In this paper, we find a way to give best simultaneous approximation of n arbitrary points in convex sets. First, we introduce a special hyperplane which is based on those n points. Then by using this hyperplane, we define best approximation of each point and achieve our purpose.
Best Approximation; Hyperplane; Best Simultaneous Approximation1. Introduction
As we known, best approximation theory has many applications. One of the best results is best simultaneous approximation of a bounded set but this target cannot be achieved easily. Frank Deutsch in [1] defined hyperplanes and gave the best approximation of a point in convex sets.
In [3,4] we can see that a hyperplane of an n-dimensional space is a flat subset with dimension.
In this paper we try to find best simultaneous approximation of n arbitrary points in convex sets. We say theorems of best approximation of a point in convex sets.
Then we give the method of finding best simultaneous approximation of n points in convex set.
2. Preliminary Notes
In this paper, we consider that X is a real inner product space. For a nonempty subset W of X and , define
.
Recall that a point is a best approximation of if
If each has at least one best approximation , then W is called proximinal.
We denote by , the set of all best approximations of x in W. Therefore
It is well-known that is a closed and bounded subset of X. If, then is located in the boundary of W.
In 2.4 lemma of [1] we can see that if K be a convex subset of X. Then each has at most one best approximation in K.
In particular, every closed convex subset K of a Hilbert space X has a unique best approximation in K.
Also in 4.1 lemma of [1] if K be a convex set and, . Then if and only if
For a nonempty subset W of X and a nonempty bounded set S in X, define
and
Each element in (If) is called a best simultaneous approximation to S from W (see [2] Preliminary Notes).
For and hyperplane H in X defined by
and we denote H by.
The Kernel of a functional f is the set
and for
we say that is in the below of hyperplane H, if .
3. Best simultaneous Approximation in Convex Sets
In this section,we consider
and
Define
Lemma 3.1. Let consider the hyperplane then
Proof. Give so we have
So by adding with equation of above, we have
Therefore have
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Note 3.2. It is obvious that . Now let , so there exist such that for all
Thus , therefore w will be in Wi, that we conclude
Theorem 3.3. Let then:
1)
2) If W be a convex subset of X, then Wi is a convex set.
3) If W be a closed set, then Wi is a closed set.
Proof. 1) Let therefore
so then we have
so by adding with equation of above, we have
therefore we have
.
Thus we have
.
Therefore.
Since all previous steps will be reversible, so for any in a fixed i, we have that consider
thus we have
so
therefore
and finally
.
2) First we proof , for all is convex set.
Give and , set
thus we have
So . Thus is convex set and since intersection of any convex set is convex, therefore Wi is convex set.
3) It is obviously that f is continuous function and we know
.
So, is closed set, this implies Wi is closed set. ■
4. Algorithm
The following theorem states that to find best simultaneous approximation of a bounded set S of W, it is enough to obtain the best approximation to any
.
Thus would be the best simultaneous approximation of S from W if is minimal.
Theorem 4.1. If W be a convex subset of X and there exist for all , then
Proof. With attention of best simultaneous approximation and (3.2) notation, we have
but according to the definition of Wi we have
thus the above equation can be written as follows
and since exist
so we have
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Corollary 4.2. With the assumptions of the previous theorem there exist i, such that is best simultaneous approximation of S in W.
Proof. With attention previous theorem, there exist such that
and by the definition of we have
after according to the above equation and define the best simultaneous approximation of the relationship will
However, the algorithm with assumes a convex set W and introduce the following.
With attention 3.1 lemma for points x1, x2 the hyperplane are possible to obtain, by 3.4 definition the points W in below H12 are V12 called.
Also for points x1, x3 the hyperplane
are formed and the points of W in below H13 are V13 called and so we do order to the points x1, xn.
By taking subscribe of any, find W1 that this set is convex (by Theorem 3.3, 2).
Therefore, if best approximation x1 exists in this set, it is called . Thus obtain for any
.
Finally, the point which has minimal distance to xi, is the best simultaneous approximation of S in W.
REFERENCESReferencesF.Deutsch, “Best Approximation in Inner Product Spaces,”Springer, Berlin, 2001.D.Fang,X.Luo and Chong Li, “Nonlinear Simultaneous Approximation in Complete Lattice Banach Spaces,”Taiwanese Journal of Mathematics, 2008.W. C. Charles, “Linear Algebra,” 1968, p. 62.V. Prasolov and V. M. Tikhomirov, “Geometry,” American Mathematical Society, 2001, p. 22.
.
, define
.
is a best approximation of 
if 
has at least one best approximation 
, then W is called proximinal.
, the set of all best approximations of x in W. Therefore
is a closed and bounded subset of X. If
, then 
is located in the boundary of W.
has at most one best approximation in K.
, 
. Then 
if and only if
(If
) is called a best simultaneous approximation to S from W (see [
and 
hyperplane H in X defined by
.
we say that 
is in the below of hyperplane H, if 
.
consider the hyperplane 
then
so we have
with equation of above, we have
■
. Now let 
, so there exist 
such that 
for all 
, therefore w will be in Wi, that we conclude
then:
therefore
so 
then we have
with equation of above, we have
.
.
.
in a fixed i, we have 
that consider
.
, for all 
is convex set.
and 
, set
. Thus 
is convex set and since intersection of any convex set is convex, therefore Wi is convex set.
.
is closed set, this implies Wi is closed set. ■
.
would be the best simultaneous approximation of S from W if 
is minimal.
for all 
, then
■
is best simultaneous approximation of S in W.
such that
we have
introduce the following.
are possible to obtain, by 3.4 definition the points W in below H12 are V12 called.
, find W1 that this set is convex (by Theorem 3.3, 2).
. Thus obtain 
for any
.
which has minimal distance to xi, is the best simultaneous approximation of S in W.