** Applied Mathematics ** Vol. 3 No. 4 (2012) , Article ID: 18877 , 3 pages DOI:10.4236/am.2012.34048

Some Problems on Best Approximation in Orlicz Spaces^{*}

^{1}College of Mathematics Science, Inner Mongolia Normal University, Huhhot, China

^{2}College of Natural Science, Inner Mongolia Agricultural University, Huhhot, China

Email: wgrd@imnu.edu.cn, 494777843@qq.com

Received January 17, 2012; revised February 19, 2012; accepted February 26, 2012

**Keywords:** Chebyshev System; Haar Subspace; Orlicz Space; Best Approximation

ABSTRACT

In this paper we studied some problems on best approximation in Orlicz spaces, for which the approximating sets are Haar subspaces, the result of this paper can be considered as the extension of the classical corresponding result.

1. Introduction

Let be a compact Hausdorff space, be all the continuous functions on. There are at least points on,. Define

as order Chebyshev system if for arbitrary vector,

has at most zero points on Q [1].

Define the linear subspace

which is spanned by order Chebyshev system as a Haar subspace of [1].

In this paper, let and be mutually complementary function. The definition and properties of function can be seen in [2]. The Orlicz space corresponding to the N function consists of all Lebesgue measurable functions on, of which the Orlicz norm

(1.1)

is finite, here

is the modulus of corresponding to. According to [2], the Orlicz norm (1.1) can also be calculated by

(1.2)

and there exists an, satisfying

shch that

here is the derivative of on the right. Equivalent to the Orlicz norm (1.1), in Orlicz space, the Luxemburg norm is defined by

. (1.3)

In the sequel and will denote the Orlicz space with Orlicz norm (1.1) and the Luxemburg norm (1.3) respectively.

It is well known that

.

2. Main Results

Now we choose and is a Haar subspace of, then we obtain Theorem 1. Let be function satisfying condition, of which the derivative on the right is continuous and strictly monotone increasing, , , if is the best approximator in the mean of in for the Orlicz norm or the Luxemburg norm, then there exist at least different zero points of in.

In order to prove this theorem, first we state the following two lemmas.

Lemma 1. [3-5]. Let be N function satisfying condition, of which the derivative on the right is continuous and strictly monotone increasing, F is a linear subspace of, then is the best approximator in the mean of in for the Luxemburg norm, if and only if for arbitrary function,

holds true.

Lemma 2. [4,5]. Under the conditions of lemma 1, is the best approximator in the mean of in for the Orlicz norm, if and only if for arbitrary function,

holds true, here satisfies

.

Proof of Theorem 1. We prove first the case of the Luxemburg norm. Here we take reduction to absurdity. Assume there exist at most different zero points of in. Based on, we choose points in, such that, here,. From lemma 1 we get

.

For, the above can be deduced as following

here every or,

According to the theory of system of linear equationswe have that, hence the transposed system of equations, also has a nonzero solution. Set , then for some. On the other hand,

Since is the derivative of function on the right, according to the properties of function (see [2]) and the hypothesis of, we obtain

.

The above shows that there exist zero points of the continuous function in every interval , that is to say, has at least different zero points in interval. Since is order Chebyshev system, we get, Together with the previous result, we get a contradiction.

In an analogous way, following lemma 2 we can also prove the case of the Orlicz norm.

In the sequel we choose, , , then the Haar subspace of is, consists of all algebraic polynomials of order not larger than. For, in order to solve the problem of best approximation of with in Orlicz space, actually we just need to consider the problem of the minimal norm of monic polynomials of order in Orlicz space, that is, to consider the extreme value problems as following

; (2.1)

. (2.2)

The similar problems in space has not been completely solved except (see [6]). In Orlicz spaces the problems have not been studied yet. Here we obtain Theorem 2. Let be function satisfying condition, its graph do not contain any straight line segment, its derivative on the right be continuous and strictly monotone increasing, then 1) The extreme value problems (2.1) and (2.2) have unique solution respectively, that is, there exist unique group and, shch that

and

satisfy

;

here and depend on

function corresponding to the Orlicz space.

2) The extremal functions and have n different zero points in respectively.

3) The odevity of extremal functions and is same to the odevity of natural number.

Proof. 1) From [2] (pp. 160-168), we know, under the conditions of theorem 2, Orlicz spaces and are strictly convex. Since is a finite dimensional linear subspace, (1) is obvious by the theory of best approximation (see [1], pp. 1-10).

2) From Theorem 1 we can easily obtain it.

3) Since function is an even function, so

Analogously,

holds true. Hence, from (1), the uniqueness of the extremal function, we obtain

;

.

By these, (3) follows.

3. Acknowledgements

This work is supported by the National Natural Science Foundation of China under the contracts No. 11161033 and the Natural Science Foundation of Inner Mongolia Autonomous Region under the contracts No. 2009MS0105.

REFERENCES

- Y. S. Sun, “Approximation Theory of Functions,” Beijing Normal University Press, Beijing, 1989.
- C. X. Wu and T. F. Wang, “Orlicz Space and Its Applications,” Hei Long Jiang Science and Technology Press, Harbin, 1983.
- D. L. Xie, “The Characteristics of Best Approximators in Orlicz Spaces,” Journal of Hangzhou University, Vol. 12, No. 3, 1985, pp. 319-322.
- Y. W. Wang and S. T. Chen, “The Best Approximating Operators in Orlicz Spaces,” Pure Mathematics and Applied Mathematics, Vol. 2, No. 2, 1986, pp. 44-51.
- C. X. Wu, T. F. Wang, S. T. Chen and Y. W. Wang, “The Geometric Theory of Orlicz Spaces,” Harbin Industrial University Press, Harbin, 1986.
- J. Gillis and G. Lewis, “Monic Polynomials with Minimal Norm,” Journal of Approximation Theory, Vol. 34, No. 2, 1982, pp. 187-193. doi:10.1016/0021-9045(82)90091-0

NOTES

^{*}Supported by the NSFC (11161033).