Exponentially Convex Functions on Hypercomplex Systems

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Volume 2011, Article ID 290403,11pages doi:10.1155/2011/290403

Research Article

Exponentially Convex Functions on

Hypercomplex Systems

Buthinah A. Bin Dehaish

Department of Mathematics, Science College of Girls, King Abdulaziz University, P.O. Box 53909, Jeddah 21593, Saudi Arabia

Correspondence should be addressed to Buthinah A. Bin Dehaish,[email protected]

Received 27 June 2010; Accepted 20 April 2011

Academic Editor: Zayid Abdulhadi

Copyrightq2011 Buthinah A. Bin Dehaish. This is an open access article distributed under

the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A hypercomplex systemh.c.s. L1Q, mis, roughly speaking, a space which is defined by a

structure measurecA, B, r,A, B ∈ BQ, such space has been studied by Berezanskii and

Krein. Our main result is to define the exponentially convex functionse.c.f.onh.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.

Mercer1909defines a continuous and symmetric real-valued functionΦona, b×

a, b⊆R2_{to be positive type if and only if}

_b

a

CxCyΦx, ydx dy≥0, 1.1

whereCx, Cy∈Ca, b.

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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 problemsee2–10.

Many studies were done on e.c.f. on diﬀerent structuressee10–18.

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→ Ris e.c.f. if and only if the kernelx, y → fxyis positive definite on the region1/2a,1/2b×1/2a,1/2b.

Now, I will give a short summary of the h.c.f.

LetQbe a complete separable locally compact metric space of pointsp, q, r, . . . , βQ be theσ-algebra of Borel subsets, andβ0Qbe the subring ofβQ, which consists of sets with

compact closure. We will consider the Borel measures; that is, positive regular measures on

βQ, finite on compact sets. The spaces of continuous functions of finite continuous function, and of bounded functions are denoted byCQ,C0Q, and,CbQ, respectively.

An h.c.s. with the basis Q is defined by its structure measure cA, B, r A, B ∈

βQ; r ∈ Q. A structure measurecA, B, ris a Borel measure inAresp.Bif we fixB, r resp.A, rwhich satisfies the following properties:

H1For allA, B∈β0Q, the functioncA, B, r∈C0Q.

H2For allA, B∈β0Qands, r∈Q, the following associativity relation holds

Q

cA, B, rdrcEr, C, s

Q

cB, C, rdrcA, Er, s, C∈βQ. 1.2

H3The structure measure is said to be commutative if

cA, B, r cB, A, r, A, B∈β0Q

1.3

A measuremis said to be a multiplicative measure if

Q

cA, B, rdmr mAmB; A, B∈β0Q. 1.4

H4We will suppose the existence of a multiplicative measure.

For anyf, g∈L1Q, m, the convolution

f∗gr

Q

fpgqdmr

p, q. 1.5

is well definedsee19.

The space L1Q, m with the convolution 1.5 is a Banach algebra which is

commutative ifH3holds. This Banach algebra is called the h.c.s. with the basis

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A nonzero measurable and bounded almost everywhere functionQr → xr∈

is said to be a character of the h.c.s.L1, if for allA, B∈β0Q

Q

cA, B, rxrdmr xAxB,

C

xrdmr xC, C∈β0Q.

1.6

H5An h.c.s. is said to be normal, if there exists an involution homomorphismQr → r∗∈Q, such thatmA mA∗, and

cA, B, C cC, B∗_{, A,} _{cA, B, C}_cA∗_{, C, B,} _{A, B}_∈_β

0Q

, 1.7

where

cA, B, C

C

cA, B, rdmr. 1.8

H6A normal h.c.s. possesses a basis unity if there exists a pointe ∈Qsuch thate∗ e

and

cA, B, e mA∗_∩_B, _{A, B}_∈_βQ. _1.9

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

L1Q, mfr−→f∗r∈L1Q, m 1.10

is an involution in the Banach algebraL1, the multiplicative measure is unique and characters

of such a system are continuous. A characterxof a normal h.c.s. is said to be Hermitian if

xr∗_xr, _r_∈_Q. _1.11

LetXandXhbe the sets of characters and Hermitian characters, respectively.

A Hermitian character of a Hermitian h.c.s. are real valuedxp xp p∈Q. LetL1Q, mbe an h.c.s. with a basisQandΦa space of complex valued functions on

Q. Assume that an operator valued functionQ p → Rp : Φ → Φis given such that the

functiongp Rpfqbelongs toΦfor anyf ∈ Φand any fixedq ∈ Q. The operators

Rp p ∈ Qare called right generalized translation operators, provided that the following

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T1Associativity axiom: the equality

Rqp

Rqf

r Rr_qRpf

r 1.12

holds for any elementsp, q∈Q.

T2There exists an elemente∈Qsuch thatReis the identity inΦ.

