Abstract

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 function is said to be a character of the h.c.s. , if for all (H5)An h.c.s. is said to be normal, if there exists an involution homomorphism , such that , and where (H6) A normal h.c.s. possesses a basis unity if there exists a point such that and If for all , then the normal h.c.s. is called Hermitian which is commutative.

We should remark that, for a normal h.c.s., the mapping is an involution in the Banach algebra , the multiplicative measure is unique and characters of such a system are continuous. A character of a normal h.c.s. is said to be Hermitian if Let and be the sets of characters and Hermitian characters, respectively.

A Hermitian character of a Hermitian h.c.s. are real valued .

Let be an h.c.s. with a basis and a space of complex valued functions on . Assume that an operator valued function is given such that the function belongs to for any and any fixed . The operators are called right generalized translation operators, provided that the following axioms are satisfied.(T1)Associativity axiom: the equality holds for any elements .(T2)There exists an element such that is the identity in .

By the bilinear form we define the left generalized translation operators , such that for almost all and with respect to the measure . and have the same properties, so that will call them generalized translation operators.

A one-to-one correspondence exists between normal h.c.s. with basis unity and weakly continuous families of bounded involutive generalized translation operators satisfying the finiteness condition, preserving positivity in the space with unimodular strongly invariant measure , and preserving the unit element. Convolution in the hypercomplex system and the corresponding family of generalized translation operators satisfy the relation Moreover, the h.c.s. is commutative if and only if the generalized translation operators are commutative (see [20]).

2. Exponentially Convex Functions

Let be a commutative normal h.c.s. with basis unity.

Definition 2.1. An essentially bounded function is called e.c.f if Note that we use the identical involution , we also present another definition of e.c.f.

A continuous bounded function is called e.c.f. if the inequality holds for all and , .

Theorem 2.2. If the generalized translation operators extended to . Then the defination (2.1) and (2.2) are equivalent for the functions .

Proof. It follows that , then the last inequality clearly implies (2.2).
Let us prove the converse assertion.
Let be an increasing sequence of compact sets covering the entire , that is, and .
We consider a function and set in (2.2).
This yields By integrating this inequality with respect to each over the set and collecting similar terms, we conclude that Further, we divide this inequality by and pass to the limit as . We get for each . By passing to the limit as and applying Lebesgue theorem, we see that (2.1) holds for all functions from . Approximating an arbitrary function from by finite continuous functions, we arrive at (2.1).

By , we shall denote the set of all bounded or continuous e.c.f.

The next theorem is an analog of the Bochner theorem for h.c.s.

Theorem 2.3. Every function admits a unique representation in the form of an integral where is a nonnegative finite regular measure on the space . Conversely, each function of the form (2.7) belongs to .

Proof. The proof is similar to that given for Theorem 3.1 of [20], so we omit it.

Corollary 2.4. If the product of any two Hermitian characters is e.c., then the product of any two continuous e.c.f. is also e.c.

Proof. It follows directly from Theorem 2.3.

Corollary 2.5. Assume that is a commutative h.c.s. with basis unity. Then a continuous bounded function is e.c. in the sense of (2.1) if and only if it is e.c. in the sense of (2.2). Moreover, it has the following properties: (i);(ii);(iii);(iv);(v).

Proof. Let is e.c.f. in the sense of (2.1) and let and . Relation (2.7) and the fact that the generalized translation operators are continuous in imply that It is also follows from relation (2.7) that(i)and (ii) are trivial.(iii)(iv) Finally,(v)

3. Exponentially Convex Functions and Kernals

Inequality (2.2) means that the kernel is positive definite function. Therefore, this kernel possesses the following properties: Now, we can use the properties of the kernel to prove the properties of the e.c.f.

Indeed, Similarly, . This implies that that is, (iv).

By setting in (iv), we obtain In view of the relation , we have Consequently, which implies (iii) and, hence, (i). Finally, (v) follows from the last inequality for , where .

4. Exponentially Convex Functions and Representations of Hypercomplex Systems

In this section, we will give the relation between the h.c.s. and e.c.f.

Let be a normal h.c.s. with basis unity . The family of bounded operators in a separable Hilbert space is called a representation of an h.c.s. if(1),(2),(3)for each , the vector is weakly continuous,(4)for all Condition (4.1) implies that the function is locally bounded.

Example 4.1. The family of generalized translation operators defines a representation of the h.c.s. in Helbert space .

Let be a representation of the h.c.s. . Below, we consider representation that satisfy conditions (1.5)–(2.2) and the following additional condition: (5)the function is bounded.

Such representation are called bounded.

Let be a representation of the Banach algebra in a separable Hilbert space .

Two representation of an h.c.s. are unitarity equivalent if and only if the corresponding representations of the algebra are equivalent h.c.s.

We recall that a representation of the Banach algebra in is said to be cyclic if there exists a vector , cyclic vector, such that the linear subspace is dense in .

Corollary 4.2. For any bounded representation of a normal h.c.s. with basis unity that satisfies the condition of separate continuity the following relation holds: For the proof (see [20]).

Theorem 4.3. Let be a normal h.c.s. with basis unity satisfying the condition of separate continuity. Then there is a bijection between the collection of continuous bounded function on e.c. in the sense of (2.1) and the set of classes of unitarily equivalent bounded cyclic representation on the h.c.s. . This bijection is given by the relation where and is the corresponding representation of the h.c.s. in a Hilbert space with cyclic vector .

Proof. If is a bounded representation of the h.c.s. with cyclic vector . Then the function is e.c.f. in the sense of (2.1). Indeed, Let . Then

Corollary 4.4. For a normal h.c.s. with basis unity that satisfies the condition of separate continuity, the concepts of e.c. in the sense of (2.1) and (2.2) are equivalent.

Proof. It suffices to show that if is e.c. in the sense of (2.1), then relation (2.2) holds for any and . Indeed, by virtue of (4.2) and (4.3), we have

Acknowledgments

This Project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 18-10/429. The author, therefore, acknowledges with thanks DSR support for Scientific Research.