Description
Spectral analysis and spectral synthesis deal with the description of translation invariant function spaces over locally compact Abelian groups. Translation invariant function spaces appear in several different contexts for example linear ordinary and partial difference and differential equations with constant coefficients, theory of group representations, classical theory of functional equations. A fundamental problem is to discover the structure of such spaces of functions, or more exactly, to find an appropriate class of basic functions, the building blocks, which serve as “typical elements” of the space, a kind of basis. It turns out that these building blocks are the exponential monomials. The problem of spectral analysis is if the function spaces in question contain any exponential monomials. However, the problem of spectral synthesis is in ascertaining if there are enough exponential monomials in these function spaces to span a dense subspace. This book studies the situation over discrete Abelian groups with wide range applications in the fields of classical functional equations, difference and differential equations, polynomial ideals, digital filtering and polynomial hypergroups.
