Spectral synthesis problems on hypergroups

Abstract. Spectral analysis and spectral synthesis deal with the description of translation invariant function spaces on topological Abelian groups. It turns out that the basic building bricks of these spaces are the exponential monomials, which store information on common eigenfunctions on the translation operators together with multiplicity. On commutative hypergroups the presence of translation operators makes it possible to formulate and study the basic problems of spectral analysis and synthesis. However, a general concept of exponential monomials is necessary to build up a theory along the lines of the group-case. Here we present some relevant problems and solutions on the subject.