Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.8, 2015

HLLY'S Theorem in With order continuous norm in term of a double sequence

Ali Hussein Battor 1, Eman Samir Bhaya 2 and Saja Fadhil Abed* 3 1. Department of Mathematics, College of Education for Girls University of Kufa, Najaf-Iraq. 2. University of Babylon college of Education for Pure Sciences. 3. Department of Mathematics, College of Education for Girls University of Kufa, Najaf-Iraq.

Abstract In this paper, we introduce Helly and Helly -Bray theorems in term double sequence in the context of with order continuous norm, and we review some of the results that are needed to prove our theorems.

We state some definitions, like as the moment double sequence and complete moment. Later we prove the corresponding generalized moment theorem and representation in term of double sequence of positive operators. 2010 Mathematics Subject Classification primary 46B42.

Keywords: vector lattice, Moment sequence, completely moment sequence, positive operator.

1. Introduction

Helly's theorem had been of some importance along time above all in the probability theory in connection with a problem of moments of distributions.

Let 푓, 푔: [훼, 훽] × [훼′, 훽′] → 푅 × 푅 , be a monotone map.

Then the following is true:

1. 푓, 푔 has countable set of discontinuity points .

′ ′ 2. If (푓푛푚, 푔푛푚) is a double sequence of functions from [훼, 훽] × [훼 , 훽 ] to 푅 × 푅 , which is uniformly bounded and monotone. The there exists a sub double sequence (푓푛푗푚푘 , 푔푛푗푚푘) of (푓푛푚, 푔푛푚) converging to a monotone map 푓, 푔.

′ ′ 3. Let (푓푛푚, 푔푛푚) be a double sequence from [훼, 훽] × [훼 , 훽 ] to 푅 × 푅 , which is monotone and converging to a map푓, 푔: [훼, 훽] × [훼′, 훽′] → 푅 × 푅.

Then for any continuous maps ℎ, 푤: [훼, 훽] × [훼, 훽] → 푅 × 푅 , we have

훽 훽′ 훽 훽′ ( ) ( ) ( ) ( ) 푛lim→∞ ∫ ∫ ℎ 푡, 푢 푑푓푛푚 푡 푑푓푛푚(푢) = ∫ ∫ ℎ 푡, 푢 푑푓 푡 푑푓(푢) 푚→∞ 훼 훼′ 훼 훼′

훽 훽′ 훽 훽′ ( ) ( ) ( ) ( ) 푛lim→∞ ∫ ∫ 푤 푡, 푢 푑푔푛푚 푡 푑푔푛푚(푢) = ∫ ∫ 푤 푡, 푢 푑푔 푡 푑푔(푢) 푚→∞ 훼 훼′ 훼 훼′

We shall investigate these three properties for function 푓, 푔 and 푓푛푚, 푔푛푚 with values in Banach lattices. We shall see that they do not remain true for any Banach lattice and that we must confine ourselves to the narrower class of Banach lattices with order continuous norm. Next, we shall give two applications of these investigations a generalized moment theorem and a representation theorem.

1 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.8, 2015

2.Helly and Helly-Bray theorem in term of a double sequence.

We recall that Banach lattice 퐸 × 퐸 is said to have an order continuous norm if lim훼,훽 ∥ 푥훼훽 ∥= 0 , lim훼훽 ∥ 푦훼훽 ∥= 0 , for every nonincreasing double net (푥훼훽, 푦훼훽) in 퐸 × 퐸 such that 푖푛푓푥훼훽 = 0 , 푖푛푓푦훼훽 = 0 .

Proposition 2.1: [1]

The Banach lattice 퐸 has order continuous norm if and only if each order interval in 퐸 is weakly compact. Moreover, a Banach lattice with order continuous norm is necessarily Dedekind complete.

The equivalence of (1) and (2) in the following proposition are the main result in [4]. We will use only (1)⇒(2) (that is a direct application of [5,III.2,Theorem 3]).

Proposition 2.2: Suppose 퐸 × 퐸 is a 𝜎 −Dedekind complete Banach lattice, the following conditions are equivalent:

1. 퐸 × 퐸 has order continuous norm;

2. Every non-decreasing function 푓: [0,1] × [0,1] → 퐸 × 퐸 has at most countably many points of discontinuity.

Proposition 2.3: Let 푓 be a non-decreasing function defined on an interval I × I of 푅 × 푅, with values in a Banach lattice 퐸 × 퐸 with order continuous norm, the folowing conditions are equivalent:

1. 푓(푥, 푦) = 푖푛푓{푓(푧, 푒)⁄푥 < 푧 ∈ 퐼 , 푦 < 푒 ∈ 퐼 } ,

2. 푓 is right-continuous at (푥, 푦) for the norm topology;

3. 푓 is right-continuous at (푥, 푦) for the .

For the left-continuity, we have the similar characterizations, and in particular for continuity.

Proof: The proof is similar in [1]

We now state and prove a Helly's theorem in term of a double sequence in the setting of Banach lattice.

Theorem 2.4: Let 퐸 × 퐸 is a Banach lattice and [훼, 훽] × [훼′, 훽′] be a closed interval in 푅 × 푅 . The following conditions are equivalent:

1. 퐸 × 퐸 has order continuous norm;

′ ′ 2. If (푓푛푚, 푔푛푚)푛,푚∈푁 is a double sequence of nondecreasing functions on [훼, 훽] × [훼 , 훽 ] , with values in some [ ] ′ ′ order interval 푎, 푏 × [푎 , 푏 ] in 퐸 × 퐸 , then there exists a double subsequence (푓푛푗푚푘 , 푔푛푗푚푘) 푛푗푚푘∈푁 of (푓푛푚, 푔푛푚) and a non-decreasing functions 푓, 푔: [훼, 훽] × [훼′, 훽′] → [푎, 푏] × [푎′, 푏′]

( ) ′ such that (푓푛푗푚푘 (푥, 푦), 푔푛푗푚푘(푥, 푦)) 푛푗푚푘∈푁 is convergent to 푓 푥, 푦 , 푔(푥, 푦) for the weak topology 𝜎(퐸 × 퐸, 퐸 × 퐸′) at each continuity point (푥, 푦) ∈]훼, 훽[×]훼′, 훽′[ of 푓, 푔 , but also for (푥 ∨ 푦) = (훼 ∨ 훼′) and for (푥 ∨ 푦) = (훽 ∨ 훽′) .

Moreover, if 퐸 × 퐸 has order continuous norm, then the functions 푓, 푔 in (2) can be assumed to be right- continuous at every points (푥, 푦) ∈]훼, 훽[×]훼′, 훽′[ .

