#### Abstract

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 be a real linear space consisting of some sequences in the field of real numbers and for each , where if , and otherwise. We endow a norm on such that the norm is monotonic; that is, for and a real sequence , the condition that for each implies that and . Let be a fixed sequence of positive numbers. Let be a complex normed linear space with the norm and . The problem considered here is, for a sequence in with , finding such that

Any element satisfying (1.1) is called a best simultaneous approximation to from . The set of all best simultaneous approximations to from is denoted by .

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 , 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 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 be as in Section 1 with the monotonic norm and let be a fixed sequence of positive numbers satisfying

(noting that such a sequence satisfying (2.1) exists, see [7]), which plays a fundamental role in the present paper. Let be a complex normed linear space with the norm . We use and to denote the duals of and , respectively. The inner product between and is denoted by while stands for the inner product of and . Also, we denote by and the closed unit ball of and , respectively. For a set in the dual of a Banach space, let signify the weak closure of the set and be endowed with the weak topology. Let , and let be endowed with the product topology. Then is a compact Hausdorff space.

Let

with the norm for each . Then , where is viewed as an element in for each . For simplicity, we write for . Thus means that . Let . Define the function on by

Furthermore, define

where denotes the family of all open neighborhoods of in . 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 . Then the following assertions hold: *(i)* is upper semicontinuous on ,*(ii)* is continuous on for all ,*(iii)* For each ,*

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

Then is a nonempty compact subset of since is upper-semicontinuous on the compact set .

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 and . is called a local best simultaneous approximation to from if there exists an open neighborhood (i.e., an open ball with center ) of such that .

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 . is called simultaneous solar point of if for each , implies that for each , here and in the sequel, . is called a simultaneous sun if each point of is a simultaneous solar point of .

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 . Then is called (i)simultaneous regular point of if, for each , and closed set satisfying and
there exists such that and
(ii)simultaneous strongly regular point of if, for each , and closed set satisfying and (2.8), there exists such that and (iii) is called a simultaneous regular set (resp., simultaneous strongly regular set ) of if each point of is a simultaneous regular point (resp., simultaneous strongly regular point) of .

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.

*Definition 2.5. *Let and . Then is said to satisfy(i)simultaneous Kolmogorov Condition if (ii)simultaneous Papini Condition if

Proposition 2.6. *Let and . *(i)*If satisfies simultaneous Kolmogorov Condition, then .*(ii)*If , then satisfies simultaneous Papini Condition.*

*Proof. *(i) Let . Then by the assumption, there exists such that . It follows from (2.6) that

This means that .

(ii) Let and . Then for each , one has that
Hence 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 . Suppose that is a simultaneous solar point of . Then for each , the condition that is a local best simultaneous approximation to from implies that .*

*Proof. *Let and is a local best simultaneous approximation to from . Then there is an open neighborhood of such that
Clearly, we may assume that . Let
We assert that . In fact, let . Then . It follows from (3.2) that
On the other hand, let be arbitrary. Suppose on the contrary that . Then
which contradicts (3.1). This completes the proof of the assertion. Since is a simultaneous solar point of , one has that . Noting that , we obtain that .

The first main result of this paper is as follows.

Theorem 3.2. *Let . Then the following statements are equivalent: *(i)* is a simultaneous solar point of ,*(ii)*For each , if and only if satisfies simultaneous Kolmogorov Condition,*(iii)* is a simultaneous regular point of .*

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

(ii) (iii) For each , and closed set satisfying and (2.8), we obtain from (2.8) that

Hence by (ii). Using the equivalence of (i) and (ii) as well as Theorem 3.1, is not local best simultaneous approximation to from . Thus there exists a sequence such that and
Let be arbitrary. Then by (2.5) and (2.6), one has that
This together with (3.6) implies that (2.9) holds, which shows that is a simultaneous regular point of .

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

Let
Then is an open subset of because the function
is continuous on by Proposition 2.1(ii). Let . Then is closed and . Furthermore,
Since is a simultaneous regular point of , one has that such that and (2.9) holds. It follows from (2.9) and Proposition 2.1 that
Therefore,
On the other hand, let . Then is a compact subset of and . Thus there is such that
For each , since
one has that
Thus when is large enough, we obtain from (3.16) and (3.14) that
This together with (2.6) and (3.13) implies that
which contradicts that . The proof is complete.

Corollary 3.3. *The following statements are equivalent: *(i)* is a simultaneous sun,*(ii)*For each and , if and only if satisfies simultaneous Kolmogorov Condition,*(iii)* is a simultaneous regular set.*

The second main result of this paper is as follows.

Theorem 3.4. *Let . Consider the following statements: *(i)* is a simultaneous strongly regular point of ,*(ii)*For each , the fact that satisfies simultaneous Papini Condition implies that satisfies simultaneous Kolmogorov Condition,*(iii)* is a simultaneous solar point of .**Then (i)(ii)(iii).*

*Proof. *(i)(ii) Let (i) holds. Suppose on the contrary that there exists such that satisfies simultaneous Papini Condition but does not satisfy simultaneous Kolmogorov Condition. Then there exist and such that (3.8) holds. Let and be as in the proof of the implication (iii)(ii) in Theorem 3.2, respectively. Then, is closed, , and (3.11) is valid. In view of the definition of simultaneous strongly regular point, there is a sequence such that and (2.10) holds. It follows from (2.10) that

since is closed. Let . Then is a closed subset of . Furthermore, let . It is easy to see that . Let and let be sufficiently large such that . Then
It follows that
This shows that , and hence . Moreover,
thanks to (3.19). This contradicts that satisfies simultaneous Papini Condition.

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

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

Corollary 3.5. *Let be a simultaneous strongly regular set of . Let and . Then if and only if 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).