On the Uniform and Simultaneous Approximations of Functions

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,band 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 F1s1* and F2s2*, s1or s2* is also a best A(1) simultaneous approximation to F1 and F2 from S with F1F2 and n=2.

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Alyazidi, M. (2021) On the Uniform and Simultaneous Approximations of Functions. Advances in Pure Mathematics, 11, 785-790. doi: 10.4236/apm.2021.1110052.

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 $‖\text{ }.\text{ }‖$.

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

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

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

For $F=\left({F}_{1},{F}_{2}\right)\in E$

${‖F‖}_{A\left(\infty \right)}=\mathrm{max}\left\{{F}_{1},{F}_{2}\right\}$

${‖F‖}_{A\left(p\right)}={\left[{‖{F}_{1}‖}^{p}+{‖{F}_{2}‖}^{p}\right]}^{\frac{1}{p}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\le p<\infty .$

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:

$\begin{array}{c}{‖F-{u}^{*}‖}_{A\left(\infty \right)}=\underset{u\in U}{\mathrm{inf}}{‖F-u‖}_{A\left(\infty \right)}\\ =\underset{s\in S}{\mathrm{inf}}\mathrm{max}\left\{‖{F}_{1}-s‖,‖{F}_{2}-s‖\right\}\\ =‖{F}_{k}-{s}^{*}‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1\text{\hspace{0.17em}}\text{ }\text{or}\text{\hspace{0.17em}}\text{ }2.\end{array}$

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 $1\le p<\infty$,

$\begin{array}{c}{‖F-{v}^{*}‖}_{A\left(p\right)}=\underset{u\in U}{\text{inf}}{‖F-u‖}_{A\left(p\right)}\\ =\underset{s\in S}{\text{inf}}\left\{{\left[{‖{F}_{1}-s‖}^{p}+{‖{F}_{2}-s‖}^{p}\right]}^{\frac{1}{p}}\right\}\\ ={\left[{‖{F}_{1}-{t}^{*}‖}^{p}+{‖{F}_{2}-{t}^{*}‖}^{p}\right]}^{\frac{1}{p}}.\end{array}$

${t}^{*}$ 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 $\sigma =±1$ such that

$‖f‖=\sigma f\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}‖g‖=-\sigma g\left(t\right).$

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 $\sigma =±1$,

$f\left({x}_{i}\right)=\sigma ‖f‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}i\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{odd}$

$g\left({x}_{i}\right)=-\sigma ‖g‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}i\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{even}$

or

$g\left({x}_{i}\right)=\sigma ‖g‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}i\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{odd}$

$f\left({x}_{i}\right)=-\sigma ‖f‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}i\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{even}$

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

$\underset{i\in {I}_{1}}{\sum }{\lambda }_{i}=\underset{i\in {I}_{2}}{\sum }{\mu }_{i}=1$

such that

${\theta }_{i}\left({F}_{1}\left({x}_{i}\right)-{s}^{*}\left({x}_{i}\right)\right)=‖{F}_{1}-{s}^{*}‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\in {I}_{1},$

${\theta }_{i}\left({F}_{2}\left({x}_{i}\right)-{s}^{*}\left({x}_{i}\right)\right)=‖{F}_{2}-{s}^{*}‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\in {I}_{2},$

$\underset{i\in {I}_{1}}{\sum }{\theta }_{i}{\lambda }_{i}s\left({x}_{i}\right)+\underset{i\in {I}_{2}}{\sum }{\theta }_{i}{\mu }_{i}s\left({x}_{i}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall s\in S,$

${\theta }_{i}=±1$.

(b) If ${s}^{*}\in {P}_{s}\left(F,1\right)$ with $‖{F}_{1}-{s}^{*}‖=‖{F}_{2}-{s}^{*}‖$ then ${s}^{*}\in {P}_{S}\left(F,p\right)$ for all p,

$1.

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 $‖{F}_{1}-{s}^{*}‖=‖{F}_{2}-{s}^{*}‖$. 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

$\left({F}_{1}-{F}_{2}\right)\left(t\right)=\left({F}_{1}-{s}^{*}\right)\left(t\right)+\left({F}_{2}-{s}^{*}\right)\left(t\right)=‖{F}_{1}-{s}^{*}‖+‖{F}_{2}-{s}^{*}‖\ge ‖{F}_{1}-{F}_{2}‖.$

This implies that $\left({F}_{1}-{F}_{2}\right)\left(t\right)=‖{F}_{1}-{F}_{2}‖$ and

$‖{F}_{1}-{s}^{*}‖+‖{F}_{2}-{s}^{*}‖=‖{F}_{1}-{F}_{2}‖\le ‖{F}_{1}-s‖+‖{F}_{2}-s‖\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall s\in S.$

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 $\left({F}_{1}\left({x}_{1}\right)-{s}_{1}^{*}\left({x}_{1}\right)\right)=‖{F}_{1}-{s}_{1}^{*}‖$ and $\left({F}_{2}\left({y}_{1}\right)-{s}_{2}^{*}\left({y}_{1}\right)\right)=‖{F}_{2}-{s}_{2}^{*}‖$, then ${s}_{1}^{*}\in {P}_{S}\left(F,1\right)$.

