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.


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.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.

International Journal of Mathematics and Mathematical Sciences
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 .
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 f : a, b → R is e.c.f.if and only if the kernel x, y → f x y is positive definite on the region 1/2 a, 1/2 b × 1/2 a, 1/2 b .Now, I will give a short summary of the h.c.f.Let Q be a complete separable locally compact metric space of points p, q, r, . . ., β Q be the σ-algebra of Borel subsets, and β 0 Q be the subring of β Q , which consists of sets with compact closure.We will consider the Borel measures; that is, positive regular measures on β Q , finite on compact sets.The spaces of continuous functions of finite continuous function, and of bounded functions are denoted by C Q , C 0 Q , and, C b Q , respectively.
An h.c.s. with the basis Q is defined by its structure measure c A, B, r A, B ∈ β Q ; r ∈ Q .A structure measure c A, B, r is a Borel measure in A resp.B if we fix B, r resp.A, r which satisfies the following properties: H3 The structure measure is said to be commutative if A measure m is said to be a multiplicative measure if H4 We will suppose the existence of a multiplicative measure.
For any f, g ∈ L 1 Q, m , the convolution f * g r Q f p g q dm r p, q .1.5 is well defined see 19 .
The space L 1 Q, m 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 Q.
A nonzero measurable and bounded almost everywhere function Q r → x r ∈ is said to be a character of the h.c.s.We should remark that, for a normal h.c.s., the mapping is an involution in the Banach algebra L 1 , the multiplicative measure is unique and characters of such a system are continuous.A character x of a normal h.c.s. is said to be Hermitian if Let X and X h be the sets of characters and Hermitian characters, respectively.A Hermitian character of a Hermitian h.c.s. are real valued x p x p p ∈ Q .Let L 1 Q, m be an h.c.s. with a basis Q and Φ a space of complex valued functions on Q. Assume that an operator valued function Q p → R p : Φ → Φ is given such that the function g p R p f q belongs to Φ for any f ∈ Φ and any fixed q ∈ Q.The operators R p p ∈ Q are called right generalized translation operators, provided that the following axioms are satisfied.

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T1 Associativity axiom: the equality 12 holds for any elements p, q ∈ Q.
T2 There exists an element e ∈ Q such that R e is the identity in Φ.
By the bilinear form we define the left generalized translation operators L p , such that L p f r R r f p for almost all p and q with respect to the measure m × m.L p and R p have the same properties, so that will call them generalized translation operators.
A one-to-one correspondence exists between normal h.c.s.L 1 Q, m with basis unity e and weakly continuous families of bounded involutive generalized translation operators L p satisfying the finiteness condition, preserving positivity in the space L 2 Q, m with unimodular strongly invariant measure m, and preserving the unit element.Convolution in the hypercomplex system L 1 Q, m and the corresponding family of generalized translation operators L p satisfy the relation Moreover, the h.c.s.L 1 Q, m is commutative if and only if the generalized translation operators L p p ∈ Q are commutative see 20 .

Exponentially Convex Functions
Let L 1 Q, m be a commutative normal h.c.s. with basis unity.
We consider a function h r ∈ C 0 Q and set λ i h r i in 2.2 .This yields n i,j 1 R r i ϕ r j h r i h r j ≥ 0.

2.4
By integrating this inequality with respect to each r 1 , . . ., r n over the set Q k k ∈ AE and collecting similar terms, we conclude that Further, we divide this inequality by n 2 and pass to the limit as n → ∞.We get for each k ∈ AE.By passing to the limit as k → ∞ and applying Lebesgue theorem, we see that 2.1 holds for all functions from C 0 Q .Approximating an arbitrary function from L 1 by finite continuous functions, we arrive at 2.1 .
By E • C Q , we shall denote the set of all bounded or continuous e.c.f.

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The next theorem is an analog of the Bochner theorem for h.c.s.i ϕ e ≥ 0; ii ϕ r ϕ r ; Proof.Let ϕ r ∈ C b Q is e.c.f. in the sense of 2.1 and let r 1 , . . ., r n ∈ Q and λ 1 , . . ., λ n ∈ Ê. Relation 2.7 and the fact that the generalized translation operators are continuous in

2.8
It is also follows from relation 2.7 that i and ii are trivial. iii

Exponentially Convex Functions and Kernals
Inequality 2.2 means that the kernel K t, s R t ϕ s is positive definite function.Therefore, this kernel possesses the following properties:

3.1
Now, we can use the properties of the kernel to prove the properties of the e.c.f.Indeed, ϕ r R e ϕ r K e, r K r, e ϕ r .

