On the Uniform and Simultaneous Approximations of Functions

Mansour Alyazidi

Department of Mathematics, King Saud University, Riyadh, Saudi Arabia.

DOI: 10.4236/apm.2021.1110052                

Abstract

We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F₁ and F₂ are continuous functions on a closed interval [a,b], S is an n-dimensional Chebyshev subspace of C [a,b] and s₁* & s₂* are the best uniform approximations to F₁ and F₂ from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F₁−s₁* and F₂−s₂*, s₁* or s₂* is also a best A(1) simultaneous approximation to F₁ and F₂ from S with F₁≥F₂ and n=2.

Keywords

Simultaneous Approximation, Uniform Approximation, Relation, Straddle Points, Alternation

1. Introduction

The interest in the simultaneous approximation started long ago [1] [2] [3] [4]. This paper concerned with the relation between the simultaneous approximation and the uniform approximation. The setting is as follows. Let $C\left[a,b\right]$ be the set of all real-valued continuous functions defined on the closed interval $\left[a,b\right]$ with the uniform norm $\Vert .\Vert$.

For $f\in C\left[a,b\right]$,

$\Vert f\Vert =\max \left\{\left|f\left(x\right)\right|,x\in \left[a,b\right]\right\}$.

The norms ${\Vert F\Vert }_{A\left(p\right)}$, $1\le p\le \infty$, on $E=C\left[a,b\right]\times C\left[a,b\right]$ are defined as follows:

For $F=\left(F_1,F_2\right)\in E$

${\Vert F\Vert }_{A\left(\infty \right)}=\max \left\{F_1,F_2\right\}$

${\Vert F\Vert }_{A\left(p\right)}={\left[{\Vert F_1\Vert }^p+{\Vert F_2\Vert }^p\right]}^{\frac{1}{p}},1\le p<\infty .$

Now if S is an n-dimensional subspace of $C\left[a,b\right]$, then $U=\left\{\left(s,s\right):s\in S\right\}$ is an n-dimensional subspace of E and there exist $u^{*}=\left(s^{*},s^{*}\right)$ and $v^{*}=\left(t^{*},t^{*}\right)$ where $s^{*},t^{*}\in S$ such that:

$\begin{array}{c}{\Vert F-u^{*}\Vert }_{A\left(\infty \right)}=\underset{u\in U}{\inf }{\Vert F-u\Vert }_{A\left(\infty \right)}\\ =\underset{s\in S}{\inf }\max \left\{\Vert F_1-s\Vert ,\Vert F_2-s\Vert \right\}\\ =\Vert F_k-s^{*}\Vert ,k=1\text{or}2.\end{array}$

Such $s^{*}$ is called a best $A\left(\infty \right)$ simultaneous approximation to $F=\left(F_1,F_2\right)$ from S. The set of all best $A\left(\infty \right)$ simultaneous approximations to F from S will be denoted by $P_S\left(F,\infty \right)$.

For $1\le p<\infty$,

$\begin{array}{c}{\Vert F-v^{*}\Vert }_{A\left(p\right)}=\underset{u\in U}{\text{inf}}{\Vert F-u\Vert }_{A\left(p\right)}\\ =\underset{s\in S}{\text{inf}}\left\{{\left[{\Vert F_1-s\Vert }^p+{\Vert F_2-s\Vert }^p\right]}^{\frac{1}{p}}\right\}\\ ={\left[{\Vert F_1-t^{*}\Vert }^p+{\Vert F_2-t^{*}\Vert }^p\right]}^{\frac{1}{p}}.\end{array}$

$t^{*}$ is called a best $A\left(p\right)$ simultaneous approximation to $F=\left(F_1,F_2\right)$ from S. The set $P_S\left(F,p\right)$ denotes the set of all best $A\left(p\right)$ simultaneous approximation to F from S. And $P_S\left(F_k\right)$ is the set of all best uniform approximation to $F_k$ from S, $k\in \left\{1,2\right\}$.

We are interested in the relation between the simultaneous approximation and the uniform approximation; in section two, we will show under certain conditions, that if $s_k^{*}\in P_S\left(F_k\right)$ then $s_k^{*}\in P_S\left(F,1\right)$, $k\in \left\{1,2\right\}$.

