# Exponentially Convex Functions on Hypercomplex Systems

Authors: Buthinah A. Bin Dehaish
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

A hypercomplex system (h.c.s.)L1(Q,m) is, roughly speaking, a space which is defined by a structure measure (c(A,B,r), (A,B∈ℬ(Q))), 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.


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## Body

## 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 functionΦ on [a,b]×[a,b]⊆R2 to be positive type if and only if(1.1)∬abC(x)C(y)Φ(x,y)dxdy≥0,
where C(x),C(y)∈C[a,b].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 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.LetQ be a complete separable locally compact metric space of points p,q,r,…,β(Q) be the σ-algebra of Borel subsets, and β0(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), C0(Q), and, Cb(Q), respectively.An h.c.s. with the basisQ 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:(H1)
For allA,B∈β0(Q), the function c(A,B,r)∈C0(Q).(H2)
For allA,B∈β0(Q) and s,r∈Q, the following associativity relation holds
(1.2)∫Qc(A,B,r)drc(Er,C,s)=∫Qc(B,C,r)drc(A,Er,s),C∈β(Q).(H3)
The structure measure is said to be commutative if(1.3)c(A,B,r)=c(B,A,r),(A,B∈β0(Q))
A measure m is said to be a multiplicative measure if
(1.4)∫Qc(A,B,r)dm(r)=m(A)m(B);A,B∈β0(Q).(H4)
We will suppose the existence of a multiplicative measure.(1.5)Foranyf,g∈L1(Q,m),theconvolution(f*g)(r)=∬Qf(p)g(q)dmr(p,q).
is well defined (see [19]).  The space L1(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)∈ℂ is said to be a character of the h.c.s. L1, if for all A,B∈β0(Q)(1.6)∫Qc(A,B,r)x(r)dm(r)=x(A)x(B),∫Cx(r)dm(r)=x(C),C∈β0(Q).(H5)
An h.c.s. is said to be normal, if there exists an involution homomorphismQ∋r↦r*∈Q, such that m(A)=m(A*), and
(1.7)c(A,B,C)=c(C,B*,A),c(A,B,C)=c(A*,C,B),(A,B∈β0(Q)),
where
(1.8)c(A,B,C)=∫Cc(A,B,r)dm(r).(H6)
A normal h.c.s. possesses a basis unity if there exists a pointe∈Q such that e*=e and
(1.9)c(A,B,e)=m(A*∩B),A,B∈β(Q).
If r*=r for all r∈Q, then the normal h.c.s. is called Hermitian which is commutative.We should remark that, for a normal h.c.s., the mapping(1.10)L1(Q,m)∋f(r)⟶f*(r)∈L1(Q,m)
is an involution in the Banach algebra L1, 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(1.11)x(r*)=x(r)¯,(r∈Q).
Let X and Xh be the sets of characters and Hermitian characters, respectively.A Hermitian character of a Hermitian h.c.s. are real valuedx(p)¯=x(p)(p∈Q).LetL1(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↦Rp:Φ→Φ is given such that the function g(p)=(Rpf)(q) belongs to Φ for any f∈Φ and any fixed q∈Q. The operators Rp(p∈Q) are called right generalized translation operators, provided that the following axioms are satisfied.(T1)
Associativity axiom: the equality(1.12)(Rpq(Rqf))(r)=(Rqr(Rpf))(r)
holds for any elements p,q∈Q.(T2)
There exists an elemente∈Q such that Re is the identity in Φ.By the bilinear form(1.13)(Lpf,g)=(g*,Rp*f*)=(Rrf,g),(f,g∈L2;r,p∈Q),
we define the left generalized translation operators Lp, such that Lpf(r)=Rrf(p) for almost all p and q with respect to the measure m×m. Lp and Rp have the same properties, so that will call them generalized translation operators.A one-to-one correspondence exists between normal h.c.s.L1(Q,m) with basis unity e and weakly continuous families of bounded involutive generalized translation operators Lp satisfying the finiteness condition, preserving positivity in the space L2(Q,m) with unimodular strongly invariant measure m, and preserving the unit element. Convolution in the hypercomplex system L1(Q,m) and the corresponding family of generalized translation operators Lp satisfy the relation
(1.14)(f*g)(p)=∫Q(Lpf)(q)g(q*)dq=(Lpf,g*)2,(f,g∈L2).
Moreover, the h.c.s. L1(Q,m) is commutative if and only if the generalized translation operators Lp(p∈Q) are commutative (see [20]).


