IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation19487610.1155/2009/194876194876Research ArticleNew Characterization for Nonlinear Weighted Best Simultaneous ApproximationLuoXianfa1WuDelin1HeJinsu2CostinRodica1Department of MathematicsChina Jiliang UniversityHangzhou 310018Chinacjlu.edu.cn2Department of MathematicsZhejiang Normal UniversityJinhua 321004Chinazjnu.edu.cn200910082009200929042009310720092009Copyright © 2009This 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.

This paper is concerned with the problem of a wide class of weighted best simultaneous approximation in normed linear spaces, and it establishes a new characterization result for the class of approximation by virtue of the notion of simultaneous regular point.

1. Introduction

The problem of best simultaneous approximation has a long history and continues to generate much interest. The problem of approximating simultaneously two continuous functions on a finite closed interval was first studied by Dunham . Since then, such problems have been extended extensively, see, for example,  and references therein. In particular, characterization and uniqueness results were obtained in  for a wide class of best simultaneously approximating problem, which includes early results as special cases.

The setting for the problems considered here is as follows. Let R be a real linear space consisting of some sequences in the field of real numbers R and ei=(δij)R for each iN, where δij=1 if j=i, and 0 otherwise. We endow a norm ·A on R such that the norm is monotonic; that is, for (av)R and a real sequence (bv), the condition that |bi||ai| for each i=1,2,, implies that (b1,b2,)R and (bv)A(av)A. Let {λi}R be a fixed sequence of positive numbers. Let X be a complex normed linear space with the norm · and GX. The problem considered here is, for a sequence (xi) in X with (λixi)R, finding g0G such that

(λixi-g0)A(λixi-g)A,gG. Any element g0 satisfying (1.1) is called a best simultaneous approximation to x̂ from G. The set of all best simultaneous approximations to x̂ from G is denoted by PG(x̂).

In order to characterize restricted Chebyshev centers of a set in normed linear spaces, the work in  introduced the concept of simultaneous regular point of a set. In this paper, we propose a same notion and the notion of simultaneous strongly regular point of a set for studying best simultaneous approximation to a sequence from the set in X, and establish new characterization results for this class of approximation problem. It should be remarked that our results are new even in the case when X is real (noting that results obtained in this paper is valid for real normed linear spaces) and when the approximated sequence is finite.

2. Preliminaries

Let R be as in Section 1 with the monotonic norm ·A and let (λi)R be a fixed sequence of positive numbers satisfying

limi(0,,0,λi,λi+1,)A=0 (noting that such a sequence (λi) satisfying (2.1) exists, see ), which plays a fundamental role in the present paper. Let X be a complex normed linear space with the norm ·. We use (R)* and X* to denote the duals of R and X, respectively. The inner product between R and (R)* is denoted by ·,· while f(x) stands for the inner product of xX and fX*. Also, we denote by V and W the closed unit ball of (R)* and X*, respectively. For a set A in the dual of a Banach space, let A¯* signify the weak* closure of the set A and be endowed with the weak* topology. Let Ω=V×W×W×, and let Ω be endowed with the product topology. Then Ω is a compact Hausdorff space.

Let

𝒳={x̂=(xi):(λixi)R} with the norm x̂𝒳=(λixi)A for each x̂𝒳. Then X𝒳, where x is viewed as an element(x,x,) in 𝒳 for each xX. For simplicity, we write f̂ for (fi). Thus (a*,f̂)Ω means that (a*,f1,f2,)Ω. Let x̂=(xi)𝒳. Define the function ϕ(x̂) on Ω by

ϕ(x̂)(ω)=a*,(Reλifi(xi)),ω=(a*,f̂)Ω. Furthermore, define

ϕ(x̂)+(ω)=infONωsupωOϕ(x̂)(ω), where Nω denotes the family of all open neighborhoods of ω in Ω. Then, by (2.1) and [9, Remark  1,  2,  4], we have the following proposition (see also [7, Proposition  2.1]).

Proposition 2.1.

Let x̂=(xi). Then the following assertions hold:

ϕ(x̂)+ is upper semicontinuous on Ω,

ϕ(x) is continuous on   Ω   for all xX,

For each xX,

ϕ(x̂-x)+=ϕ(x̂)+-ϕ(x),maxωΩϕ(x̂-x)+(ω)=maxωΩ[ϕ(x̂)+(ω)-ϕ(x)(ω)]=supωΩϕ(x̂-x)(ω)=x̂-x𝒳.

