Best Simultaneous Approximation of Finite Set in Inner Product Space

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 .


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 . 1 n  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 .

Preliminary Notes
In this paper, we consider that X is a real inner product space .For a nonempty subset W of X and x X  , define Recall that a point is a best approximation of 0 In 2.4 lemma of [1] we can see that if K be a convex subset of X.Then each x X  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

 
, infsup ) is called a best simultaneous approximation to S from W (see [2] Preliminary Notes).

Best simultaneous Approximation in Convex Sets
In this section , we consider ; : So by adding 2 y with equation of above , we have It is obvious that .Now let , so there exist such that , therefore w will be in W i , that we conclude If W be a closed set , then W i is a closed set .
2 v so by adding with equation of above , we have Thus we have Therefore Since all p ous steps will be reversible , so for any i W w in a fixed i , we have 1, ,

So
Thus is convex set and since intersectio set.

or
orem states that to find best simultaneof a bounded set S of W , it is enough to 3) It is obviously that f is continuous function and we know

Alg ithm
The following the ous approximation obtain the best approximation to any x W i e P x . Thus but according to the definition of W i we have thus the above equation can be written a lows s fol ow H 12 are V 12 called .Also for points x 1 , x 3 the hyperplane definition the points W in bel are formed and the points of W in below H 13 are V 13 called W 1 that this set is co ation x exists in this set , it is and so we do order to the points x 1 , x n .By taking subscribe of any 1 n V , find nvex (by Theorem 3.3, 2).