Exponentially Convex Functions on Hypercomplex Systems

A hypercomplex system (h.c.s.) is, roughly speaking, a space which is defined by a structure measure , , such space has been studied by Berezanskii and Krein. Our main result is to define the exponentially convex functions (e.c.f.) on (h.c.s.), and we will study their properties. The definition of such functions is a natural generalization of that defined on semigroup. 1. Introduction Harmonic Analysis theory and its relation with positive definite kernels is one of the most important subjects in functional analysis, which has different applications in mathematics and physics branches. Mercer (1909) defines a continuous and symmetric real-valued function on to be positive type if and only if where . Positive definite kernels generate a different kinds of functions, for example, positive, negative, and e.c.f. For more details you can see the work done by Stewart [1] in 1976 who gave a survey of these functions. Harmonic analysis of these functions on finite and infinite spaces or groups, semigroups, and hypergroups have a long history and many applications in probability theory, operator theory, and moment problem (see [2–10]). Many studies were done on e.c.f. on different structures (see [10–18]). Our aim in this study is to carry over the harmonic analysis of the e.c.f to the case of the h.c.s. These functions were first introduced by Berg et al., cf. [2]. The continuous functions is e.c.f. if and only if the kernel is positive definite on the region . Now, I will give a short summary of the h.c.f. Let be a complete separable locally compact metric space of points be the -algebra of Borel subsets, and be the subring of , which consists of sets with compact closure. We will consider the Borel measures; that is, positive regular measures on , finite on compact sets. The spaces of continuous functions of finite continuous function, and of bounded functions are denoted by , , and, , respectively. An h.c.s. with the basis is defined by its structure measure . A structure measure is a Borel measure in (resp. ) if we fix (resp. ) which satisfies the following properties:(H1)For all , the function . (H2)For all and , the following associativity relation holds (H3)The structure measure is said to be commutative if A measure is said to be a multiplicative measure if (H4) We will suppose the existence of a multiplicative measure. is well defined (see [19]). The space with the convolution (1.5) is a Banach algebra which is commutative if (H3) holds. This Banach algebra is called the h.c.s. with the basis . A nonzero measurable and bounded almost everywhere