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,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.

1. Introduction

The interest in the simultaneous approximation started long ago    . 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     for the notations and the terminology of this section which will be used throughout this paper. The uniform approximation theory can be found in  . 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 .

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.

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