In what follows, we always assume that G is a nonempty subset of X. Let x̂𝒳 and gG. Write

Ωx̂-g={ωΩ:ϕ(x̂)+(ω)-ϕ(g)(ω)=x̂-g𝒳}. Then Ωx̂-g+ is a nonempty compact subset of Ω since ϕ(x̂)+-ϕ(g) is upper-semicontinuous on the compact set Ω.

The following concept is an extension of the notion of local best approximation given in  to the case of best simultaneous approximation.

Definition 2.2.

Let g0G and x̂𝒳. g0 is called a local best simultaneous approximation to x̂ from G if there exists an open neighborhood U(g0) (i.e., an open ball with center g0) of g0 such that g0PGU(g0)(x̂).

The notion of suns, introduced by Efimov and Stečhkin in , has proved to be very important in nonlinear approximation theory in normed linear spaces. The following definition is an extension of the notion of suns to the case of simultaneous approximations, see .

Definition 2.3.

Let g0G. g0 is called simultaneous solar point of G if for each x̂𝒳, g0PG(x̂) implies that g0PG(x̂α) for each α>0, here and in the sequel, x̂α=g0+α(x̂-g0). G is called a simultaneous sun if each point of G is a simultaneous solar point of G.

The following notion stated in Definition 2.4 (i) is similar to the notion of simultaneous regular point in , which was used to characterize restricted Chebyshev centers of a set in a normed linear space.

Definition 2.4.

Let g0G. Then g0 is called

simultaneous regular point of G if, for each x̂𝒳, gG and closed set A satisfying Ωx̂-g0AΩ and

min(a*,f̂)Aa*,(Reλifi(g-g0))>0, there exists {gn}G such that gn-g00 and ϕ(gn-g0)(a*,f̂)>ϕ(x̂-g0)+(a*,f̂)-x̂-g0𝒳,(a*,f̂)A,nN,

simultaneous strongly regular point of G if, for each x̂𝒳, gG and closed set A satisfying Ωx̂-g0AΩ and (2.8), there exists {gn}G such that gn-g00 and

ϕ(gn-g0)(a*,f̂)>0,(a*,f̂)A,nN,

G is called a simultaneous regular set (resp., simultaneous strongly regular set ) of X if each point of G is a simultaneous regular point (resp., simultaneous strongly regular point) of G.

The following notions are respectively analogues to Kolmogorov Condition (cf. [10, 12]) and Papini Condition (cf. ) in nonlinear approximation theory in normed linear spaces.

Definition 2.5.

Let g0G and x̂𝒳. Then g0 is said to satisfy

simultaneous Kolmogorov Condition if

max{a*,(Reλifi(g0-g)):(a*,f̂)Ωx̂-g0}0,gG,

simultaneous Papini Condition if

max{a*,(Reλifi(g-g0)):(a*,f̂)Ωx̂-g}0,gG.

Proposition 2.6.

Let g0G and x̂𝒳.

If (x̂,g0) satisfies simultaneous Kolmogorov Condition, then g0PG(x̂).

If g0PG(x̂), then (x̂,g0) satisfies simultaneous Papini Condition.

Proof.

(i) Let gG{g0}. Then by the assumption, there exists (a*,f̂)Ωx̂-g0 such that a*,(Reλifi(g0-g))0. It follows from (2.6) that

x̂-g𝒳a*,(Reλifi(xi-g))=a*,(Reλifi(xi-g0))+a*,(Reλifi(g0-g))x̂-g0𝒳. This means that g0PG(x̂).

(ii) Let g0PG(x̂) and gG. Then for each (a*,f̂)Ωx̂-g, one has that a*,(Reλifi(g-g0))=a*,(Reλifi(g-xi))+a*,(Reλifi(xi-g0))-x̂-g𝒳+x̂-g0𝒳0. Hence (x̂,g0) satisfies simultaneous Papini Condition. The proof is complete.

3. Characterizations for Best Simultaneous Approximations

The relationship of best simultaneous approximations and local best simultaneous approximations is as follows.

Theorem 3.1.

Let g0G. Suppose that g0 is a simultaneous solar point of G. Then for each x̂𝒳, the condition that g0 is a local best simultaneous approximation to x̂ from G implies that g0PG(x̂).