## 2. Exponentially Convex Functions

LetL1(Q,m) be a commutative normal h.c.s. with basis unity.Definition 2.1.
An essentially bounded functionφ(r)(r∈Q) is called e.c.f if
(2.1)∫φ(r)(x**x)(r)dr≥0,∀x∈L1.
Note that we use the identical involution x*=x, we also present another definition of e.c.f.A continuous bounded functionφ(r)(r∈Q) is called e.c.f. if the inequality(2.2)∑i,j=1nλiλj(Rrjφ)(rj)≥0
holds for all r1,…,rn∈Q and λ1,…,λn∈ℝ, (n∈ℕ).Theorem 2.2.
If the generalized translation operatorsRt extended to L∞:Cb(G)→Cb(G×G). Then the defination (2.1) and (2.2) are equivalent for the functions φ(r)∈Cb(Q).Proof.
(2.3)∫φ(t)(x*x)(t)dt=∫∫(Ltx)(s)φ(t)x(s)dsdt=∫∫(Lsx)(t)φ(t)x(s)dsdt=∫(x,Ls*φ̅)2x(s)ds=∫∫x(t)(Ls*φ̅)(t)¯x(s)ds=∫∫(Lsφ)(t)x(t)x(s)dsdt=∫∫(Rtφ)(s)x(s)x(t)dsdt≥0.
It follows that (Rtφ)(s)∈Cb(Q×Q), then the last inequality clearly implies (2.2).
Let us prove the converse assertion.
LetQn be an increasing sequence of compact sets covering the entire Q, that is, Q1⊆Q2⊆⋯⊆Qn and Q=⋃i=1nQi.
We consider a functionh(r)∈C0(Q) and set λi=h(ri) in (2.2).
This yields(2.4)∑i,j=1n(Rriφ)(rj)h(ri)h(rj)≥0.
By integrating this inequality with respect to each r1,…,rn over the set Qk(k∈ℕ) and collecting similar terms, we conclude that
(2.5)nm(Qk)∫Qk(Rrφ)(r)h2(r)dr+n(n-1)∬Qk(Rrφ)(s)h(r)h(s)drds≥0.
Further, we divide this inequality by n2 and pass to the limit as n→∞. We get
(2.6)∬Qk(Rrφ)(s)h(r)h(s)¯drds≥0,
for each k∈ℕ. By passing to the limit as k→∞ and applying Lebesgue theorem, we see that (2.1) holds for all functions from C0(Q). Approximating an arbitrary function from L1 by finite continuous functions, we arrive at (2.1).ByE·C(Q), 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 functionφ∈E·C(Q) admits a unique representation in the form of an integral
(2.7)φ(r)=∫Xhx(r)dμ(x),(r∈Q),
where μ is a nonnegative finite regular measure on the space Xh. Conversely, each function of the form (2.7) belongs to E·C(Q).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 Theorem2.3.Corollary 2.5.
Assume thatL1(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: (i)
φ(e)≥0;(ii)
φ(r)=φ(r)¯;(iii)
|φ(r)|≤φ(e);(iv)
|(Rsφ)(t)|2≤(Rsφ)(s)(Rtφ)(t);(v)
|φ(s)-φ(t)|2≤2φ(e)[φ(e)-(Rsφ)(t)].Proof.
Letφ(r)∈Cb(Q) is e.c.f. in the sense of (2.1) and let r1,…,rn∈Q and λ1,…,λn∈ℝ. Relation (2.7) and the fact that the generalized translation operators are continuous in L∞(Q,m) imply that
(2.8)∑i,j=1nλiλj(Rriφ)(rj)=∑i,j=1nλiλj∫Xh(Rrix)(rj)dμ(x)=∫Xh∑i,j=1nλiλjx(ri)x(rj)dμ(x)=∫Xh(∑k=1nλkx(rk))2dμ(x)≥0.
It is also follows from relation (2.7) that(i)
and (ii) are trivial.(iii)
(2.9)|φ(r)|≤∫Xh|x(r)|dμ(x)≤μ(X)=φ(e).(iv)
(2.10)|(Rsφ)(t)|2=|∫Xhx(s)x(t)dμ(x)|2≤∫Xh|x(s)|2dμ(x)∫Xh|x(t)|2dμ(x)=∫Xhx(s)x(s*)dμ(x)∫Xhx(t)x(t*)dμ(x)=(Rsφ)(s)(Rtφ)(t).
Finally,(v)
(2.11)|φ(s)-φ(t)|2≤|∫Xh(x(s)-x(t))dμ(x)|2≤φ(e)∫Xh(|x(s)|2+|x(t)|2-2x(s)x(t))dμ(x)≤2φ(e)∫Xh(1-Rsx)(t)dμ(x)=2φ(e)[φ(e)-(Rsφ)(t)].