Proof: ퟏ ⟹ ퟐ

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′ ′ ′ ′ ′ Let (훼푘푙, 훼 푘푙)푘,푙∈푁 be a dense double sequence in [훼, 훽] × [훼 , 훽 ] including (훼, 훼 ) and (훽, 훽 ). Since the order ′ ′ interval [푎, 푏] × [푎 , 푏 ] is weakly compact (Proposition 1), the double sequence (푓푛푚, 푔푛푚) has a double subsequence (1) (1) (1) ′ (1) ′ ′ ′ ′ (푓푛푚 , 푔푛푚) such that (푓푛푚 (훼1, 훼1), 푔푛푚(훼1, 훼1)) 푛,푚∈푁 is weakly convergent to some (푎1, 푎1) ∈ [푎, 푏] × [푎 , 푏 ] (by Eberlein-Smulian theorem [2, The.10.13]). By induction , for p=2,3,… , q=2,3,…, the double sequence ((푝−1),(푞−1)) ((푝−1),(푞−1)) (푝,푞) (푝,푞) (푝,푞) ′ (푝,푞) ′ (푓푛푚 , 푔푛푚 ) has a subdouble sequence (푓푛푚 , 푔푛푚 ) such that (푓푛푚 (훼푝, 훼푞), 푔푛푚 (훼푝, 훼푞)) 푛,푚∈푁 is ′ ′ ′ weakly convergent to some (푎푝, 푎푞) ∈ [푎, 푏] × [푎 , 푏 ] . Using the well known diagonal process, we define a sub double (푗,푘) (푗,푘) sequence (푓푛푗푚푘 , 푔푛푗푚푘) 푗,푘∈푁 of (푓푛푚, 푔푛푚) by (푓푛푗푚푘 , 푔푛푗푚푘 ) = (푓푗푘 , 푔푗푘 ) ′ ′ ′ ( It is clear that the double sequence (푓푛푗푚푘(훼푝, 훼푞), 푔푛푗푚푘 (훼푝, 훼푞))푗,푘∈푁 is weakly convergent to (푎푝, 푎푞) 푝 = ) ′ ′ 1,2, … , 푞 = 1,2, … , and, the considered functions being non-decreasing, that (훼푟1 ∨ 훼푟2) ≤ (훼푠1 ∨ 훼푆2) implies ′ ′ (푎푟1 ∨ 푎푟2) ≤ (푎푠1 ∨ 푎푆2).

Recalling that 퐸 × 퐸 is Dedekind complete (Proposition 1), we now define a non-decreasing function 푓, 푔: [훼, 훽] × [훼′, 훽′] → [푎, 푏] × [푎′, 푏′] by

푓(푥, 푦) = 푖푛푓 {(푎푟1, 푏푟2)│푥 ≤ 훼푟1, 푦 ≤ 푏푟2}

푔(푥, 푦) = 푖푛푓 {(푎푟1, 푏푟2)│푥 ≤ 훼푟1, 푦 ≤ 푏푟2}

( ) ( ) It is clear that lim푗,푘→∞ 푓푛푗푚푘 푥, 푦 = 푓(푥, 푦), lim푗,푘→∞ 푔푛푗푚푘 푥, 푦 = 푔(푥, 푦) for 𝜎(퐸 × 퐸, 퐸′ × 퐸′), if (푥, 푦) = (훼, 훼′) or (푥, 푦) = (훽, 훽′) . Let us show that this equality remains true for any (푥, 푦) ∈ [훼, 훽] × [훼′, 훽′] such that 푓, 푔 is continuous at(푥, 푦). To this end , we recall first that the topological dual 퐸′ × 퐸′ of 퐸 × 퐸 is the set of all differences of two positive linear functionals on 퐸 × 퐸 [2,Corollary 12.5].Let 휑be any ′ positive linear functionals on 퐸 × 퐸. For any (훼푝 ∨ 훼푞) ≤ (푥 ∨ 푦) , we have

′ ( ) ′ ( ) 휑(푓푛푗푚푘 (훼푝, 훼푞)) ≤ 휑(푓푛푗푚푘 푥, 푦 ) , 휑(푔푛푗푚푘 (훼푝, 훼푞)) ≤ 휑(푔푛푗푚푘 푥, 푦 )

′ ′ ′ and, since (푓푛푗푚푘(훼푝, 훽푞))푗,푘∈푁 is weakly convergent to 푓(훼푝, 훼푞) = (푎푝, 푎푞) and (푔푛푗푚푘 (훼푝, 훼푞))푗,푘∈푁 is weakly ′ ′ convergent to (훼푝, 훼푞) = (푎푝, 푎푞) , we obtain :

′ 휑(푓(훼푝, 훼푞)) ≤ lim 푖푛푓휑(푓푛 푚 (푥, 푦)) 푗,푘→∞ 푗 푘

′ 휑(푔(훼푝, 훼푞)) ≤ lim 푖푛푓휑(푔푛 푚 (푥, 푦)) 푗,푘→∞ 푗 푘

′ ′ But {(훼푝, 훼푞)|훼푝 ≤ 푥 , 훼푞 ≤ 푦} is a directed upwards set converging to (푥, 푦) and, by continuity of 푓, 푔 at (푥, 푦) , ′ ′ 휑(푓(푥, 푦)) , 휑(푔(푥, 푦)) is the limit of the double net {휑 (푓(훼푝, 훼푞)) |훼푝 ≤ 푥 , 훼푞 ≤ 푦 },

′ ′ {휑 (푔(훼푝, 훼푞)) │훼푝 ≤ 푥 , 훼푞 ≤ 푦 }. It follows that,

휑(푓(푥, 푦)) ≤ lim 푖푛푓 휑 (푓푛 푚 (푥, 푦)) 푗,푘→∞ 푗 푘

휑(푔(푥, 푦)) ≤ lim 푖푛푓 휑 (푔푛 푚 (푥, 푦)) 푗,푘→∞ 푗 푘

′ and , similarly , by considering (훼푝 ∨ 훼푞) ≥ (푥 ∨ 푦 ), we also obtain

lim 푠푢푝 휑 (푓푛 푚 (푥, 푦)) ≤ 휑(푓(푥, 푦)) 푗,푘→∞ 푗 푘

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lim 푠푢푝 휑 (푔푛 푚 (푥, 푦)) ≤ 휑(푔(푥, 푦)). 푗,푘→∞ 푗 푘

( ) ( ) ( ) ( ) We conclude that 휑(푓 푥, 푦 ) = lim푗,푘→∞ 휑(푓푛푗푚푘 푥, 푦 ) , 휑(푔 푥, 푦 ) = lim푗,푘→∞ 휑(푔푛푗푚푘 푥, 푦 ) and, finally, that ( ) 푓 푥, 푦 , 푔(푥, 푦) is the weak limit of the double sequence (푓푛푗푚푘 (푥, 푦), 푔푛푗푚푘(푥, 푦)) 푗,푘∈푁 .