Proof.

Let x̂𝒳 and g0 is a local best simultaneous approximation to x̂ from G. Then there is an open neighborhood U(g0,δ) of g0 such that x̂-g0𝒳x̂-g𝒳,gU(g0,δ)G. Clearly, we may assume that x̂(g0). Let α=min{1,δ(λi)A(2x̂-g0𝒳)}. We assert that g0PG(x̂α). In fact, let gGU(g0,δ). Then g-g0δ. It follows from (3.2) that x̂α-g0𝒳g0-g𝒳-x̂α-g0𝒳=(λi)Ag0-g-αx̂-g0𝒳δ(λi)A-δ2(λi)Aαx̂-g0𝒳=x̂α-g0𝒳. On the other hand, let gGU(g0,δ) be arbitrary. Suppose on the contrary that x̂-g𝒳<x̂-g0𝒳. Then x̂-g𝒳x̂-x̂α𝒳+x̂α-g0𝒳<x̂-x̂α𝒳+x̂α-g0𝒳=(1-α)x̂-g0𝒳+αx̂-g0𝒳=x̂-g0𝒳, which contradicts (3.1). This completes the proof of the assertion. Since g0 is a simultaneous solar point of G, one has that g0PG(g0+(1/α)(x̂α-g0)). Noting that x̂=g0+(1/α)(x̂α-g0), we obtain that g0PG(x̂).

The first main result of this paper is as follows.

Theorem 3.2.

Let g0G. Then the following statements are equivalent:

g0 is a simultaneous solar point of G,

For each x̂𝒳, g0PG(x̂) if and only if (x̂,g0) satisfies simultaneous Kolmogorov Condition,

g0 is a simultaneous regular point of G.

Proof.

The equivalence of (i)(ii) is exactly [7, Theorem  3.1].

(ii) (iii) For each x̂𝒳, gG and closed set A satisfying Ωx̂-g0AΩ and (2.8), we obtain from (2.8) that

max(a*,f̂)Ωx̂-g0a*,(Reλifi(g0-g))<0. Hence g0PG(x̂) by (ii). Using the equivalence of (i) and (ii) as well as Theorem 3.1, g0 is not local best simultaneous approximation to x̂ from G. Thus there exists a sequence {gn}G such that gn-g00 and x̂-gn𝒳<x̂-g0𝒳,nN. Let (a*,f̂)A be arbitrary. Then by (2.5) and (2.6), one has that x̂-gn𝒳ϕ(x̂-gn)+(a*,f̂)=ϕ(x̂-g0)+(a*,f̂)-ϕ(gn-g0)(a*,f̂). This together with (3.6) implies that (2.9) holds, which shows that g0 is a simultaneous regular point of G.

(iii)(ii) Let (iii) hold. By Proposition 2.6, it suffices to prove that (x̂,g0) satisfies simultaneous Kolmogorov Condition for each x̂𝒳 with g0PG(x̂). For this end, suppose on the contrary that there exists x̂𝒳 with g0PG(x̂) such that (x̂,g0) does not satisfy simultaneous Kolmogorov Condition. Then there exist gG and ϵ>0 such that

max(a*,f̂)Ωx̂-g0a*,(Reλifi(g0-g))=-ϵ<0. Let U={(a*,f̂)Ω:a*,(Reλifi(g0-g))<-ϵ2}. Then U is an open subset of Ω because the function ω=(a*,f̂)ϕ(g0-g)(ω)=a*,(Reλifi(g-g0)) is continuous on   Ω by Proposition 2.1(ii). Let A=U¯. Then A is closed and Ωx̂-g0AΩ. Furthermore, min(a*,f̂)Aa*,(Reλifi(g-g0))ϵ2. Since g0 is a simultaneous regular point of G, one has that {gn}G such that gn-g00 and (2.9) holds. It follows from (2.9) and Proposition 2.1 that ϕ(x̂-gn)+(a*,f̂)<x̂-g0𝒳,(a*,x̂)A,nN. Therefore, max(a*,x̂)Aϕ(x̂-gn)+(a*,f̂)<x̂-g0𝒳,nN. On the other hand, let K=ΩU. Then K is a compact subset of Ω and KΩx̂-g0=. Thus there is δ>0 such that max(a*,x̂)Kϕ(x̂-g0)+(a*,f̂)<x̂-g0𝒳-δ. For each (a*,f̂)Ω, since a*,(Reλifi(xi-gn))=a*,(Reλifi(xi-g0))+a*,(Reλifi(g0-gn))a*,(Reλifi(xi-g0))+(λi)Ag0-gn, one has that ϕ(x̂-gn)+(a*,f̂)ϕ(x̂-g0)+(a*,f̂)+(λi)Ag0-gn. Thus when n is large enough, we obtain from (3.16) and (3.14) that max(a*,x̂)Kϕ(x̂-gn)+(a*,f̂)max(a*,x̂)Kϕ(x̂-gn)+(a*,f̂)+(λi)Ag0-gn<x̂-g0𝒳. This together with (2.6) and (3.13) implies that x̂-gn𝒳=max(a*,x̂)Ωϕ(x̂-gn)+(a*,f̂)<x̂-g0𝒳, which contradicts that g0PG(x̂). The proof is complete.

