Spectral synthesis and functional equations

Abstract

We recall from the previous part that for a given locally compact Abelian group G any proper closed translation invariant subspace of C(G) is called a variety. The set of all exponentials in a variety is called the spectrum of the variety, and the set of all exponential monomials in a variety is called the spectral set of the variety. If V is a variety, then sp V denotes the spectrum of V and we write sp f for sp τ (f). If μ is in 蒙 _c (G), then we use the notation sp μ for the spectrum of the annihilator of the ideal generated by μ, and for any subset Λ of 蒙 _c (G) the spectrum, or spectral set of Λ, is the spectrum, or the spectral set of the annihilator of the ideal generated by Λ.