To prove that we may assume 푓, 푔 right-continuous at every point (푥, 푦) ∈ [훼, 훽] × [훼′, 훽′] , it suffices to replace 푓, 푔 by the functions ℎ, 푤: [훼, 훽] × [훼′, 훽′] → [푎, 푏] × [푎′, 푏′] , defined by

ℎ(훼, 훼′) = 푓(훼, 훼′) , ℎ(훽, 훽′) = 푓(훽, 훽′) { ℎ(푥, 푦) = 푖푛푓 {푓(푧, 푒)│푥 ≤ 푧 ∈ [훼, 훽] , for (푥, 푦) ∈ [훼, 훽] × [훼′, 훽′] 푦 ≤ 푒 ∈ [훼′, 훽′]}

푤(훼, 훼′) = 푔(훼, 훼′), 푤(훽, 훽′) = 푔(훽, 훽′) {푤(푥, 푦) = 푖푛 푓 푔(푧, 푒) │푥 ≤ 푧 ∈ [훼, 훽], for (푥, 푦) ∈ [훼, 훽] × [훼′, 훽′] 푦 ≤ 푒 ∈ [훼′, 훽′]}

It is obvious that 푓 ≤ ℎ , 푔 ≤ 푤 , ℎ, 푤 are non-decreasing, right-continuous on ]훼, 훽[×]훼′, 훽′[ and also that 푓(푥, 푦) = ℎ(푥, 푦) , 푔(푥, 푦) = 푤(푥, 푦) for some (푥, 푦) ∈]훼, 훽[×]훼′, 훽′[ if and only if 푓, 푔 are right-continuous at (푥, 푦) . We now show that 푓, 푔 is continuous at (푥, 푦) ∈]훼, 훽[×]훼′, 훽′[ if and only if ℎ, 푤 is continuous at (푥, 푦) , we have successively

ℎ(푥, 푦) = 푓(푥, 푦) = 푠푢푝 {푓(푧, 푒)│푧 < 푥, 푒 < 푦} ≤ 푠푢푝 {ℎ(푧, 푒)│푧 < 푥, 푒 < 푦}

푤(푥, 푦) = 푔(푥, 푦) = 푠푢푝{푔(푧, 푒)│푧 < 푥, 푒 < 푦} ≤ 푠푢푝{푤(푧, 푒)│푧 < 푥, 푒 < 푦} and, consequently

ℎ(푥, 푦) = 푠푢푝{ℎ(푧, 푒)│푧 < 푥, 푒 < 푦} ,

푤(푥, 푦) = 푠푢푝 {푤(푧, 푒)│푧 < 푥, 푒 < 푦}

Conversely, if ℎ, 푤 is continuous at (푥, 푦) , we then have:

푓(푥, 푦) ≤ ℎ(푥, 푦)=푠푢푝푧<푥 ℎ(푧, 푒) = 푠푢푝푧<푥 (푖푛푓푧<푧1푓(푧1, 푒1)) 푒<푦 푒<푦 푒<푒1

≤ 푠푢푝푧<푥 (푖푛푓푧<푧1<푥푓(푧1, 푒1) ≤ 푓(푥, 푦) 푒<푦 푒<푒1<푦

푔(푥, 푦) ≤ 푤(푥, 푦)=푠푢푝푧<푥 푤(푧, 푒) = 푠푢푝푧<푥 (푖푛푓푧<푧1 푔(푧1, 푒1)) 푒<푦 푒<푦 푒<푒1

≤ 푠푢푝푧<푥 (푖푛푓푧<푧1<푥 푔 (푧1, 푒1) ≤ 푔(푥, 푦) 푒<푦 푒<푒1<푦

Hence 푓(푥, 푦) = ℎ(푥, 푦) , 푔(푥, 푦) = 푤(푥, 푦) and 푓, 푔 is right-continuous at (푥, 푦). Moreover, by Proposition 2, there ′ ′ exists an increasing double sequence (푧푛푚, 푒푛푚) in [훼, 훽] × [훼 , 훽 ], converging to (푥, 푦), such that 푓, 푔 is continuous at each (푧푛푚, 푒푛푚) .It follows that:

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푓(푥, 푦) = ℎ(푥, 푦) = 푠푢푝푛,푚 (푖푛푓푧푛푚<푧1 푓(푧1, 푒1)) = 푠푢푝푛,푚 푓(푧푛푚, 푒푛푚) 푒푛푚<푒1

≤ 푠푢푝푧<푥 푓(푧, 푒 푒<푦

푔(푥, 푦) = 푤(푥, 푦) = 푠푢푝푛,푚 (푖푛푓푧푛푚<푧1 푔(푧1, 푒1)) = 푠푢푝푛,푚 푔(푧푛푚, 푒푛푚) 푒푛푚<푒1

≤ 푠푢푝푧<푥 푔(푧, 푒 ) 푒<푦

We conclude that 푓(푥, 푦) = 푠푢푝푧<푥 푓(푥, 푦) , 푔(푥, 푦) = 푠푢푝푧<푥 푔(푥, 푦) and, finally, 푓, 푔 are also left-continuous at 푒<푦 푒<푦 (푥, 푦) .

ퟐ ⇒ ퟏ.

Suppose 2 is true , then for every order interval [푎, 푏] × [푎′, 푏′] in 퐸 × 퐸 , for every double sequence in [푎, 푏] × [푎′, 푏′] must have a double subsequence weakly converging to some point in [푎, 푏] × [푎′, 푏′] . Thus order interval is weakly compact (by Eberlein-Smulian theorem again [1, The. 10.13]). By Proposition 1 that 퐸 has order continuous norm. ◙

The following example shows that functions ℎ, 푤 in the previous proof is not necessarily to be right-continuous at (훼, 훼′) or left-continuous at (훽, 훽′).

Example 2.5: For n=1,2,…, define

푓푛 , 푔푛: [0,1] × [0,1] → [0,1] × [0,1] by

1 1 푓 (푥, 푦) = (0,0) , 푔 (푥, 푦) = (0,0) if 푥, 푦 ∈ [0, ] ∧ [0, ] 푛 푛 푛 푛

1 1 1 1 1 1 1 1 푓 (푥, 푦) = ( , ) , 푔 (푥, 푦) = ( , ) if 푥, 푦 ∈] , 1 − [ ∧ [ , 1 − [ 푛 2 2 푛 2 2 푛 푛 푛 푛

1 1 푓 (푥, 푦) = (1,1) , 푔 (푥, 푦) = (1,1) if 푥, 푦 ∈ [1 − , 1] ∧ [1 − ] 푛 푛 푛 푛

Thus, the functions ℎ, 푤 (which coincide with 푓, 푔 ) is clearly not right-continuous at (0,0) , nor left-continuous at (1,1) .

