﻿Best Simultaneous Approximation of Finite Set in Inner Product Space

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

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

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

Keywords: Best Approximation‎; ‎Hyperplane; ‎Best Simultaneous Approximation

ABSTRACT

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

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

(1.1)

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:

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

■

‎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.

REFERENCES

1. F‎. ‎Deutsch‎, “‎Best Approximation in Inner Product Spaces,” ‎ Springer, Berlin, 2001.
2. D‎. ‎Fang‎, ‎X‎. ‎Luo and Chong Li‎, “‎Nonlinear Simultaneous Approximation in Complete‎ Lattice Banach Spaces‎,”‎ Taiwanese Journal of Mathematics‎, 2008.
3. W‎. ‎C‎. ‎Charles‎, “Linear Algebra,” 1968, p. 62.
4. V‎. ‎Prasolov and V‎. ‎M‎. ‎Tikhomirov‎‎, “Geometry,” ‎American Mathematical Society‎, 2001, p. 22.