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 [1]. Since then, such problems have been extended extensively, see, for example, [1–7] and references therein. In particular, characterization and uniqueness results were obtained in [7] 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 i∈N, 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 G⊂X. The problem considered here is, for a sequence (xi) in X with (λi∥xi∥)∈R∞, finding g0∈G such that

∥(λi∥xi-g0∥)∥A≤∥(λi∥xi-g∥)∥A,∀g∈G.
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 [8] 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 [7]), 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 x∈X and f∈X*. 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):(λi∥xi∥)∈R∞}
with the norm ∥x̂∥𝒳=∥(λi∥xi∥)∥A for each x̂∈𝒳. Then X⊂𝒳, where x is viewed as an element(x,x,…) in 𝒳 for each x∈X. 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)=infO∈Nωsupω′∈Oϕ(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)∈ℱ. Then the following assertions hold:

In what follows, we always assume that G is a nonempty subset of X. Let x̂∈𝒳 and g∈G. Write

Ωx̂-g={ω∈Ω:ϕ(x̂)+(ω)-ϕ(g)(ω)=∥x̂-g∥𝒳}.
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 [10] to the case of best simultaneous approximation.

Definition 2.2.

Let g0∈G and x̂∈𝒳. 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 g0∈PG∩U(g0)(x̂).

The notion of suns, introduced by Efimov and Stečhkin in [11], 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 [4].

Definition 2.3.

Let g0∈G. g0 is called simultaneous solar point of G if for each x̂∈𝒳, g0∈PG(x̂) implies that g0∈PG(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 [8], which was used to characterize restricted Chebyshev centers of a set in a normed linear space.

Definition 2.4.

Let g0∈G. Then g0 is called

simultaneous regular point of G if, for each x̂∈𝒳, g∈G and closed set A satisfying Ωx̂-g0⊂A⊂Ω and

min(a*,f̂)∈A〈a*,(Reλifi(g-g0))〉>0,
there exists {gn}⊂G such that ∥gn-g0∥→0 and
ϕ(gn-g0)(a*,f̂)>ϕ(x̂-g0)+(a*,f̂)-∥x̂-g0∥𝒳,∀(a*,f̂)∈A,n∈N,

simultaneous strongly regular point of G if, for each x̂∈𝒳, g∈G and closed set A satisfying Ωx̂-g0⊂A⊂Ω and (2.8), there exists {gn}⊂G such that ∥gn-g0∥→0 and

ϕ(gn-g0)(a*,f̂)>0,∀(a*,f̂)∈A,n∈N,

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. [12]) in nonlinear approximation theory in normed linear spaces.

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

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

Proof.

(i) Let g∈G∖{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-g∥𝒳≥〈a*,(Reλifi(xi-g))〉=〈a*,(Reλifi(xi-g0))〉+〈a*,(Reλifi(g0-g))〉≥∥x̂-g0∥𝒳.
This means that g0∈PG(x̂).

(ii) Let g0∈PG(x̂) and g∈G. Then for each (a*,f̂)∈Ωx̂-g, one has that
〈a*,(Reλifi(g-g0))〉=〈a*,(Reλifi(g-xi))〉+〈a*,(Reλifi(xi-g0))〉≤-∥x̂-g∥𝒳+∥x̂-g0∥𝒳≤0.
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 g0∈G. Suppose that g0 is a simultaneous solar point of G. Then for each x̂∈𝒳, the condition that g0 is a local best simultaneous approximation to x̂ from G implies that g0∈PG(x̂).

Proof.

Let x̂∈𝒳 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-g0∥𝒳≤∥x̂-g∥𝒳,∀g∈U(g0,δ)⋂G.
Clearly, we may assume that x̂≠(g0). Let
α=min{1,δ∥(λi)∥A(2∥x̂-g0∥𝒳)}.
We assert that g0∈PG(x̂α). In fact, let g∈G∖U(g0,δ). Then ∥g-g0∥≥δ. It follows from (3.2) that
∥x̂α-g0∥𝒳≥∥g0-g∥𝒳-∥x̂α-g0∥𝒳=∥(λi)∥A∥g0-g∥-α∥x̂-g0∥𝒳≥δ∥(λi)∥A-δ2∥(λi)∥A≥α∥x̂-g0∥𝒳=∥x̂α-g0∥𝒳.
On the other hand, let g∈G∩U(g0,δ) be arbitrary. Suppose on the contrary that ∥x̂-g∥𝒳<∥x̂-g0∥𝒳. Then
∥x̂-g∥𝒳≤∥x̂-x̂α∥𝒳+∥x̂α-g0∥𝒳<∥x̂-x̂α∥𝒳+∥x̂α-g0∥𝒳=(1-α)∥x̂-g0∥𝒳+α∥x̂-g0∥𝒳=∥x̂-g0∥𝒳,
which contradicts (3.1). This completes the proof of the assertion. Since g0 is a simultaneous solar point of G, one has that g0∈PG(g0+(1/α)(x̂α-g0)). Noting that x̂=g0+(1/α)(x̂α-g0), we obtain that g0∈PG(x̂).

The first main result of this paper is as follows.

Theorem 3.2.