In order to consider a Helly-Bray theorem in the context of banach lattices, we need the following lemma.

Lemma2.6: Suppose 퐸 × 퐸 be a 𝜎 −Dedekind complete vector lattice, [훼, 훽] × [훼′, 훽′] a closed interval in 푅 × 푅 and 푓 a non-decreasing functions from [훼, 훽] × [훼′, 훽′] into 퐸 × 퐸.

′ ′ Let also 훼 = 푥0 < 푥1 < ⋯ < 푥푝 = 훽 , 훼 = 푦0 < 푦0 < ⋯ < 푦푞 = 훽 assume that 푓 is order continuous at

(푥1, 푦1), … , (푥푝−1, 푦푞−1) , and consider a functions

ℎ: [훼, 훽] × [훼′, 훽′] → 푅 × 푅 such that

ℎ(푥, 푦) = 푐표푛푠푡푎푛푡 (푧푗, 푒푖) 푖푓 푥 ∈ [푥푗−1, 푥푗[ , 푦 ∈ [푦푖−1, 푦푖[ { (1 ≤ 푗 ≤ 푝) , (1 ≤ 푖 ≤ 푞) ′ ℎ(훽, 훽 ) = (푧푝, 푒푞)

5 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.8, 2015

Then ℎ is integrable with respect to 푓 and

′ 훽 훽 푝 푞

∫ ∫ ℎ(푡, 푢)푑푓(푡)푑푓(푢) = ∑ ∑(푧푗 + 푒푖)[푓(푥푗, 푦푖) − 푓(푥푗−1, 푦푖−1)] 훼 훼′ 푗=1 푖=1

Proof: For simplicity, we only prove that

푥1 푦1 ′ ∫ ∫ ℎ(푡, 푢)푑푓(푡)푑푓(푢) = (푧1, 푒1)[푓(푥1, 푦1) − 푓(훼, 훼 )] 훼 훼′

′ ′ ′ [ ] Given any partition 훼 = 푥0 < 푥1 < ⋯ < 푥푟1 = 푥1 of 훼, 푥1 ,

′ ′ ′ ′ [ ′ ] ′ ′ ′ ′ 훼 = 푦0 < 푦1 < ⋯ < 푦푟2 = 푦1 of 훼 , 푦1 and points 푡푖1 ∈ [푥푖1−1, 푥푖1], (1 ≤ 푖1 ≤ 푟1), 푢푖2 ∈ [푦푖2−1, 푦푖2] , (1 ≤ 푖2 ≤ 푟2) , we have :

푟1 푟2 ′ ′ ′ ′ ( ) ( ′) |∑ ∑ ℎ(푡푖1, 푢푖2)[푓(푥푖1, 푦푖2) − 푓(푥푖1−1, 푦푖2−1)] − (푧1 + 푒1)[푓 푥1, 푦1 − 푓 훼, 훼 ]| 푖1=1 푖2=1 ( ) ( ) ′ ′ = │ℎ(푡푟1, 푢푟2) − 푧1 + 푒1 │[푓 푥1, 푦1 − 푓(푥푟1−1, 푦푟2−1)] ( ) ( ) ( ) ′ ′ ≤ │ 푧2 + 푒2 − 푧1 + 푒1 │[푓 푥1, 푦1 − 푓(푥푟1−1, 푦푟2−1)]

′ We now choose an increasing double sequence (훼푛푚, 훽푛푚) in [훼, 푥1[ × [훼 , 푦1[ , converging to (푥1, 푦1) . Since 푓 order continuous at (푥1, 푦1) , we have

′ 푓(푥1, 푦1) = sup{푓(푥, 푦)│훼 ≤ 푥 < 푥1 , 훼 ≤ 푦 < 푦1} = 푠푢푝푛,푚(훼푛푚, 훽푛푚)

′ Then (𝜌푛푚, 𝜌푛푚) = │(푧2 + 푒2) − (푧1 + 푒1)│[푓(푥1, 푦1) − 푓(훼푛푚, 훽푛푚)] is a nonincreasing double sequence in 퐸 such that

′ 푖푛푓 𝜌푛푚 ∨ 푖푛푓 𝜌푛푚 = 0 . 푛,푚 푛,푚

′ On the other hand, letting (훿푛푚 ∨ 훿푛푚) = ((푥1) − 𝜌푛푚) > 0, we see that

′ ′ ′ ′ ′ ( ) ( ) ( ) ′ ′ ′ max ( (푥푖1, 푦푖2) − (푥푖1−1, 푦푖2−1)) ≤ (훿푛푚, 훿푛푚) ⇒ │ 푧2, 푒2 − 푧1, 푒1 │[푓 푥1, 푦1 − 푓(푥푟1−1, 푦푟2−1)] ≤ (𝜌푛푚, 𝜌푛푚) 1≤푖1≤푟1 1≤푖2≤푟2

The proof is complete. ◙

We now state and prove a Helly-Bray theorem in term of a double sequence:

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Theorem 2.7: Consider a closed interval [훼, 훽] × [훼′, 훽′] in 푅 × 푅 , a Banach lattice 퐸 × 퐸 with order continuous ′ ′ norm and an order interval [푎, 푏] × [푎 , 푏 ] in 퐸 × 퐸 . Let (푓푛푚, 푔푛푚) be a double sequence of nondecreasing functioms from [훼, 훽] × [훼′, 훽′] into [푎, 푏] × [푎′, 푏′] and assume there exists a nondecreasing functions 푓, 푔: [훼, 훽] × [훼′, 훽′] → [푎, 푏] × [푎′, 푏′] such that

lim 푓푛푚(푥, 푦) = 푓(푥, 푦) , lim 푔푛푚(푥, 푦) = 푔(푥, 푦) 푛,푚→∞ 푛,푚→∞

For the weak topology 𝜎(퐸 × 퐸, 퐸′ × 퐸′) at each continuity point (푥, 푦) ∈ [훼, 훽] × [훼′, 훽′] of 푓, 푔 , but also for (푥, 푦) = (훼, 훼′) and (푥, 푦) = (훽, 훽′) .Then, for each continuous function ℎ, 푤: [훼, 훽] × [훼′, 훽′] → 푅 × 푅 , we have

훽 훽′ 훽 훽′

lim ∫ ∫ ℎ(푡, 푢)푑푓푛푚(푡)푑푓푛푚(푢) = ∫ ∫ ℎ(푡, 푢)푑푓(푡)푑푓(푢), 푛,푚→∞ 훼 훼′ 훼 훼

훽 훽′ 훽 훽′

lim ∫ ∫ 푤(푡, 푢)푑푔푛푚(푡)푑푔푛푚(푢) = ∫ ∫ 푤(푡, 푢)푑푔(푡)푑푔(푢) 푛,푚→∞ 훼 훼′ 훼 훼′

For 𝜎(퐸 × 퐸, 퐸′ × 퐸′).