Corollary 3.3.

The following statements are equivalent:

G is a simultaneous sun,

For each g0G and x̂𝒳, g0PG(x̂) if and only if (x̂,g0) satisfies simultaneous Kolmogorov Condition,

G is a simultaneous regular set.

The second main result of this paper is as follows.

Theorem 3.4.

Let g0G. Consider the following statements:

g0 is a simultaneous strongly regular point of G,

For each x̂𝒳, the fact that (x̂,g0) satisfies simultaneous Papini Condition implies that (x̂,g0) satisfies simultaneous Kolmogorov Condition,

g0 is a simultaneous solar point of G.

Then (i)(ii)(iii).

Proof.

(i)(ii) Let (i) holds. Suppose on the contrary that there exists x̂𝒳 such that (x̂,g0) satisfies simultaneous Papini Condition but does not satisfy simultaneous Kolmogorov Condition. Then there exist gG and ϵ>0 such that (3.8) holds. Let U and A be as in the proof of the implication (iii)(ii) in Theorem 3.2, respectively. Then, A is closed, Ωx̂-g0AΩ, and (3.11) is valid. In view of the definition of simultaneous strongly regular point, there is a sequence {gn}G such that gn-g00 and (2.10) holds. It follows from (2.10) that

max(a*,f̂)Aa*,(Reλifi(gn-g0))>0,nN, since A is closed. Let K=ΩU. Then K is a closed subset of Ω. Furthermore, let β=max(a*,f̂)Ka*,(Reλifi(xi-g0)). It is easy to see that β<x̂-g0𝒳. Let β0=(1/2)(x̂-g0𝒳-β) and let n be sufficiently large such that gn-g0𝒳<β0. Then x̂-gn𝒳x̂-g0𝒳-g0-gn𝒳>x̂-g0𝒳-β0. It follows that max(a*,f̂)Ka*,(Reλifi(xi-gn))max(a*,f̂)Ka*,(Reλifi(xi-g0))+g0-gn𝒳=β+g0-gn𝒳=[x̂-g0𝒳-β0]-[β0-g0-gn𝒳]<x̂-gn𝒳. This shows that KΩΩx̂-gn, and hence Ωx̂-gnU. Moreover, max(a*,f̂)Ωx̂-gna*,(Reλifi(gn-g0))inf(a*,f̂)Ua*,(Reλifi(gn-g0))=min(a*,f̂)Aa*,(Reλifi(gn-g0))>0 thanks to (3.19). This contradicts that (x̂,g0) satisfies simultaneous Papini Condition.

(ii)(iii) Let x̂𝒳 and g0PG(x̂). Then (x̂,g0) satisfies simultaneous Papini Condition by Proposition 2.6. Hence (x̂,g0) satisfies simultaneous Kolmogorov Condition thanks to (ii). Using Theorem 3.2 and Proposition 2.6, one has that g0 is a simultaneous solar point of G. The proof is complete.

By Theorem 3.4 and Proposition 2.6, we have the following result.

Corollary 3.5.

Let G be a simultaneous strongly regular set of X. Let x̂𝒳 and g0G. Then g0PG(x̂) if and only if (x̂,g0) satisfies simultaneous Papini Condition.

Acknowledgment

The authors would like to thank the support in part by the NNSF of China (Grant no. 90818020) and the NSF of Zhejiang Province of China (Grant no. Y7080235).

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