On the existence and uniqueness of extensions of semi-Hölder real-valued functions

Costică Mustăţa$^\ast $

April 13, 2010.

$^\ast $“T. Popoviciu” Institute of Numerical Analysis, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro, cmustata2001@yahoo.com.

Let $(X,d)$ be a quasi-metric space, $y_0\in X$ a fixed element and $Y$ a subset of $X$ such that $y_0\in Y$. Denote by $(\Lambda _{\alpha ,0}(Y,d),\| \cdot |^{\alpha }_{Y,d})$ the asymmetric normed cone of real-valued $d$-semi-H$\ddot{o}$lder functions defined on $Y$ of exponent $\alpha \in (0,1]$, vanishing in $y_0$, and by $(\Lambda _{\alpha ,0}(Y,\bar{d}),\| \cdot |^{\alpha }_{Y,\bar{d}})$ the similar cone if $d$ is replaced by conjugate $\bar{d}$ of $d$.

One considers the following claims:

For every $f$ in the linear space $\Lambda _{\alpha ,0}(Y)=\Lambda _{\alpha ,0}(Y,d)\cap \Lambda _{\alpha ,0}(Y,\bar{d})$ there exist $F\in \Lambda _{\alpha ,0}(X,d)$ such that $F|_Y=f$ and $\| F|^{\alpha }_{Y,d}=\| f|^{\alpha }_{Y,d}$;
For every $f\in \Lambda _{\alpha ,0}(Y)$ there exists $\bar{F}\in \Lambda _{\alpha ,0}(X,\bar{d})$ such that $\bar{F}|_Y=f$ and $\| \bar{F}|^{\alpha }_{Y,\bar{d}}=\| f|^{\alpha }_{Y,\bar{d}}$;
The extension $F$ in $(a)$ is unique;
The extension $\bar{F}$ in $(b)$ is unique;
The annihilator $Y^{\bot }_{\bar{d}}$ of $Y$ in $\Lambda _{\alpha ,0}(X,\bar{d})$ is proximinal for the elements of $\Lambda _{\alpha ,0}(X)$ with respect to the distance generated by $\| \cdot |^{\alpha }_{Y,d}$;
The annihilator $Y^{\bot }_{d}$ of $Y$ in $\Lambda _{\alpha ,0}(X,d)$ is proximinal for the elements of $\Lambda _{\alpha ,0}(X)$ with respect to the distance generated by $\| \cdot |^{\alpha }_{Y,\bar{d}}$;
$Y^{\bot }_{\bar{d}}$ in the claim $(e)$ is Chebyshevian;
$Y^{\bot }_{d}$ in the claim $(f)$ is Chebyshevian.

Then the following equivalences hold:
$(a)\Leftrightarrow (e)$; $(b)\Leftrightarrow (f)$; $(c)\Leftrightarrow (g)$; $(d)\Leftrightarrow (h)$.

MSC. 46A22, 41A50, 41A52.

Keywords. Extensions, semi-Lipschitz functions, semi-H$\ddot{o}$lder functions, best approximation, quasi-metric spaces

1 Introduction

Let $X$ be a nonempty set and $d:X\times X\rightarrow \lbrack 0,\infty )$ a function with the properties:

$d(x,y)=d(y,x)=0\ $iff $x=y,$
$d(x,y)\leq d(x,z)+d(z,y),$

for all $x,y,z\in X.$

Then the function $d$ is called a quasi-metric on $X$ and the pair $(X,d)$ is called quasi-metric space ( [ 13 ] ).

Because, in general, $d(x,y)\neq d(y,x),$ for $x,y\in X$ one defines the conjugate quasi-metric $\overline{d}$ of $d,$ by the equality $\bar{d}(x,y)=d(y,x),$ for all $x,y\in X.$

Let $Y$ be a nonvoid subset of $(X,d)$ and $\alpha \in (0,1]$ a fixed number.