Proof: By Proposition 2, we have know that 푓, 푔 has at most countable many points of discontinuity. For 푝 = 푞 = 1,2, , … ,, let 훼 = 푥(푝) < 푥(푝) < ⋯ < 푥(푝) = 훽, 훼′ = 푦(푞) < 푦(푞) < ⋯ < 푥(푞) = 훽′ be points of continuity of 푓, 푔 , excepted 0 1 푘푝 0 1 푘푞 1 1 perhaps 훼 and 훽 , such that │ℎ(푥, 푦) − ℎ(푧, 푒)│ ≤ , │푤(푥, 푦) − 푤(푧, 푒)│ ≤ 푝+푞 푝+푞

(푝) (푝) (푞) (푞) if 푥, 푧 ∈ [푥푗−1, 푥푗 ] , 푦, 푒 ∈ [푦푖−1, 푦푖 ] (1 ≤ 푗 ≤ 푘푝), (1 ≤ 푖 ≤ 푘푞).

′ ′ (푝) (푞) (푝) (푝) (푞) (푞) Define ℎ푝,푞, 푤푝,푞 on [훼, 훽] × [훼 , 훽 ] by ℎ푝,푞(푥, 푦) = ℎ(푥푗−1, 푦푖−1) if 푥 ∈ [푥푗−1, 푥푗 [, 푦 ∈ [푦푖−1, 푦푖 [ , (1 ≤ 푗 ≤

푘푝), (1 ≤ 푖 ≤ 푘푞) and ℎ (훽, 훽′) = ℎ(푥(푝) , 푦(푞) ) , 푤 (훽, 훽) = 푤(푥(푝) , 푦(푞) ) . 푝,푞 푘푝−1 푘푞−1 푝,푞 푘푝−1 푘푞−1

By Lemma 6 , we have

′ 훽 훽 푘푝 푘푞 (푝) (푞) (푝) (푞) (푝) (푞) ∫ ∫ ℎ푝,푞(푡, 푢)푑푓푛푚(푡)푑푓푛푚(푢) = ∑ ∑ ℎ (푥푗−1, 푦푖−1) [푓푛푚 (푥푗 , 푦푖 ) − 푓푛푚 (푥푗−1, 푦푖−1)] 훼 훼′ 푖=1 푗=1

′ 훽 훽 푘푝 푘푞 (푝) (푞) (푝) (푞) (푝) (푞) ∫ ∫ 푤푝,푞(푡, 푢)푑푔푛푚(푡)푑푔푛푚(푢) = ∑ ∑ 푤 (푥푗−1, 푦푖−1) [푔푛푚 (푥푗 , 푦푖 ) − 푔푛푚 (푥푗−1, 푦푖−1)] 훼 훼′ 푖=1 푗=1

And this weakly converges in 푛, 푚 to

′ 푘푝 푘푞 훽 훽 (푝) (푞) (푝) (푞) (푝) (푞) ∑ ∑ ℎ (푥푗−1, 푦푖−1) [푓 (푥푗 , 푦푖 ) − 푓 (푥푗−1, 푦푖−1)] = ∫ ∫ ℎ푝,푞(푡, 푢)푑푓(푡)푑푓(푢) . 푗=1 푖=1 훼 훼′

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′ 푘푝 푘푞 훽 훽 (푝) (푞) (푝) (푞) (푝) (푞) ∑ ∑ 푤 (푥푗−1, 푦푖−1) [푔 (푥푗 , 푦푖 ) − 푔 (푥푗−1, 푦푖−1)] = ∫ ∫ 푤푝,푞(푡, 푢)푑푔(푡)푑푔(푢) . 푗=1 푖=1 훼 훼′

On the other hand, it follows from

훽 훽′ 1 |∫ ∫ (ℎ − ℎ )푑푓 | ≤ ((푏 + 푏′) − (푎 + 푎′)) 푎푛푑 푝,푞 푛푚 푝 + 푞 훼 훼′

훽 훽′ 1 |∫ ∫ (ℎ − ℎ )푑푓| ≤ ((푏 + 푏′) − (푎 + 푎′)) 푝,푞 푝 + 푞 훼 훼′

훽 훽′ 1 |∫ ∫ (푤 − 푤 )푑푔 | ≤ ((푏 + 푏′) − (푎 + 푎′)) 푎푛푑 푝,푞 푛푚 푝 + 푞 훼 훼′

훽 훽′ 1 |∫ ∫ (푤 − 푤 )푑푔| ≤ ((푏 + 푏′) − (푎 + 푎′)) 푝,푞 푝 + 푞 훼 훼′ that

훽 훽′ 훽 훽′

lim ∫ ∫ (ℎ − ℎ푝,푞)푑푓푛푚 = lim ∫ ∫ (ℎ − ℎ푝,푞)푑푓 = 0 푝,푞→∞ 푝,푞→∞ 훼 훼′ 훼 훼′

훽 훽′ 훽 훽′

lim ∫ ∫ (푤 − 푤푝,푞)푑푔푛푚 = lim ∫ ∫ (푤 − 푤푝,푞)푑푔 = 0 푝,푞→∞ 푝,푞→∞ 훼 훼′ 훼 훼′ uniformly in 푛, 푚 for the norm topology. The result follows. ◙

Next, we give the following Corollary:

Corollary 2.8: With the same assumptions as in Theorem 2, we have also

훽 훽′ 훽 훽′

lim ∫ ∫ 푓푛푚(푡, 푢)푑ℎ(푡)푑ℎ(푢) = ∫ ∫ 푓(푡, 푢)푑ℎ(푡)푑ℎ(푢) 푛,푚→∞ 훼 훼′ 훼′ 훼′

훽 훽′ 훽 훽′

lim ∫ ∫ 푔푛푚(푡, 푢)푑푤(푡)푑푤(푢) = ∫ ∫ 푔(푡, 푢)푑푤(푡)푑푤(푢) 푛,푚→∞ 훼 훼′ 훼′ 훼′

for 𝜎(퐸 × 퐸, 퐸′ × 퐸′)