(ii) If $\left({F}_{1}\left({x}_{1}\right)-{s}_{1}^{*}\left({x}_{1}\right)\right)=-‖{F}_{1}-{s}_{1}^{*}‖$ and $\left({F}_{2}\left({y}_{1}\right)-{s}_{2}^{*}\left({y}_{1}\right)\right)=-‖{F}_{2}-{s}_{2}^{*}‖$, then ${s}_{2}^{*}\in {P}_{S}\left(F,1\right)$.

Proof

(i) suppose that $\left({F}_{1}\left({x}_{1}\right)-{s}_{1}^{*}\left({x}_{1}\right)\right)=‖{F}_{1}-{s}_{1}^{*}‖$ and $\left({F}_{2}\left({y}_{1}\right)-{s}_{2}^{*}\left({y}_{1}\right)\right)=‖{F}_{2}-{s}_{2}^{*}‖$, since $-{F}_{2}\ge -{F}_{1}$ then

$\left({s}_{1}^{*}\left({x}_{2}\right)-{F}_{2}\left({x}_{2}\right)\right)\ge \left({s}_{1}^{*}\left({x}_{2}\right)-{F}_{1}\left({x}_{2}\right)\right)=‖{F}_{1}-{s}_{1}^{*}‖$

and if $x\in \left[a,b\right]$ is such that $\left({F}_{2}-{s}_{1}^{*}\right)\left(x\right)\ge 0$, then

$‖{F}_{1}-{s}_{1}^{*}‖\ge \left({F}_{1}-{s}_{1}^{*}\right)\left(x\right)\ge \left({F}_{2}-{s}_{1}^{*}\right)\left(x\right)\ge 0$.

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

$‖{F}_{2}-{s}_{1}^{*}‖=-\left({F}_{2}-{s}_{1}^{*}\right)\left(\gamma \right)$.

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 then taking ${x}_{1}={z}_{1},\gamma ={z}_{2},{x}_{3}={z}_{3}$ we have:

$\left({F}_{1}-{s}_{1}^{*}\right)\left({z}_{1}\right)=\left({F}_{1}-{s}_{1}^{*}\right)\left({z}_{3}\right)=‖{F}_{1}-{s}_{1}^{*}‖$,

$-\left({F}_{2}-{s}_{1}^{*}\right)\left({z}_{2}\right)=‖{F}_{2}-{s}_{1}^{*}‖$,

$a\le {z}_{1}<{z}_{2}<{z}_{3}\le b$.

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

${\mu }_{1}s\left({z}_{1}\right)-{\mu }_{2}s\left({z}_{2}\right)+{\mu }_{3}s\left({z}_{3}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall s\in S$

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 ${\omega }_{1}s\left({z}_{1}\right)-{\omega }_{2}s\left({z}_{2}\right)+{\omega }_{3}s\left({z}_{3}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall s\in S$ where ${\omega }_{2}={\omega }_{1}+{\omega }_{3}=1$ and from theorem 1 ${s}_{1}^{*}\in {P}_{S}\left(F,1\right)$.

ii) If $\left({F}_{1}\left({x}_{1}\right)-{s}_{1}^{*}\left({x}_{1}\right)\right)=-‖{F}_{1}-{s}_{1}^{*}‖$ and $\left({F}_{2}\left({y}_{1}\right)-{s}_{2}^{*}\left({y}_{1}\right)\right)=-‖{F}_{2}-{s}_{2}^{*}‖$, since ${F}_{1}\ge {F}_{2}$ then

$\left({F}_{1}\left({y}_{2}\right)-{s}_{2}^{*}\left({y}_{2}\right)\right)\ge \left({F}_{2}\left({y}_{2}\right)-{s}_{2}^{*}\left({y}_{2}\right)\right)=‖{F}_{2}-{s}_{2}^{*}‖$

and if $x\in \left[a,b\right]$ is such that $\left({s}_{2}^{*}-{F}_{1}\right)\left(x\right)\ge 0$, then

$‖{F}_{2}-{s}_{2}^{*}‖\ge \left({s}_{2}^{*}-{F}_{2}\right)\left(x\right)\ge \left({s}_{2}^{*}-{F}_{1}\right)\left(x\right)\ge 0$.

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

$‖{F}_{1}-{s}_{2}^{*}‖=\left({F}_{1}-{s}_{2}^{*}\right)\left(\gamma \right)$.

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 then taking ${y}_{1}={z}_{1},\gamma ={z}_{2},{y}_{3}={z}_{3}$ we have:

$\left({F}_{2}-{s}_{2}^{*}\right)\left({z}_{1}\right)=\left({F}_{2}-{s}_{2}^{*}\right)\left({z}_{3}\right)=-‖{F}_{1}-{s}_{1}^{*}‖$,

$\left({F}_{1}-{s}_{2}^{*}\right)\left({z}_{2}\right)=‖{F}_{1}-{s}_{2}^{*}‖$,

$a\le {z}_{1}<{z}_{2}<{z}_{3}\le b$.

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

${\mu }_{i}>0,\text{\hspace{0.17em}}i\in \left\{1,2,3\right\}$ such that

$-{\mu }_{1}s\left({z}_{1}\right)+{\mu }_{2}s\left({z}_{2}\right)-{\mu }_{3}s\left({z}_{3}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall s\in S$

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 $-{\omega }_{1}s\left({z}_{1}\right)+{\omega }_{2}s\left({z}_{2}\right)-{\omega }_{3}s\left({z}_{3}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall s\in S$ 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 $A\left( \infty \right)$

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.

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