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*
1College of Mathematics Science, Inner Mongolia Normal University, Huhhot, China
2College 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
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- C. X. Wu and T. F. Wang, “Orlicz Space and Its Applications,” Hei Long Jiang Science and Technology Press, Harbin, 1983.
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NOTES
*Supported by the NSFC (11161033).