8 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.8, 2015

Proof :By the formula of integration by parts [3] , we have

훽 훽′ ′ ′ ′ ′ ∫ ∫ 푓푛푚(푡, 푢)푑ℎ(푡, 푢) = 푓푛푚(훽, 훽 )ℎ(훽, 훽 ) − 푓푛푚(훼, 훼 )ℎ(훼, 훼 ) 훼 훼′

훽 훽′

− ∫ ∫ ℎ(푡, 푢)푑푓푛푚(푡)푑푓푛푚(푢) … . (1) 훼 훼′

훽 훽′ ′ ′ ′ ∫ ∫ 푔푛푚(푡, 푢)푑푤(푡, 푢) = 푔푛푚(훽, 훽 )푤(훽, 훽 ) − 푔푛푚(훼, 훼)푤(훼, 훼 ) 훼 훼′

훽 훽′

− ∫ ∫ 푤(푡, 푢)푑푔푛푚(푡)푑푔푛푚(푢) … (2) 훼 훼′

In the same way, we can prove that

훽 훽′ ∫ ∫ 푓(푡, 푢)푑ℎ(푡, 푢) = 푓(훽, 훽′)ℎ(훽, 훽′) − 푓(훼, 훼′)ℎ(훼, 훼′) 훼 훼′

훽 훽′ − ∫ ∫ ℎ(푡, 푢)푑푓(푡)푑푓(푢) 훼 훼′

훽 훽′ ∫ ∫ 푔(푡, 푢)푑푤(푡, 푢) = 푔(훽, 훽′)푤(훽, 훽′) − 푔(훼, 훼′)푤(훼, 훼′) 훼 훼′

훽 훽′ − ∫ ∫ 푤(푡, 푢)푑푔(푡)푑푔(푢) 훼 훼′

If we take the 푙푖푚푖푡 in (1) and (2) as 푛, 푚 → ∞ , we have

훽 훽′ ′ ′ ′ lim (∫ ∫ 푓푛푚(푡, 푢)푑ℎ(푡, 푢) = 푓푛푚(훽, 훽 )ℎ(훽, 훽 ) − 푓푛푚(훼, 훼)ℎ(훼, 훼 ) 푛,푚→∞ 훼 훼′

훽 훽′

− ∫ ∫ ℎ(푡, 푢)푑푓푛푚(푡)푑푓푛푚(푢)) 훼 훼′

훽 훽′ ′ ′ ′ ′ lim (∫ ∫ 푔푛푚(푡, 푢)푑푤(푡, 푢) = 푔푛푚(훽, 훽 )푤(훽, 훽 ) − 푔푛푚(훼, 훼 )푤(훼, 훼 ) 푛,푚→∞ 훼 훼′

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훽 훽′

− ∫ ∫ 푤(푡, 푢)푑푔푛푚(푡)푑푔푛푚(푢)) 훼 훼′

Thus

훽 훽′ 훽 훽′

lim ∫ ∫ 푓푛푚(푡, 푢)푑ℎ(푡)푑ℎ(푢) = ∫ ∫ 푓(푡, 푢)푑ℎ(푡)푑ℎ(푢) 푛,푚→∞ 훼 훼′ 훼 훼′

훽 훽′ 훽 훽′

lim ∫ ∫ 푔푛푚(푡, 푢)푑푤(푡)푑푤(푢) = ∫ ∫ 푔(푡, 푢)푑푤(푡)푑푤(푢) ◙ 푛,푚→∞ 훼 훼′ 훼 훼′

3. Applications

As in the classical case, we are now able to prove a moment theorem and to deduce from it a representation theorem for operators on the space of continuous function on [0,1] , by means of a nondecreasing function . Our proofs are easy adaptions of the classical proofs. We include these proofs for sake of completeness.

Let us set some definitions, which will be used for prove the theorems following.

Definition 3.1: Let 퐸 be a Banach lattice and Let 푓: [훼, 훽] × [훼, 훽] → 퐸 , [훼, 훽] ⊂ 푅 .

푓 is called o-bounded variation if there exist 푀 ∈ 퐸 such that for any partition (푥0, 푥1, … , 푥푛), (푦0, 푦1, … , 푦푚) of [훼, 훽] × [훼, 훽] , the following inequality holds :

푛 푚

∑ ∑|푓(푥푖+1, 푦푗+1) − 푓(푥푖, 푦푗)| ≤ 푀 푖=1 푗=1

Where 푀 = (푀1, 푀2).

Definition 3.2: Let 퐸 be a Banach lattice. The double sequence (푎푘푙, 푏푘푙) in 퐸 × 퐸 is called moment double sequence if there exist two functions

푓, 푔 ∶ [0,1] × [0,1] → 퐸 with o-bounded variation , such that for any 푘, 푙 , we have

1 1 푘 푙 푎푘푙 = ∫ ∫(푡 +푢 ) 푑푓(푡)푑푓(푢) 0 0

1 1 푘 푙 푏푘푙 = ∫ ∫(푡 +푢 ) 푑푔(푡)푑푔(푢) 0 0

Defintion 3.3: Let 퐸 a Banach lattice. The double sequence (푎푘푙, 푏푘푙) in 퐸 × 퐸 is completely monotone, if for any , 푚, 푘, 푙 , we have

10 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.8, 2015

푛 푛 푛 푛 ∆푛푎 = ∑ ∑(−1)푖+푗 ( ) ( ) 푎 ≥ 0 푘푙 푖 푗 푘+푖,푙+푗 푖=0 푗=0

푚 푚 푚 푚 ∆푚푏 = ∑ ∑(−1)푖+푗 ( ) ( ) 푏 ≥ 0 푘푙 푖 푗 푘+푖,푙+푗 푖=0 푗=0

We now state and prove a Moment theorem in term of a double sequence:

Theorem 3.4: Let 퐸 be a Banach lattice with order continuous norm and (푎푘푙, 푏푘푙) is a double sequence in 퐸 × 퐸 . (푎푘푙, 푏푘푙) is a moment double sequence of a nondecreasing functions 푓 and 푔 if and only if (푎푘푙, 푏푘푙) is completely monotone .

Proof: From definition (3) and (4) we get that, if the double sequence (푎푘푙, 푏푘푙) is the moment double sequence of nondecreasing functions 푓 and 푔 then it is completely monotone.

