · · · + |𝑡𝑛| : = = 𝑡1𝑣1 + · · · + 𝑡𝑛𝑣𝑛, ∈ Z, = 1,..., 𝑛}. The function |𝑥| is a special example of a quasinorm on 𝐺. For every > 0, we denote by S′ 𝑘(𝐺) the set of all compex-valued functions 𝑓(𝑥) on that satisfy |𝑓(𝑥)|(1 + |𝑥|)-𝑘 → 0 as |𝑥| → ∞. The set S′(𝐺) is a Banach space with respect to the norm ‖𝑓‖𝐺,𝑘 = ‖𝑓‖𝑘 := sup𝑥∈𝐺 |𝑓(𝑥)|(1 + +|𝑥|)-𝑘. Clearly, S′ 𝑘1(𝐺) ⊆ S′ 𝑘2(𝐺) if 𝑘1 0S′ 𝑘(𝐺) with the topology of the inductive limit of the Banach spaces S′ 𝑘(𝐺). Thus 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 S′(𝐺) for any finitely generated Abelian group 𝐺.