Definition 1

A function $f:Y\rightarrow \mathbb {R}$ is called $d$-semi-Hölder (of exponent $\alpha $) if there exists a constant $K_{Y}(f)\geq 0$ such that

\begin{equation} f(x)-f(y)\leq K_{Y}(f)d^{\alpha }(x,y), \label{f.1.1}\end{equation}
1

for all $x,y\in Y.$
$f:Y\rightarrow \mathbb {R}$ is called $\overline{d}$-semi-Hölder (of exponent $\alpha )$ if there exists a constant $\overline{K}_{Y}(f)\geq 0$ such that

\begin{equation} f(x)-f(y)\leq \overline{K}_{Y}(f)\cdot d^{\alpha }(y,x), \label{f.1.2.}\end{equation}
2

for all $x,y\in Y.$

The smallest constant $K_{Y}(f)$ in (1) is denoted by $\left\Vert f\right\vert _{Y,d}^{\alpha }$ and one shows that

\begin{equation} \left\Vert f\right\vert _{Y,d}^{\alpha }:=\sup \left\{ \tfrac {(f(x)-f(y))\vee 0}{d^{\alpha }(x,y)}:d(x,y)>0;x,y\in Y\right\} .\label{f.1.3.}\end{equation}

Analogously one defines $\left\Vert f\right\vert _{Y,\overline{d}}^{\alpha }.$

Observe that the function $f$ is $d$-semi-Hölder on $Y$ iff $-f$ is $\overline{d}$-semi-Hölder on $Y.$ Moreover

\begin{equation} \left\Vert f\right\vert _{Y,d}^{\alpha }=\left\Vert -f\right\vert _{Y,\overline{d}}^{\alpha }. \label{f.1.4.}\end{equation}

Definition 2

( [ 14 ] ). Let $(X,d)$ be a quasi-metric space and $Y\subseteq X$ a nonempty set. The function $f:Y\rightarrow \mathbb {R}$ is called $\leq _{d}$-increasing on $Y$ if $f(x)\leq f(y)$ whenever $d(x,y)=0,$ $x,y\in Y.$

The set of all $\leq _{d}$-increasing functions on $Y$ is denoted by $\mathbb {R}_{\leq _{d}}^{Y}$ and it is a cone in the linear space $\mathbb {R}^{Y}$ of all real-valued functions on $Y.$

The set

\begin{equation} \Lambda _{\alpha }(Y,d):=\{ f\in \mathbb {R}_{\leq _{d}}^{Y};f\ \text{is}\ d\text{-semi-H\"{o}lder and} \ \left\Vert f\right\vert _{Y,d}^{\alpha }<\infty \} \label{f.1.5.}\end{equation}

is also a cone, called the cone of $d$-semi-Hölder functions on $Y.$

If $y_{0}\in Y$ is arbitrary, but fixed, one considers the cone

\begin{equation} \Lambda _{\alpha ,0}(Y,d):=\{ f\in \Lambda _{\alpha }(Y,d):f(y_{0})=0\} .\label{f.1.6}\end{equation}

Then the functional $\left\Vert \ \right\vert _{Y,d}^{\alpha }:\Lambda _{\alpha ,0}(Y,d)\rightarrow \lbrack 0,\infty )$ is subadditive, positively homogeneous and the equality $\left\Vert f\right\vert _{Y,d}^{\alpha }=\left\Vert -f\right\vert _{Y,d}^{\alpha }=0\ $implies$\ f\equiv 0.\ $This means that $\left\Vert \cdot \right\vert _{Y,d}^{\alpha }$ is an asymmetric norm (see [ 13 ] , [ 14 ] ), on the cone $\Lambda _{\alpha ,0}(Y,d)$.

The pair $\left( \Lambda _{\alpha ,0}(Y,d),\left\Vert \ \right\vert _{Y,d}^{\alpha }\right) $ is called the asymmetric normed cone of $d$-semi-Hölder real-valued function on $Y$ (compare with [ 14 ] ).