## 3. Exponentially Convex Functions and Kernals

Inequality (2.2) means that the kernel K(t,s)=(Rtφ)(s) is positive definite function. Therefore, this kernel possesses the following properties:(3.1)K(t,t)≥0,K(t,s)=K(s,t)¯,|K(t,s)|2≤K(t,t)K(s,s),|K(t,r)-K(s,r)|2≤K(r,r)(K(t,t)-2ReK(t,s)+K(s,s)).
Now, we can use the properties of the kernel to prove the properties of the e.c.f.Indeed,(3.2)φ(r)=(Reφ)(r)=K(e,r)=K(r,e)¯=φ(r)¯.
Similarly, (Rrφ)(s)=(Rsφ)(r)¯. This implies that(3.3)|(Rtφ)(s)|2≤(Rtφ)(t)(Rsφ)(s),
that is, (iv).By settings=e in (iv), we obtain(3.4)|(Rtφ)(e)|2≤(Rtφ)(t)(Reφ)(e),|(Iφ)(t)|2≤(Rtφ)(t)φ(e),|φ(t)|2≤φ(e)(Rtφ)(t).
In view of the relation |(Lpf)(q)|≤∥f∥∞(f∈Cb(Q)), we have(3.5)(Rrφ)(r)≤‖φ‖∞.
Consequently,(3.6)|φ(r)|2≤φ(e),
which implies (iii) and, hence, (i). Finally, (v) follows from the last inequality for K(t,s), where r=e.(3.7)|φ(s)-φ(t)|2≤|(Rtφ)(e)-(Rsφ)(e)|2≤(Reφ)(e)[(Rtφ)(t)-2Re(Rtφ)(s)+(Rsφ)(s)]=φ(e)[(Rtφ)(t)-2(Rtφ)(s)+(Rsφ)(s)]≤2φ(e)[φ(e)-(Rsφ)(t)].


## 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.LetL1(Q,m) be a normal h.c.s. with basis unity e. The family of bounded operators U=(Up)p∈Q in a separable Hilbert space ℋ is called a representation of an h.c.s. if(1)
Ue=1,(2)
Up*=Up*(p∈Q),(3)
for eachζ∈ℋ, the vector Q∋p↦Upζ∈ℋ is weakly continuous,(4)
for allA,B∈ℬ0(Q)(4.1)∫c(A,B,r)Urdr=∫AUpdp∫BUqdq.
Condition (4.1) implies that the function Q∋p↦∥Up∥ is locally bounded.Example 4.1.
The family of generalized translation operatorsLp(p∈Q) defines a representation of the h.c.s. L1(Q,m) in Helbert space L2(Q,m).LetQ∋p↦Up be a representation of the h.c.s. L1(Q,m). Below, we consider representation that satisfy conditions (1.5)–(2.2) and the following additional condition: (5)
the functionQ∋p↦∥Up∥ is bounded.Such representation are called bounded.LetL1(Q,m)∋x↦Ux be a representation of the Banach algebra L1(Q,m) in a separable Hilbert space ℋ.Two representation of an h.c.s.L1(Q,m) are unitarity equivalent if and only if the corresponding representations of the algebra L1(Q,m) are equivalent h.c.s.We recall that a representation of the Banach algebraL1(Q,m) in ℋ is said to be cyclic if there exists a vector ζ∈ℋ, cyclic vector, such that the linear subspace {Uxζ:x∈L1(Q,m)} is dense in ℋ.Corollary 4.2.
For any bounded representationUr of a normal h.c.s. with basis unity that satisfies the condition of separate continuity the following relation holds:
(4.2)Rs(Urζ,η)H=(UrUsζ,η)H,Ls(Urζ,η)H=(UsUrζ,η)H,(r,s∈Q;ζ,η∈H).
For the proof (see [20]).Theorem 4.3.
LetL1(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. L1(Q,m). This bijection is given by the relation
(4.3)φ(r)=(Urζ0,ζ0)H,(r∈Q),
where Q∈E·C(Q) and Ur is the corresponding representation of the h.c.s. L1(Q,m) in a Hilbert space ℋ with cyclic vector ζ0.Proof.
IfUr is a bounded representation of the h.c.s. L1(Q,m) with cyclic vector ζ0. Then the function φ(r)=(Urζ0,ζ0)ℋ is e.c.f. in the sense of (2.1). Indeed, Let x∈L1(Q,m). Then
(4.4)∫φ(r)(x*x)(r)dr=(∫(x*x)(r)Urdrζ0,ζ0)H=(Ux*xζ0,ζ0)H=‖Uxζ0‖22≥0.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 ifφ(r) is e.c. in the sense of (2.1), then relation (2.2) holds for any λ1,…,λn∈ℝ and r1,…,rn∈Q. Indeed, by virtue of (4.2) and (4.3), we have
(4.5)∑i,j=1nλiλj(Rriφ)(rj)=∑i,jnλiλjRri(Urjζ0,ζ0)H=∑i,j=1nλiλj(UrjUriζ0,ζ0)H=‖∑λiUriζ0‖H2≥0.



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