Examples of Hypercomplex Systems

Abstract

As indicated in the introduction to Section 2 of Chapter 1, the appearance of generalized translation operators is, as a rule, connected with the existence of a Fourier-type transformation satisfying the Plancherel theorem and the inversion formula. These generalized translation operators often possess additional properties which enable one to construct a hypercomplex system. In view of the existence of developed harmonic analysis for hypercomplex systems, it is possible to consider, from the general point of view, numerous results of harmonic analysis obtained in various special cases. Note that the application of duality theory to these cases sometimes gives new results (see, e.g., Vainerman’s version of the inverse problem for the Sturm-Liouville equation [Vai3] in Subsection 4.5 of this chapter).