Definition 1 A point $t\in \left[a,b\right]$ is called a straddle point for two functions f and g in $C\left[a,b\right]$ if there exists $\sigma =\pm 1$ such that

$\Vert f\Vert =\sigma f\left(t\right),\Vert g\Vert =-\sigma g\left(t\right).$

Definition 2 The functions f and $g\in C\left[a,b\right]$ are said to have d alternations on $\left[a,b\right]$ if there exists $d+1$ distinct points $x_1<\cdots <x_{d+1}$ in $\left[a,b\right]$ such that for some $\sigma =\pm 1$,

$f\left(x_i\right)=\sigma \Vert f\Vert ,\text{if}i\text{is}\text{odd}$

$g\left(x_i\right)=-\sigma \Vert g\Vert ,\text{if}i\text{is}\text{even}$

or

$g\left(x_i\right)=\sigma \Vert g\Vert ,\text{if}i\text{is}\text{odd}$

$f\left(x_i\right)=-\sigma \Vert f\Vert ,\text{if}i\text{is}\text{even}$

We follow [5] [6] [7] [8] for the notations and the terminology of this section which will be used throughout this paper. The uniform approximation theory can be found in [9] [10]. Theorems 1 and 2 of this section and the remark thereafter which are needed for our analysis are direct consequences of theorems 1 and 3 of [6].

Theorem 1 Let S be an n-dimensional subspace of $C\left[a,b\right]$ which contains a nonzero constant, $F=\left(F_1,F_2\right)\in E$ then:

(a) $s^{*}\in P_S\left(F,1\right)$ if and only if there exists subsets $X_1=\left\{x_i,i\in I_1\right\}$, $X_2=\left\{x_i,i\in I_2\right\}$ of $\left[a,b\right]$ and positive numbers ${\lambda }_i,i\in I_1,{\mu }_i\in I_2$ with

${\displaystyle \underset{i\in I_1}{\sum }{\lambda }_i}={\displaystyle \underset{i\in I_2}{\sum }{\mu }_i}=1$

such that

${\theta }_i\left(F_1\left(x_i\right)-s^{*}\left(x_i\right)\right)=\Vert F_1-s^{*}\Vert ,i\in I_1,$

${\theta }_i\left(F_2\left(x_i\right)-s^{*}\left(x_i\right)\right)=\Vert F_2-s^{*}\Vert ,i\in I_2,$

${\displaystyle \underset{i\in I_1}{\sum }{\theta }_i{\lambda }_{i}s\left(x_i\right)}+{\displaystyle \underset{i\in I_2}{\sum }{\theta }_i{\mu }_{i}s\left(x_i\right)}=0\forall s\in S,$

${\theta }_i=\pm 1$.

(b) If $s^{*}\in P_s\left(F,1\right)$ with $\Vert F_1-s^{*}\Vert =\Vert F_2-s^{*}\Vert$ then $s^{*}\in P_S\left(F,p\right)$ for all p,

$1<p\le \infty$.

Theorem 2 LetS be an n-dimensional Haar subspace of $C\left[a,b\right]$, if $F_1\ge F_2$ on $\left[a,b\right]$ then $s^{*}\in P_s\left(F,\infty \right)$ if and only if $F_1-s^{*}$ & $F_2-s^{*}$ have a straddle point or n alternations on $\left[a,b\right]$ with $\Vert F_1-s^{*}\Vert =\Vert F_2-s^{*}\Vert$. Furthermore, if $F_1-s^{*}$ & $F_2-s^{*}$ have n alternations on $\left[a,b\right]$ then $s^{*}$ is unique.

Remark If $t\in \left[a,b\right]$ is a straddle point for $F_1-s^{*}$ & $F_2-s^{*}$, $F_1\ge F_2$ on $\left[a,b\right]$ then

$\left(F_1-F_2\right)\left(t\right)=\left(F_1-s^{*}\right)\left(t\right)+\left(F_2-s^{*}\right)\left(t\right)=\Vert F_1-s^{*}\Vert +\Vert F_2-s^{*}\Vert \ge \Vert F_1-F_2\Vert .$

This implies that $\left(F_1-F_2\right)\left(t\right)=\Vert F_1-F_2\Vert$ and

$\Vert F_1-s^{*}\Vert +\Vert F_2-s^{*}\Vert =\Vert F_1-F_2\Vert \le \Vert F_1-s\Vert +\Vert F_2-s\Vert \forall s\in S.$

Hence $s^{*}\in P_S\left(F,1\right)$.

2. The Main Result

Theorem 3 Let $s_k^{*}\in P_S\left(F_k\right)$, where $F_k\in C\left[a,b\right]$, $k\in \left\{1,2\right\}$, $F_1\ge F_2$ on $\left[a,b\right]$, $F=\left(F_1,F_2\right)$ and S is a 2-dimensional Chebyshev subspace of $C\left[a,b\right]$ containing a nonzero constant function. And let $X=\left\{a=x_1<x_2<x_3=b\right\}$ be the alternating set for $F_1-s_1^{*}$, $Y=\left\{a=y_1<y_2<y_3=b\right\}$ be the alternating set for $F_2-s_2^{*}$.

(i) If $\left(F_1\left(x_1\right)-s_1^{*}\left(x_1\right)\right)=\Vert F_1-s_1^{*}\Vert$ and $\left(F_2\left(y_1\right)-s_2^{*}\left(y_1\right)\right)=\Vert F_2-s_2^{*}\Vert$, then $s_1^{*}\in P_S\left(F,1\right)$.