Analogously, one defines the asymmetric normed cone $(\Lambda _{\alpha ,0}(Y,\overline{d}),\| \cdot | ^{\alpha }_{Y,\overline{d}}). $ of all $\overline{d}$-semi-Hölder real-valued functions on $Y$, vanishing at the fixed point $y_{0}\in Y.$

By the above definitions it follows that

\[ f\in (\Lambda _{\alpha ,0}(Y,d),\left\Vert \ \right\vert _{Y,d}^{\alpha })\ \text{iff\ }-f\in (\Lambda _{\alpha ,0}(Y,\overline{d}),\left\Vert \ \right\vert _{Y,\overline{d}}^{\alpha }) \]

and, moreover, $\left\Vert f\right\vert _{Y,d}^{\alpha }=\left\Vert -f\right\vert _{Y,\overline{d}}^{\alpha }.$

Defining $\Lambda _{\alpha ,0}(Y)$ by

\begin{equation} \Lambda _{\alpha ,0}(Y)=\Lambda _{\alpha ,0}(Y,d)\cap \Lambda _{\alpha ,0}(Y,\overline{d}),\label{f.1.7}\end{equation}

It follows that $\Lambda _{\alpha ,0}(Y)$ is a linear subspace. The following, theorem holds.

Theorem 3

For every $f\in \Lambda _{\alpha ,0}(Y)$ there exist at least one function $F\in \Lambda _{\alpha ,0}(Y,d)$ and at least one function $\overline{F}\in \Lambda _{\alpha ,0}(Y,\overline{d})$ such that

$\left. F\right\vert _{Y}=$ $\left. \overline{F}\right\vert _{Y}=f.$
$\left\Vert F\right\vert _{Y,d}^{\alpha }=\left\Vert f\right\vert _{Y,d}^{\alpha }$ and $\left\Vert \overline{F}\right\vert _{Y,\overline{d}}^{\alpha }=\left\Vert f\right\vert _{Y,\overline{d}}^{\alpha }.$

Proof â–¼

By Theorem 2 and Remark 3 in [ 11 ] it follows that the functions defined by the formulae:

\begin{align} F(f)(x) & =\inf _{y\in Y}\{ f(y)+\left\Vert f\right\vert _{Y,d}^{\alpha }d^{\alpha }(x,y)\} ,\ x\in X,\label{f.1.8}\\ G(f)(x) & =\sup _{y\in Y}\{ f(y)-\left\Vert f\right\vert _{Y,d}^{\alpha }d^{\alpha }(y,x\} ,\ x\in X,\nonumber \end{align}

are elements of $\Lambda _{\alpha ,0}(X,d)$ and, respectively, the functions given by

\begin{align} \overline{F}(f)(x) & =\inf _{y\in Y}\{ f(y)+\left\Vert f\right\vert _{Y,\overline{d}}^{\alpha }d^{\alpha }(y,x)\} ,\ x\in X,\label{f.1.9}\\ \overline{G}(f)(x) & =\sup _{y\in Y}\{ f(y)-\left\Vert f\right\vert _{Y,\overline{d}}^{\alpha }d^{\alpha }(x,y)\} ,\ x\in X\nonumber \end{align}

are elements of $\Lambda _{\alpha ,0}(X,\overline{d})$ such that

\begin{equation} \left. F(f)\right\vert _{Y}=\left. G(f)\right\vert _{Y}=f\ \text{and\ }\left\Vert F(f)\right\vert _{Y,d}^{\alpha }=\left\Vert G(f)\right\vert _{Y,d}^{\alpha }=\left\Vert f\right\vert _{Y,d}^{\alpha },\label{f.1.10}\end{equation}

respectively

\begin{equation} \left. \overline{F}(f)\right\vert _{Y}=\left. \overline{G}(f)\right\vert _{Y}=f\ \text{and\ }\left\Vert \overline{F}(f)\right\vert _{Y,\overline{d}}^{\alpha }=\left\Vert \overline{G}(f)\right\vert _{Y,\overline{d}}^{\alpha }=\left\Vert f\right\vert _{Y,\overline{d}}^{\alpha }.\label{f.1.11}\end{equation}

