Advances in Pure Mathematics
Vol.3 No.5(2013), Article ID:35429,4 pages DOI:10.4236/apm.2013.35069

Best Simultaneous Approximation of Finite Set in Inner Product Space

Sied Hossein Akbarzadeh, Mahdi Iranmanesh

Department of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran


Copyright © 2013 Sied Hossein Akbarzadeh, Mahdi Iranmanesh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received May 12, 2013; revised June 13, 2013; accepted July 15, 2013

Keywords: Best Approximation‎; ‎Hyperplane; ‎Best Simultaneous Approximation


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.

1. 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


‎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




Lemma 3.1. ‎Let consider the hyperplane then

Proof. ‎Give so we have‎

So by adding with equation of above‎, ‎we have

Therefore have


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:


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



‎Since all previous steps will be reversible‎, ‎so for any in a fixed i‎, we have ‎‎ ‎that consider‎

thus we have‎




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


‎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.


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