By the bilinear form

Lpf, g

g∗, Rp∗f∗ R_rf, g, f, g∈L₂; r, p∈Q, 1.13

we define the left generalized translation operatorsLp, such thatLpfr Rrfpfor almost

allpandqwith respect to the measurem×m.LpandRp have the same properties, so that

will call them generalized translation operators.

A one-to-one correspondence exists between normal h.c.s.L1Q, mwith basis unity

e and weakly continuous families of bounded involutive generalized translation operators

Lp satisfying the finiteness condition, preserving positivity in the space L2Q, m with

unimodular strongly invariant measurem, and preserving the unit element. Convolution in the hypercomplex systemL1Q, mand the corresponding family of generalized translation

operatorsLpsatisfy the relation

f∗gp

Q

Lpf

qgq∗dq Lpf, g∗

2,

f, g∈L2

. 1.14

Moreover, the h.c.s. L1Q, m is commutative if and only if the generalized translation

operatorsLp p∈Qare commutativesee20.

2. Exponentially Convex Functions

LetL1Q, mbe a commutative normal h.c.s. with basis unity.

Definition 2.1. An essentially bounded functionϕr r∈Qis called e.c.f if

ϕrx∗_∗_xrdr_≥₀_, _∀_x_∈_L

1. 2.1

Note that we use the identical involutionx∗ x, we also present another definition of e.c.f.

A continuous bounded functionϕr r∈Qis called e.c.f. if the inequality

n

i,j 1

λiλj

Rrjϕ

rj

≥0 2.2

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Theorem 2.2. If the generalized translation operatorsRtextended toL∞ : CbG → CbG×G.

Then the defination2.1and2.2are equivalent for the functionsϕr∈CbQ.

Proof.

ϕtx∗xtdt Ltxsϕtxsds dt

Lsxtϕtxsds dt

x, Ls∗ϕ₂xsds

xtLs∗ϕ

txsds

Lsϕ

txtxsds dt

Rtϕ

sxsxtds dt≥0.

2.3

It follows thatRtϕs∈CbQ×Q, then the last inequality clearly implies2.2.

Let us prove the converse assertion.

LetQnbe an increasing sequence of compact sets covering the entireQ, that is,Q1 ⊆

Q2⊆ · · · ⊆QnandQ

_n

i 1Qi.

We consider a functionhr∈C0Qand setλi hriin2.2.

This yields

n

i,j 1

Rriϕ

rj

hrih

rj

≥0. 2.4

By integrating this inequality with respect to eachr1, . . . , rn over the set Qk k ∈ and

collecting similar terms, we conclude that

nmQk

Qk

Rrϕ

rh2_rdr_nn₋₁

Qk

Rrϕ

shrhsdr ds≥0. 2.5

Further, we divide this inequality byn2and pass to the limit asn → ∞. We get

Qk

Rrϕ

shrhsdr ds≥0, 2.6

for eachk ∈ . By passing to the limit ask → ∞and applying Lebesgue theorem, we see

that2.1holds for all functions fromC0Q. Approximating an arbitrary function fromL1by

finite continuous functions, we arrive at2.1.

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

Theorem 2.3. Every functionϕ∈E·CQadmits a unique representation in the form of an integral

ϕr

Xh

xrdμx, r ∈Q, 2.7

whereμis a nonnegative finite regular measure on the spaceXh. Conversely, each function of the form

2.7belongs toE·CQ.

Proof. The proof is similar to that given for Theorem 3.1 of20, 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 fromTheorem 2.3.

Corollary 2.5. Assume thatL1Q, mis a commutative h.c.s. with basis unity. Then a continuous

bounded functionϕris e.c. in the sense of2.1if and only if it is e.c. in the sense of 2.2. Moreover,

it has the following properties:

iϕe≥0;

iiϕr ϕr;

iii|ϕr| ≤ϕe;

iv|Rsϕt|2≤RsϕsRtϕt;

v|ϕs−ϕt|2≤2ϕeϕe−Rsϕt.

Proof. Letϕr ∈ CbQis e.c.f. in the sense of2.1and letr1, . . . , rn ∈ Qandλ1, . . . , λn ∈

. Relation2.7and the fact that the generalized translation operators are continuous in

L∞Q, mimply that

n

i,j 1

λiλj

Rriϕ

rj

n

i,j 1

λiλj

Xh

Rrix

rj

dμx

Xh n

i,j 1

λiλjxrix

rj

dμx

Xh n

k 1

λkxrk

2

dμx≥0.

2.8

It is also follows from relation2.7that

iandiiare trivial.

iii

_ϕr_≤

Xh

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iv

_R_s_ϕ

t2

Xh

xsxtdμx

2

≤

Xh

|xs|2_dμx

Xh

|xt|2_dμx

Xh

xsxs∗_dμx

Xh

xtxt∗_dμx

Rsϕ

sRtϕ

t.

