This book studies the situation over discrete Abelian groups with wide range applications. It covers classical functional equations, difference and differential equations, polynomial ideals, digital filtering and polynomial hypergroups, giving unified treatment of several different problems. There is no other comprehensive work in this field. The book will be of interest to graduate students, research workers in harmonic analysis, spectral analysis, functional equations and hypergroups.

1 Introduction (p. 1)

The basic tools for the investigation of different algebraic and analytical structures are representation and duality."Representation" means that we establish a correspondence between our abstract structure and a similar, more particular one. Usually this more particular structure, the"representing" structure is formed by functions, de.ned on a set which is the so-called"dual" object.

In order to get a"faithful" representation, it seems to be reasonable that the correspondence in question is one-to-one. Another reasonable requirement is that if the same procedure is applied to the dual object, then its dual can be identified with the original structure. In order to do that, the dual object should have an"internal" characterization. Finally, a characterization of the"representing" structure is also desirable : which functions on the dual object belong to the"representing" structure?

The method of representation and duality appears in several different fields of algebra, analysis, etc. For instance, linear spaces can be represented as linear spaces of linear functionals, topological spaces can be represented as topological spaces of continuous functions, topological groups can be represented as topological groups of special homomorphisms, and so on. However, in all these cases one can assure the faithfulness via different assumptions only.

In the case of linear spaces the injectivity of the representing mapping holds only if the linear functionals of the original linear space form a separating family, which leads to Hahn– Banach type theorems. In the case of topological spaces the same requirement leads to conditions similar to those in Uryshon’s Lemma. In the theory of algebras the corresponding representation process can be described by the Gelfand transformation.

Let A be a complex algebra and let H denote a set of algebra homomorphisms of A onto C, the algebra of complex numbers. Such homomorphisms are called multiplicative linear functionals . We remark that the assumption on the surjectivity of a complex algebra homomorphism is obviously equivalent to it being nonidentically zero. Evidently, H is a subset of the algebraic dual of A, however, in general, H has no natural algebraic structure.