Spectral Analysis and Spectral Synthesis

Spectral analysis and spectral synthesis deal with the description of translation invariant function spaces over locally compact Abelian groups. One considers the space of all complex valued continuous functions on a locally compact Abelian group G, which is a locally convex topological linear space with respect to the point-wise linear operations (addition, multiplication with scalars) and to the topology of uniform convergence on compact sets. A variety is a closed translation invariant subspace of this space. Continuous homomorphisms of G into the additive topological group of complex numbers and into the multiplicative topological group of nonzero complex numbers, respectively, are called additive and exponential functions, respectively. A function is a polynomial if it belongs to the algebra generated by the continuous additive functions. An exponential monomial is the product of a polynomial and an exponential. It turns out that exponential functions, or more generally, exponential monomials can be considered as basic building bricks of varieties. A given variety may or may not contain any exponential function or exponential monomial. If it contains an exponential function, then we say that spectral analysis holds for the variety. An exponential function in a variety can be considered as a kind of spectral value and the set of all exponential functions in a variety is called the spectrum of the variety. It follows that spectral analysis for a variety means that the spectrum of the variety is nonempty. On the other hand, the set of all exponential monomials contained in a variety is called the spectral set of the variety. It turns out that if an exponential monomial belongs to a variety, then the exponential function appearing in the representation of this exponential monomial belongs to the variety, too. Hence, if the spectral set of a variety is nonempty, then also the spectrum of the variety is nonempty and spectral analysis holds. There is, however, an even stronger property of some varieties, namely, if the spectral set of the variety spans a dense subspace of the variety. In this case, we say that spectral synthesis holds for the variety. It follows that for nonzero varieties spectral synthesis implies spectral analysis. If spectral analysis (resp., spectral synthesis) holds for every variety on an Abelian group, then we say that spectral analysis (resp., spectral synthesis) holds on the Abelian group. A famous and pioneer result of L. Schwartz exhibits the situation by stating that if the group is the reals with the Euclidean topology, then spectral values do exist, that is, any nonzero variety contains an exponential function. In other words, in this case the spectrum is nonempty, spectral analysis holds. Furthermore, spectral synthesis also holds in this situation: there are sufficiently many exponential monomials in the variety in the sense that their linear hull is dense in the subspace. In this survey paper, we present a summary of the relevant results in spectral analysis and spectral synthesis including the most recent developments.