2.10

Finally,

v

_ϕs₋_ϕt2_≤

Xh

xs−xtdμx

2

≤ϕe

Xh

|xs|2_|_xt_|2₋₂_xsxt_dμx

≤2ϕe

Xh

1−Rsxtdμx 2ϕe

ϕe−Rsϕ

t.

2.11

3. Exponentially Convex Functions and Kernals

Inequality 2.2 means that the kernel Kt, s Rtϕs is positive definite function.

Therefore, this kernel possesses the following properties:

Kt, t≥0, Kt, s Ks, t,

|Kt, s|2_≤_{Kt, tKs, s,}

|Kt, r−Ks, r|2_≤_{Kr, rKt, t}₋_{2 Re}_{Kt, s}_{Ks, s.}

3.1

Now, we can use the properties of the kernel to prove the properties of the e.c.f. Indeed,

ϕr Reϕ

r Ke, r Kr, e ϕr. 3.2

Similarly,Rrϕs Rsϕr. This implies that

_R_t_ϕ

s2 _≤

Rtϕ

tRsϕ

s, 3.3

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By settings einiv, we obtain

Rtϕ

e2_≤

Rtϕ

tReϕ

e,

_Iϕ

t2_≤

Rtϕ

tϕe,

_ϕt2_≤_ϕe

Rtϕ

t.

3.4

In view of the relation|Lpfq| ≤ f_∞f ∈CbQ, we have

Rrϕ

r≤ϕ_∞. 3.5

Consequently,

_ϕr2_≤_ϕe, _3.6

which implies iiiand, hence, i. Finally, v follows from the last inequality for Kt, s, wherer e.

_ϕs₋_ϕt2_≤

Rtϕ

e−Rsϕ

e2

≤Reϕ

eRtϕ

t−2Re

Rtϕ

s Rsϕ

s ϕeRtϕ

t−2Rtϕ

s Rsϕ

s

≤2ϕeϕe−Rsϕ

t.

3.7

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.

LetL1Q, mbe a normal h.c.s. with basis unity e. The family of bounded operators

U Up_p∈Qin a separable Hilbert spaceHis called a representation of an h.c.s. if

1Ue 1,

2U∗_p Up∗ p∈Q,

3for eachζ∈ H, the vectorQp→Upζ∈ His weakly continuous,

4for allA, B∈ B0Q

cA, B, rUrdr

A

Updp

B

Uqdq. 4.1

Condition4.1implies that the functionQp→ Upis locally bounded.

Example 4.1. The family of generalized translation operators Lpp ∈ Q defines a

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Let Q p → Up be a representation of the h.c.s. L1Q, m. Below, we consider

representation that satisfy conditions1.5–2.2and the following additional condition:

5the functionQp→ Upis bounded.

Such representation are called bounded.

Let L1Q, m x → Ux be a representation of the Banach algebra L1Q, m in a

separable Hilbert spaceH.

Two representation of an h.c.s. L1Q, m are unitarity equivalent if and only if the

corresponding representations of the algebraL1Q, mare equivalent h.c.s.

We recall that a representation of the Banach algebraL1Q, minHis said to be cyclic

if there exists a vectorζ∈ H, cyclic vector, such that the linear subspace{Uxζ:x∈L1Q, m}

is dense inH.

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:

Rs

Urζ, η

H

UrUsζ, η

H,

Ls

Urζ, η

H

UsUrζ, η

H,

r, s∈Q; ζ, η∈ H. 4.2

For the proof (see [20]).

Theorem 4.3. LetL1Q, mbe 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 onQe.c.

in the sense of2.1and the set of classes of unitarily equivalent bounded cyclic representation on the

h.c.s.L1Q, m. This bijection is given by the relation

ϕr Urζ0, ζ0_H, r∈Q, 4.3

whereQ ∈E·CQandUr is the corresponding representation of the h.c.s.L1Q, min a Hilbert

spaceHwith cyclic vectorζ0.

Proof. IfUr is a bounded representation of the h.c.s.L1Q, mwith cyclic vectorζ0. Then the

functionϕr Urζ0, ζ0His e.c.f. in the sense of2.1. Indeed, Letx∈L1Q, m. Then

ϕrx∗xrdr

x∗xrUrdrζ0, ζ0

H

Ux∗xζ0, ζ0_H

Uxζ022≥0.

4.4

Corollary 4.4. For a normal h.c.s. with basis unity that satisfies the condition of separate continuity,

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Proof. It suﬃces to show that ifϕris e.c. in the sense of2.1, then relation2.2holds for anyλ1, . . . , λn∈andr1, . . . , rn∈Q. Indeed, by virtue of4.2and4.3, we have

n

i,j 1

λiλj

Rriϕ

rj

n

i,j

λiλjRri

Urjζ0, ζ0

H

n

i,j 1

λiλj

UrjUriζ0, ζ0

H

λiUriζ0

2

H≥0.

4.5

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.

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