(ii) If $\left(F_1\left(x_1\right)-s_1^{*}\left(x_1\right)\right)=-\Vert F_1-s_1^{*}\Vert$ and $\left(F_2\left(y_1\right)-s_2^{*}\left(y_1\right)\right)=-\Vert F_2-s_2^{*}\Vert$, then $s_2^{*}\in P_S\left(F,1\right)$.

Proof

(i) suppose that $\left(F_1\left(x_1\right)-s_1^{*}\left(x_1\right)\right)=\Vert F_1-s_1^{*}\Vert$ and $\left(F_2\left(y_1\right)-s_2^{*}\left(y_1\right)\right)=\Vert F_2-s_2^{*}\Vert$, since $-F_2\ge -F_1$ then

$\left(s_1^{*}\left(x_2\right)-F_2\left(x_2\right)\right)\ge \left(s_1^{*}\left(x_2\right)-F_1\left(x_2\right)\right)=\Vert F_1-s_1^{*}\Vert$

and if $x\in \left[a,b\right]$ is such that $\left(F_2-s_1^{*}\right)\left(x\right)\ge 0$, then

$\Vert F_1-s_1^{*}\Vert \ge \left(F_1-s_1^{*}\right)\left(x\right)\ge \left(F_2-s_1^{*}\right)\left(x\right)\ge 0$.

Hence there exists a $\gamma \in \left[a,b\right]$ such that

$\Vert F_2-s_1^{*}\Vert =-\left(F_2-s_1^{*}\right)\left(\gamma \right)$.

If $\gamma =a$ or $\gamma =b$ then $\gamma$ is a straddle point for $F_1-s_1^{*}$ & $F_2-s_1^{*}$ which implies that $s_1^{*}\in P_S\left(F,1\right)$.

If $a<\gamma <b$ then taking $x_1=z_1,\gamma =z_2,x_3=z_3$ we have:

$\left(F_1-s_1^{*}\right)\left(z_1\right)=\left(F_1-s_1^{*}\right)\left(z_3\right)=\Vert F_1-s_1^{*}\Vert$,

$-\left(F_2-s_1^{*}\right)\left(z_2\right)=\Vert F_2-s_1^{*}\Vert$,

$a\le z_1<z_2<z_3\le b$.

Now, since S is a Chebyshev subspace of dimension 2, there exists ${\mu }_i>0,i\in \left\{1,2,3\right\}$ such that

${\mu }_{1}s\left(z_1\right)-{\mu }_{2}s\left(z_2\right)+{\mu }_{3}s\left(z_3\right)=0\forall s\in S$

because $1\in S$, ${\mu }_2={\mu }_1+{\mu }_3$ and setting ${\omega }_i=\frac{{\mu }_i}{{\mu }_2},i\in \left\{1,2,3\right\}$ we have ${\omega }_{1}s\left(z_1\right)-{\omega }_{2}s\left(z_2\right)+{\omega }_{3}s\left(z_3\right)=0\forall s\in S$ where ${\omega }_2={\omega }_1+{\omega }_3=1$ and from theorem 1 $s_1^{*}\in P_S\left(F,1\right)$.

ii) If $\left(F_1\left(x_1\right)-s_1^{*}\left(x_1\right)\right)=-\Vert F_1-s_1^{*}\Vert$ and $\left(F_2\left(y_1\right)-s_2^{*}\left(y_1\right)\right)=-\Vert F_2-s_2^{*}\Vert$, since $F_1\ge F_2$ then

$\left(F_1\left(y_2\right)-s_2^{*}\left(y_2\right)\right)\ge \left(F_2\left(y_2\right)-s_2^{*}\left(y_2\right)\right)=\Vert F_2-s_2^{*}\Vert$

and if $x\in \left[a,b\right]$ is such that $\left(s_2^{*}-F_1\right)\left(x\right)\ge 0$, then

$\Vert F_2-s_2^{*}\Vert \ge \left(s_2^{*}-F_2\right)\left(x\right)\ge \left(s_2^{*}-F_1\right)\left(x\right)\ge 0$.

Hence there exists a $\gamma \in \left[a,b\right]$ such that

$\Vert F_1-s_2^{*}\Vert =\left(F_1-s_2^{*}\right)\left(\gamma \right)$.

If $\gamma =a$ or $\gamma =b$ then $\gamma$ is a straddle point for $F_1-s_2^{*}$ & $F_2-s_2^{*}$ which implies that $s_2^{*}\in P_S\left(F,1\right)$.