For the opposite side we have for 푛 = 0,1,2, … define two functions

푓푛, 푔푛 ∶ [0,1] × [0,1] → 퐸 by

푚−1 푚−1 푛 푛 푓 (푥, 푦) = ∑ ∑ ( ) ( ) ∆푛−푖−푗푎 푛 푖 푗 푖푗 푖=0 푗=0

푚−1 푚 푚−1 푚 for (푥, 푦) ∈] , [ ∧] , [ 푛 푛 푛 푛

푚−1 푚−1 푛 푛 푔 (푥, 푦) = ∑ ∑ ( ) ( ) ∆푛−푖−푗푏 푛 푖 푗 푖푗 푖=0 푗=0

푚 푚 푚 푚 푛 푛 푓 ( , ) = ∑ ∑ ( ) ( ) ∆푛−푖−푗푎 푛 푛 푛 푖 푗 푖푗 푖=0 푗=0

푓푛(0,0) = (0,0) , 푓푛(1,1) = 푎(0,0)

푚 푚 푚 푚 푛 푛 푔 ( , ) = ∑ ∑ ( ) ( ) ∆푛−푖−푗푏 푛 푛 푛 푖 푗 푖푗 푖=0 푗=0

푔푛(0,0) = (0,0) , 푔푛(1,1) = 푏(0,0)

The functions 푓푛 , 푔푛 is nondecreasing and has values in the order interval [0, 푎0] × [0, 푏0] of 퐸 × 퐸 .

If we define the operator Λ on the space of polynomials by

푛 푛 푛 푛 푖 푗 Λ(∑ ∑(푐푖푥 + 푑푗푦 )) = ∑ ∑(푐푖푎푖푗 + 푑푗푎푖푗) 푖=0 푗=0 푖=0 푗=0

푛 푛 푛 푛 푖 푗 Λ(∑ ∑(푐푖푥 + 푑푗푦 )) = ∑ ∑(푐푖푏푖푗 + 푑푗푏푖푗) 푖=0 푗=0 푖=0 푗=0

11 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.8, 2015

It is clear that the Bernstein polynomials

푛 푛 푛 푛 푖 푘 푗 푙 퐵 (푥, 푦) = ∑ ∑ ( ) ( ) ( ) ( ) (푥푖 + 푦푗)(1 − (푥 + 푦))푛−푖−푗 (푘,푙),푛 푖 푗 푛 푛 푖=0 푗=0 verify

1 1 푘 푙 Λ(퐵푛,(푘,푙)) = ∫ ∫(푡 +푢 )푑푓푛(푡)푑푓푛(푢) 0 0

1 1 푘 푙 (퐵푛,(푘,푙)) = ∫ ∫(푡 +푢 )푑푔푛(푡)푑푔푛(푢) 0 0

Using Theorem 1,we can choose a sub double sequence (푓 , 푔 ) of (푓 , 푔 ) and a nondecreasing 푛푗푚푘 푛푘1푚푘2 푗,푘∈푁 푛푚 푛푚 ( ) functions 푓 , 푔 such that (푓푛푗푚푘 (푥, 푦), 푔푛푗푚푘(푥, 푦))푗,푘∈푁 weakly converges to 푓 푥, 푦 , 푔(푥, 푦) at each continuity point (푥, 푦) ∈]0,1[ ∧ ]0,1[ of 푓, 푔 for 푥, 푦 = 0 and 푥, 푦 = 1 . By Theorem 2, for every 푘, 푙 we have:

1 1 1 1 푘 푙 푘 푙 lim ∫ ∫(푡 +푢 )푑푓푛 푚 (푡)푑푓푛 푚 (푢) = ∫ ∫(푡 +푢 )푑푓(푡)푑푓(푢), 푗,푘→∞ 푗 푘 푗 푘 0 0 0 0

1 1 1 1 푘 푙 푘 푙 lim ∫ ∫(푡 +푢 )푑푔푛 푚 (푡)푑푔푛 푚 (푢) = ∫ ∫(푡 +푢 )푑푔(푡)푑푔(푢), 푗,푘→∞ 푗 푘 푗 푘 0 0 0 0 for 𝜎(퐸, 퐸′) .

We now show that lim푛→∞ Λ(퐵(푘,푙),푛) = 푎푘푙 , lim푛→∞(퐵(푘,푙),푛) = 푏푘푙 for the norm topology of 퐸 × 퐸 , and the conclusion will follow . By classical algebraic computation , it is easy to show that 푎(0,0) = Λ(퐵(0,0),푛) ,

푏(0,0) = Λ(퐵(0,0),푛) and that

푛 푛 (푖 + 푗)((푖 + 푗) − 1) … . ((푖 + 푗) − 푘 − 푙 + 1) 푛 푛 푎 = ∑ ∑ ( ) ( ) ∆푛−푖−푗푎 푘푙 푛(푛 − 1) … (푛 − 푘 − 푙 + 1) 푖 푗 푖푗 푖=푘 푗=푙

푛 푛 (푖 + 푗)((푖 + 푗) − 1) … . ((푖 + 푗) − 푘 − 푙 + 1) 푛 푛 푏 = ∑ ∑ ( ) ( ) ∆푛−푖−푗푏 . 푘푙 푛(푛 − 1) … (푛 − 푘 − 푙 + 1) 푖 푗 푖푗 푖=푘 푗=푙

Consequentely , we obtain :

푛 푛 (푖 + 푗)((푖 + 푗) − 1) … . ((푖 + 푗) − 푘 − 푙 + 1) 푖 푘 푗 푙 푛 푛 푎 − Λ(퐵 ) = ∑ ∑ ( − ( ) ( ) ) ( ) ( ) ∆푛−푖−푗푎 푘푙 (푘,푙),푛 푛(푛 − 1) … (푛 − 푘 − 푙 + 1) 푛 푛 푖 푗 푘푙 푖=푘 푗=푙 푘−1 푙−1 푛 푛 푖 푗 − ∑ ∑ ( ) ( ) ( )푘 ( )푙∆푛−푖−푗푎 푖 푗 푛 푛 푖푗 푖=0 푗=0

12 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.8, 2015

푛 푛 (푖 + 푗)((푖 + 푗) − 1) … . ((푖 + 푗) − 푘 − 푙 + 1) 푖 푘 푗 푙 푛 푛 푏 − Λ(퐵 ) = ∑ ∑ ( − ( ) ( ) ) ( ) ( ) ∆푛−푖−푗푏 푘푙 (푘,푙),푛 푛(푛 − 1) … (푛 − 푘 − 푙 + 1) 푛 푛 푖 푗 푘푙 푖=푘 푗=푙 푘−1 푙−1 푛 푛 푖 푗 − ∑ ∑ ( ) ( ) ( )푘 ( )푙∆푛−푖−푗푏 푖 푗 푛 푛 푖푗 푖=0 푗=0

푖 푗 Let (푧, 푒) = ( , ) , and observe that 푛 푛

푘−1 푙−1 (푖 + 푗)((푖 + 푗) − 1) … . ((푖 + 푗) − 푘 − 푙 + 1) 푖 푘 푗 푙 (푛푧 − 푖) + (푛푒 − 푗) − ( ) ( ) = ∏ − (푧푘 + 푒푙) 푛(푛 − 1) … (푛 − 푘 − 푙 + 1) 푛 푛 (푛 − 푖) + (푛 − 푗) 푖=0 푗=0