Let g0∈G. Then the following statements are equivalent:

g0 is a simultaneous solar point of G,

For each x̂∈𝒳, g0∈PG(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 x̂∈𝒳, g∈G and closed set A satisfying Ωx̂-g0⊂A⊂Ω and (2.8), we obtain from (2.8) that

max(a*,f̂)∈Ωx̂-g0〈a*,(Reλifi(g0-g))〉<0.
Hence g0∉PG(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-g0∥→0 and
∥x̂-gn∥𝒳<∥x̂-g0∥𝒳,∀n∈N.
Let (a*,f̂)∈A be arbitrary. Then by (2.5) and (2.6), one has that
∥x̂-gn∥𝒳≥ϕ(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 x̂∈𝒳 with g0∈PG(x̂). For this end, suppose on the contrary that there exists x̂∈𝒳 with g0∈PG(x̂) such that (x̂,g0) does not satisfy simultaneous Kolmogorov Condition. Then there exist g∈G and ϵ>0 such that

max(a*,f̂)∈Ωx̂-g0〈a*,(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-g0⊂A⊂Ω. Furthermore,
min(a*,f̂)∈A〈a*,(Reλifi(g-g0))〉≥ϵ2.
Since g0 is a simultaneous regular point of G, one has that {gn}⊂G such that ∥gn-g0∥→0 and (2.9) holds. It follows from (2.9) and Proposition 2.1 that
ϕ(x̂-gn)+(a*,f̂)<∥x̂-g0∥𝒳,∀(a*,x̂)∈A,n∈N.
Therefore,
max(a*,x̂)∈Aϕ(x̂-gn)+(a*,f̂)<∥x̂-g0∥𝒳,∀n∈N.
On the other hand, let K=Ω∖U. Then K is a compact subset of Ω and K∩Ωx̂-g0=∅. Thus there is δ>0 such that
max(a*,x̂)∈Kϕ(x̂-g0)+(a*,f̂)<∥x̂-g0∥𝒳-δ.
For each (a*,f̂)∈Ω, since
〈a*,(Reλifi(xi-gn))〉=〈a*,(Reλifi(xi-g0))〉+〈a*,(Reλifi(g0-gn))〉≤〈a*,(Reλifi(xi-g0))〉+∥(λi)∥A∥g0-gn∥,
one has that
ϕ(x̂-gn)+(a*,f̂)≤ϕ(x̂-g0)+(a*,f̂)+∥(λi)∥A∥g0-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)∥A∥g0-gn∥<∥x̂-g0∥𝒳.
This together with (2.6) and (3.13) implies that
∥x̂-gn∥𝒳=max(a*,x̂)∈Ωϕ(x̂-gn)+(a*,f̂)<∥x̂-g0∥𝒳,
which contradicts that g0∈PG(x̂). The proof is complete.

Corollary 3.3.

The following statements are equivalent:

G is a simultaneous sun,

For each g0∈G and x̂∈𝒳, g0∈PG(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 g0∈G. Consider the following statements:

g0 is a simultaneous strongly regular point of G,

For each x̂∈𝒳, 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 x̂∈𝒳 such that (x̂,g0) satisfies simultaneous Papini Condition but does not satisfy simultaneous Kolmogorov Condition. Then there exist g∈G 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-g0⊂A⊂Ω, and (3.11) is valid. In view of the definition of simultaneous strongly regular point, there is a sequence {gn}⊂G such that ∥gn-g0∥→0 and (2.10) holds. It follows from (2.10) that

max(a*,f̂)∈A〈a*,(Reλifi(gn-g0))〉>0,∀n∈N,
since A is closed. Let K=Ω∖U. Then K is a closed subset of Ω. Furthermore, let β=max(a*,f̂)∈K〈a*,(Reλifi(xi-g0))〉. It is easy to see that β<∥x̂-g0∥𝒳. Let β0=(1/2)(∥x̂-g0∥𝒳-β) and let n be sufficiently large such that ∥gn-g0∥𝒳<β0. Then
∥x̂-gn∥𝒳≥∥x̂-g0∥𝒳-∥g0-gn∥𝒳>∥x̂-g0∥𝒳-β0.
It follows that
max(a*,f̂)∈K〈a*,(Reλifi(xi-gn))〉≤max(a*,f̂)∈K〈a*,(Reλifi(xi-g0))〉+∥g0-gn∥𝒳=β+∥g0-gn∥𝒳=[∥x̂-g0∥𝒳-β0]-[β0-∥g0-gn∥𝒳]<∥x̂-gn∥𝒳.
This shows that K⊂Ω∖Ωx̂-gn, and hence Ωx̂-gn⊂U. Moreover,
max(a*,f̂)∈Ωx̂-gn〈a*,(Reλifi(gn-g0))〉≥inf(a*,f̂)∈U〈a*,(Reλifi(gn-g0))〉=min(a*,f̂)∈A〈a*,(Reλifi(gn-g0))〉>0
thanks to (3.19). This contradicts that (x̂,g0) satisfies simultaneous Papini Condition.

(ii)⇒(iii) Let x̂∈𝒳 and g0∈PG(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 x̂∈𝒳 and g0∈G. Then g0∈PG(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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