On the Principal Ideal Theorem and spectral synthesis on discrete abelian groups (Q1677529)

Let \(G\) be an abelian group equipped with the discrete topology, \(C(G)\) be the set of all complex valued functions on \(G\). The set \(C(G)\) is a locally convex topological vector space with the topology of pointwise convergence. A closed linear subspace \(V\subseteq C(G)\) is called a variety on \(G\), if it is translation invariant that is if \(f(x)\in V\) then \(f(x+y)\in V\) for each \(y\in G\). A homomorphism of \(G\) into the multiplicative group of nonzero complex numbers is called an exponential, and a homomorphism of \(G\) into the additive group of complex numbers is called an additive function. A function is a polynomial if it belongs to the algebra generated by the additive functions. A product of a polynomial and an exponential is called an exponential monomial. If a variety is spanned by exponential monomials, then we say that the spectral synthesis holds for this variety. In the paper under review the author considers the case when a variety \(V\) is the set of solutions of the homogeneous convolution equations system on \(G\): \(V=\{ f\in C(G) : f\ast\mu_i=0, \,\, i=1,2,\dots,n\}\), where \(\mu_i\) are finitely supported complex valued functions on \(G\). The main result of the paper is the theorem that the spectral synthesis holds for this variety.