ABSTRACT
Spectral analysis and spectral synthesis deal with the description of translation invariant function spaces on topological groups. The basic building blocks of this description are the exponential functions and exponential monomials. The case of non-discrete Abelian groups is more sophisticated. In fact, some counterexamples of D. I. Gurevich in Gurevic show that a direct extension of Laurent Schwartz’s result is not possible even for functions in several real variables. Affine isometries obviously form a group with respect to composition. The linear groups are homogeneous transformation groups, acting on the finite dimensional vector space and leaving the origin fixed. The presence of translation operators makes it possible to study classical functional equations on affine groups. The chapter discusses different basic classes of functions which are also fundamental from the point of view of spectral synthesis.