If $a<\gamma <b$ then taking $y_1=z_1,\gamma =z_2,y_3=z_3$ we have:

$\left(F_2-s_2^{*}\right)\left(z_1\right)=\left(F_2-s_2^{*}\right)\left(z_3\right)=-\Vert F_1-s_1^{*}\Vert$,

$\left(F_1-s_2^{*}\right)\left(z_2\right)=\Vert F_1-s_2^{*}\Vert$,

$a\le z_1<z_2<z_3\le b$.

Now, since S is a Chebyshev subspace of dimension 2, there exists

${\mu }_i>0,i\in \left\{1,2,3\right\}$ such that

$-{\mu }_{1}s\left(z_1\right)+{\mu }_{2}s\left(z_2\right)-{\mu }_{3}s\left(z_3\right)=0\forall s\in S$

because $1\in S$, ${\mu }_2={\mu }_1+{\mu }_3$ and setting ${\omega }_i=\frac{{\mu }_i}{{\mu }_2},i\in \left\{1,2,3\right\}$ we have $-{\omega }_{1}s\left(z_1\right)+{\omega }_{2}s\left(z_2\right)-{\omega }_{3}s\left(z_3\right)=0\forall s\in S$ where ${\omega }_2={\omega }_1+{\omega }_3=1$ and from theorem 1 $s_2^{*}\in P_S\left(F,1\right)$ and the theorem is proved.

The following example shows that conditions (i) & (ii) in theorem 3 are necessary conditions.

Example 1 $S=\text{span}\left\{1,x\right\}$ is a Chebyshev subspace of $C\left[0,1\right]$ and

$s_1^{*}=\frac{1}{8}+x$ is the best uniform approximation to $F_1=\sqrt{x}$, $s_2^{*}=\frac{-1}{8}+x$ is the

best uniform approximation to $F_2=x^2$, $F_1\ge F_2$ on $\left[0,1\right]$, $s_1^{*}\notin P_S\left(F,1\right)$ and $s_2^{*}\notin P_S\left(F,1\right)$.

It is possible, under the assumptions of theorem 3 that both $s_1^{*}$ and $s_2^{*}$ belong to the set of best A(1) simultaneous approximation as illustrated in the following example

Example 2 $S=\text{span}\left\{1,x\right\}$ is a Chebyshev subspace of $C\left[0,1\right]$ and

$s_1^{*}=\frac{-1}{8}+x$ is the best uniform approximation to $F_1=x^2$, $s_2^{*}=\frac{-1}{3\sqrt{3}}+x$ is the

best uniform approximation to $F_2=x^3$, $F_1\ge F_2$ on $\left[0,1\right]$.

$s_1^{*},s_2^{*}\in P_S\left(F,1\right)$. Furthermore $s_2^{*}=\frac{-1}{3\sqrt{3}}+x$ is the unique best $A(\; \infty \; )$

simultaneous approximation to $F=\left(F_1,F_2\right)$ from S.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Dunham, C.B. (1967) Simultaneous Chebyshev Approximation of Two Functions on an Interval. Proceedings of the AMS, 18, 472-477. https://doi.org/10.1090/S0002-9939-1967-0212463-6
[2]	Watson, G.A. (1993) A Characterization of Best Simultaneous Approximation. Journal of Approximation Theory, 75, 175-182. https://doi.org/10.1006/jath.1993.1097
[3]	Huotari, R. and Shi, J. (1995) Simultaneous Approximation from Convex Sets. Computers & Mathematics with Applications, 30, 197-206. https://doi.org/10.1016/0898-1221(95)00097-6
[4]	Pinkus, A. (1997) Uniqueness in Vector-Valued Approximation. Journal of Approximation Theory, 91, 17-92. https://doi.org/10.1006/jath.1993.1030
[5]	Asiry, M. and Watson, G.A. (1999) On Solution of a Class of Best Simultaneous Approximation Problems. International Journal of Computer Mathematics, 75, 413-425. https://doi.org/10.1080/00207169908804818
[6]	Asiry, M. and Watson, G.A. (2000) Simultaneous Approximation from Chebyshev and Weak Chebyshev Spaces. Communications in Applied Analysis, 4, No. 3.
[7]	Alyazidi-Asiry, M. (2016) Adjoining a Constant Function to N-Dimensional Chebyshev Space. Journal of Function Spaces, 2016, Article ID: 4813979. https://doi.org/10.1155/2016/4813979
[8]	Alyazidi-Asiry, M. (2017) Extending a Chebyshev Subspace to a Weak Chebyshe Subspace of Higher Dimension and Related Results. Journal of Applied and Computational Mathematics, 6, Article ID: 1000347. https://doi.org/10.4172/2168-9679.1000347
[9]	Cheney, E.W. (1966) Introduction to Approximation Theory. McGraw-Hill, New York, London.
[10]	Watson, G.A. (1980) Approximation Theory and Numerical Methods. John Wiley & Sons, Chichester.