It follows that , given 휖 > 0, there exists 푛0 such that

(푖 + 푗)((푖 + 푗) − 1) … . ((푖 + 푗) − 푘 − 푙 + 1) 푖 푘 푗 푙 − ( ) ( ) ≤ ϵ 푛(푛 − 1) … (푛 − 푘 − 푙 + 1) 푛 푛 for 푛 ≥ 푛0 and

푛 푛 (푖 + 푗)((푖 + 푗) − 1) … . ((푖 + 푗) − 푘 − 푙 + 1) 푖 푘 푗 푙 푛 푛 |∑ ∑ ( − ( ) ( ) ) ( ) ( ) ∆푛−푖−푗푎 | ≤ 휖 푛(푛 − 1) … (푛 − 푘 − 푙 + 1) 푛 푛 푖 푗 푘푙 푖=푘 푗=푙

푛 푛 (푖 + 푗)((푖 + 푗) − 1) … . ((푖 + 푗) − 푘 − 푙 + 1) 푖 푘 푗 푙 푛 푛 |∑ ∑ ( − ( ) ( ) ) ( ) ( ) ∆푛−푖−푗푏 | ≤ 휖 푛(푛 − 1) … (푛 − 푘 − 푙 + 1) 푛 푛 푖 푗 푘푙 푖=푘 푗=푙 for 푛 ≥ 푛0 .

It is also clear that

푛 푛 푛 푛 푖 푘 푗 푙 푘 푘 푙 푙 |∑ ∑ ( ) ( ) ( ) ( ) ∆푛−푖−푗푎 | ≤ ( ) ( ) 푎 푖 푗 푛 푛 푘푙 푛 푛 (0,0) 푖=푘 푗=푙

푛 푛 푛 푛 푖 푘 푗 푙 푘 푘 푙 푙 |∑ ∑ ( ) ( ) ( ) ( ) ∆푛−푖−푗푏 | ≤ ( ) ( ) 푏 푖 푗 푛 푛 푘푙 푛 푛 (0,0) 푖=푘 푗=푙

Now, it easy to conclude that if 푛 is large enough we have

│푎푘푙 − Λ(퐵(푘,푙),푛)│ ≤ 2휖푎(0,0) ∨ │푏푘푙 − Λ(퐵(푘,푙),푛)│ ≤ 2휖푏(0,0), which proves the theorem. ◙

We use only an ((Helly theorem)) and not a representation theorem of [6] in prove theorem above, Making it simpler than those [7,8], such that theorem is only a special case of the ((moment theorem)) of [4,5] , And improve the ((moment theorem)) in [9] .we conclude this section by showing that , in the special setting of banach lattice with order continuous norm , is a corollary of the above theorem.

13 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.8, 2015

We now state and prove a Representation theorem in term of a double sequence:

Theorem 3.5: Every positive linear operator 퐿 on 퐶([훼, 훽] × [훼, 훽], 푅 × 푅) with values in a Banach lattice 퐸 with order continuous norm, is representable in the from

훽 훽 퐿(ℎ) = ∫ ∫ ℎ(푡, 푢)푑푓(푡)푑푓(푢) 훼 훼

훽 훽 퐿(푤) = ∫ ∫ 푤(푡, 푢)푑푔(푡)푑푔(푢) , 훼 훼 where 푓, 푔 is a nondecreasing functions from [훼, 훽] × [훼, 훽] into 퐸 × 퐸.

Proof: It is clear that it suffices to prove the result for the interval [0,1] .The double sequence (푎푘푙, 푏푘푙)푘,푙∈푁 defined by 푘 푙 푘 푙 the formula 푎푘푙 = 퐿(푡 + 푢 ) , 푏푘푙 = 퐿(푡 + 푢 ) is completely monotone.

In fact, we have

푛 푛 푛 푛 푛 푛 푛 푛 ∆푛푎 = ∑ ∑ ( ) ( ) 푎 = ∑ ∑ ( ) ( ) 퐿(푡푘+푖 + 푢푙+푗) = 퐿(푡푘(1 − 푡)푛 + 푢푙(1 − 푢)푛) ≥ 0 푘푙 푖 푗 푘+푖,푙+푗 푖 푗 푖=0 푗=0 푖=0 푗=0

푚 푚 푚 푚 푚 푚 푚 푚 ∆푚푏 = ∑ ∑ ( ) ( ) 푏 = ∑ ∑ ( ) ( ) 퐿(푡푘+푖 + 푢푙+푗) = 퐿(푡푘(1 − 푡)푚 + 푢푙(1 − 푢)푚) ≥ 0 . 푘푙 푖 푗 푘+푖,푙+푗 푖 푗 푖=0 푗=0 푖=0 푗=0

By the preceeding theorem, there exists a nondecreasing functions 푓, 푔 such that

1 1 1 1 퐿(푡푘 + 푢푙) = ∫ ∫(푡푘 + 푢푙)푑푓(푡)푑푓(푢) , 퐿(푡푘 + 푢푙) = ∫ ∫(푡푘 + 푢푙)푑푔(푡)푑푔(푢) 0 0 0 0 and , by Weierstrass theorem this equality extends to every continuous functions. We recall that a positive linear mapping from a Banach lattice into a is automatically continuous ([2,Theorem 12.3]). ◙

References [1] Debieve,C.,Duchon,M.,Duhoux,M.,Helly's theorem in Banach Lattice with order continuous norms,199 [2] Aliprantis, C.D. and Burkinshaw,O.,Ordered vector spaces and linear operators , Academic Press, New York,1985. [3] Debieve, C., Duchon, M., Integration by parts in vector lattices, Tatra Mountains Math.Puble.6 (1995).13-18. [4] Lavric, B., A Characterisation of Banach lattice with order continuous norm, Radovi Matematicki 8 (1992), 37-41. [5] Schwartz, L.,Analyes mathematique I , Princeton University Press , Hermann, Paris, 1967. [6] Cristescu,R.,Order vector space and linear operators, Abacus Press,Kent,1976. [7] Debieve,C.,Duchon,M.,Riecan,B.,Moment problem in some ,Tatra Mountains Math .Publ.12(1997),1-5. [8] Debieve,C.,Duchon,M.,Duhoux,M.,,A generalized moment problem in vector lattice (to appear Tatra Mountains Math.Publ.). [9] Duchon,M.,Riecan,B.,Moment problem in vector lattice, zb.Prir.-Mat.Fak.Univ.v Novom Sadu,Ser.Mat.26(1996),53- 61.

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