3.2
Similarly, R r ϕ s R s ϕ r .This implies that that is, iv .

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By setting s e in iv , we obtain Iϕ t 2 ≤ R t ϕ t ϕ e , ϕ t 2 ≤ ϕ e R t ϕ t .

3.4
In view of the relation
Mercer 1909 defines a continuous and symmetric real-valued function Φ on a, b × a, b ⊆ R 2 to be positive type if and only if b a C x C y Φ x, y dx dy ≥ 0, 1.1 where C x , C y ∈ C a, b .

Theorem 4 . 3 .U x ζ 0 2 2
Let L 1 Q, m 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 Q e.c. in the sense of 2.1 and the set of classes of unitarily equivalent bounded cyclic representation on the h.c.s.L 1 Q, m .This bijection is given by the relation ϕ r U r ζ 0 , ζ 0 H , r ∈ Q , 4.3 where Q ∈ E • C Q and U r is the corresponding representation of the h.c.s.L 1 Q, m in a Hilbert space H with cyclic vector ζ 0 .Proof.If U r is a bounded representation of the h.c.s.L 1 Q, m with cyclic vector ζ 0 .Then the function ϕ r U r ζ 0 , ζ 0 H is e.c.f. in the sense of 2.1 .Indeed, Let x ∈ L 1 Q, m .Then ϕ r x * x r dr x * x r U r drζ 0 , ζ 0 H U x * x ζ 0 , ζ 0 H ≥ 0.
Note that we use the identical involution x * x, we also present another definition of e.c.f.If the generalized translation operators R t extended toL ∞ : C b G → C b G × G .Then the defination 2.1 and 2.2 are equivalent for the functions ϕ r ∈ C b Q .
Definition 2.1.An essentially bounded functionϕ r r ∈ Q is called e.c.f if ϕ r x * * x r dr ≥ 0, ∀x ∈ L 1 .2.1 holds for all r 1 , . .., r n ∈ Q and λ 1 , . .., λ n ∈ Ê, n ∈ AE .Theorem 2.2.t ϕ s ∈ C b Q × Q , then the last inequality clearly implies 2.2 .Let us prove the converse assertion.Let Q n be an increasing sequence of compact sets covering the entire The proof is similar to that given for Theorem 3.1 of 20 , so we omit it.
Theorem 2.3.Every function ϕ ∈ E • C Q admits a unique representation in the form of an integral Proof.It follows directly from Theorem 2.3.Corollary 2.5.Assume that L 1 Q, m is a commutative h.c.s. with basis unity.Then a continuous bounded function ϕ r 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:

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 L 1 Q, m be a normal h.c.s. with basis unity e.The family of bounded operators U U p p∈Q in a separable Hilbert space H is called a representation of an h.c.s.ifThe family of generalized translation operators L p p ∈ Q defines a representation of the h.c.s.L 1 Q, m in Helbert space L 2 Q, m .Let Q p → U p be a representation of the h.c.s.L 1 Q, m.Below, we consider representation that satisfy conditions 1.5 -2.2 and the following additional condition: 5 the function Q p → U p is bounded.U x be a representation of the Banach algebra L 1 Q, m in a separable Hilbert space H.Two representation of an h.c.s.L 1 Q, m are unitarity equivalent if and only if the corresponding representations of the algebra L 1 Q, m are equivalent h.c.s.We recall that a representation of the Banach algebra L 1 Q, m in H is said to be cyclic if there exists a vector ζ ∈ H, cyclic vector, such that the linear subspace {Ux ζ : x ∈ L 1 Q, m } is dense in H.For any bounded representation U r of a normal h.c.s. with basis unity that satisfies the condition of separate continuity the following relation holds: International Journal of Mathematics and Mathematical SciencesProof.It suffices to show that if ϕ r is e.c. in the sense of 2.1 , then relation 2.2 holds for any λ 1 , . . ., λ n ∈ Ê and r 1 , . . ., r n ∈ Q.Indeed, by virtue of 4.2 and 4.3 , we have 4.4Corollary 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.