For $f\in \Lambda _{\alpha ,0}(Y)$ let us consider the following (nonempty) sets of extensions:

\begin{equation} \mathcal{E}_{d}(f):=\{ H\in \Lambda _{\alpha ,0}(X,d):\left. H\right\vert _{Y}=f\ \text{and }\left\Vert H\right\vert _{Y,d}^{\alpha }=\left\Vert f\right\vert _{Y,d}^{\alpha }\} \label{f.1.12}\end{equation}

and

\begin{equation} \mathcal{E}_{\overline{d}}(f):=\{ \overline{H}\in \wedge _{\alpha ,0}(X,\overline{d}):\left. \overline{H}\right\vert _{Y}=f\ \text{and\ }\left\Vert \overline{H}\right\vert _{Y,\overline{d}}^{\alpha }=\left\Vert f\right\vert _{Y,\overline{d}}^{\alpha }\} .\label{f.1.13}\end{equation}

The sets $\mathcal{E}_{d}(f)$ and $\mathcal{E}_{\overline{d}}(f)$ are convex and

\begin{equation} F(f)(x)\geq H(x)\geq G(f)(x),\ x\in X\label{f.1.14}\end{equation}

for all $H\in \mathcal{E}_{d}(f);$

\begin{equation} \overline{F}(f)(x)\geq \overline{H}(x)\geq \overline{G}(f)(x),\ x\in H,\label{f.1.15}\end{equation}

for all $\overline{H}\in \mathcal{E}_{\overline{d}}(f)$.

Also, for $F\in \Lambda _{\alpha ,0}(X),$ $\left. F\right\vert _{Y}\in \Lambda _{\alpha ,0}(Y)$ and

\begin{align*} F-H & \in \Lambda _{\alpha ,0}(X,\overline{d}),\ \text{for all }H\in \mathcal{E}_{d}(\left. F\right\vert _{Y}),\\ F-\overline{H} & \in \Lambda _{\alpha ,0}(X,d)\ \text{for all }\overline{H}\in \mathcal{E}_{\overline{d}}(\left. F\right\vert _{Y}). \end{align*}

Let $(X,d)$ be a quasi-metric space, $y_{0}\in X$ fixed and $Y\subseteq X$ such that $y_{0}\in Y.$ Let

\begin{equation} Y_{d}^{\perp }:=\{ G\in \Lambda _{\alpha ,0}(X,d):\left. G\right\vert _{Y}=0\} \label{f.1.16}\end{equation}

and

\begin{equation} Y_{\overline{d}}^{\perp }:=\{ \overline{G}\in \wedge _{\alpha ,d}(X,\overline{d}):\left. \overline{G}\right\vert _{Y}=0\} .\label{f.1.17}\end{equation}

Obviously, for $F\in \Lambda _{\alpha ,0}(X)$

\begin{equation} F-\mathcal{E}_{d}(\left. F\right\vert _{Y})\subset \Lambda _{\alpha ,0}(X,\overline{d})\label{f.1.18}\end{equation}

and

\begin{equation} F-\mathcal{E}_{\overline{d}}(\left. F\right\vert _{Y})\subset \Lambda _{\alpha ,0}(X,d).\label{f.1.19}\end{equation}

In the sequel we prove a result of Phelps type ( [ 1 ] , [ 10 ] , [ 12 ] ) concerning the existence and uniqueness of the extensions preserving the smallest semi-Hölder constants and a problem of best approximation by elements of $Y_{d}^{\perp }$ and $Y_{\overline{d}}^{\perp },$ respectively.

Let $(X,\left\Vert \ \right\vert )$ be an asymmetric norm (see [ 13 ] , [ 14 ] ) and let $M$ be a nonempty set of $X.$ The set $M$ is called proximinal for $x\in X$ iff there exists at least one element $m_{0}\in M$ such that

\[ \left\Vert x-m_{0}\right\vert =\inf \{ \left\Vert x-m\right\vert :m\in M\} =\rho (x,M). \]

If $M$ is proximinal for $x,$ then the set $P_{M}(x)=\{ m_{0}\in M:\left\Vert x-m_{0}\right\vert =\rho (x,M)\} $ is called the set of elements of best approximations for $x$ in $M.$ If card $P_{M}(x)=1$ then the set $M$ is called Chebyshevian for $x.$

The set $M$ is called proximinal if $M$ is proximinal for every $x\in X,$ and Chebyshevian if $M$ is Chebyshevian for every $x\in X.$

Now, consider the following two problems of best approximation:

P$_{\overline{\text{d}}}$(F). For $F\in \Lambda _{\alpha ,0}(X)$ find $G_{0}\in Y_{d}^{\perp }$ such that

\begin{equation} \left\Vert F-G_{0}\right\vert _{Y,\overline{d}}^{\alpha }=\inf \{ \left\Vert F-G\right\vert _{Y,\overline{d}}^{\alpha }:G\in Y_{d}^{\perp }\} =\rho _{\overline{d}}(F,Y_{d}^{\perp })\label{f.1.20}\end{equation}

and

P$_{\text{d}}$(F). For $F\in \Lambda _{\alpha ,0}(X)$ find $\overline{G}_{0}\in Y_{\overline{d}}^{\perp }$ such that

\begin{equation} \left\Vert F-\overline{G}_{0}\right\vert _{Y,d}^{\alpha }=\inf \{ \left\Vert F-\overline{G}\right\vert _{X,d}^{\alpha }:\overline{G}\in Y_{\overline{d}}^{\perp }\} =\rho _{d}(F,Y_{\overline{d}}^{\perp }).\label{f.1.21}\end{equation}

Let

\begin{equation} P_{Y_{\overline{d}}^{\perp }}(F):=\{ \overline{G}_{0}\in Y_{\overline{d}}^{\perp }:\left\Vert F-\overline{G}_{0}\right\vert _{X,d}^{\alpha }=\rho _{d}(F,Y_{\overline{d}}^{\perp })\} \label{f.1.22}\end{equation}

and

\begin{equation} P_{Y_{d}^{\perp }}(F):=\{ G_{0}\in Y_{d}^{\perp }:\left\Vert F-G_{0}\right\vert _{X,\overline{d}}^{\alpha }=\rho _{\overline{d}}(F,Y_{d}^{\perp })\} .\label{f.1.23}\end{equation}

The following theorem holds.

Theorem 4

If $F\in \Lambda _{\alpha ,0}(X)$ then

\begin{equation} P_{Y_{\overline{d}}^{\perp }}(F)=F-\mathcal{E}_{d}(\left. F\right\vert _{Y}),\label{f.1.24}\end{equation}

\begin{equation} P_{Y_{d}^{\perp }}(F)=F-\mathcal{E}_{\overline{d}}(\left. F\right\vert _{Y})\label{f.1.25}\end{equation}

and

\begin{equation} \rho _{d}(F,Y_{\overline{d}}^{\perp })=\left\Vert F\right\vert _{Y}\left\vert _{Y,d}^{\alpha }\right. ,\label{f.1.26}\end{equation}

\begin{equation} \rho _{\overline{d}}(F,Y_{d}^{\perp })=\left\Vert F\right\vert _{Y}\left\vert _{Y,\overline{d}}^{\alpha }\right. \ .\label{f.1.27}\end{equation}

Proof â–¼

Let $F\in \Lambda _{\alpha ,0}(X)(=\Lambda _{\alpha ,0}(X,d)\cap \Lambda _{\alpha ,0}(X,\overline{d}))$

Then, for every $\overline{G}\in Y_{\overline{d}}^{\perp },$

\[ \left\Vert F\right\vert _{Y}\left\vert _{Y,d}^{\alpha }\right. =\left\Vert F\right\vert _{Y}-\left. \overline{G}\right\vert _{Y}\left\vert _{Y,d}^{\alpha }\right. \leq \left\Vert F-\overline{G}\right\vert _{X,d}^{\alpha }. \]

Taking the infimum with respect to $\overline{G}\in Y_{\overline{d}}^{\perp },$ one obtains $\left\Vert F\right\vert _{Y}| _{Y,d}^{\alpha } \leq \rho _{d}(F,Y_{\overline{d}}^{\perp }).$ On the other hand, for every $H\in \mathcal{E}_{d}(\left. F\right\vert _{Y}),$

\[ \left\Vert F\right\vert _{Y}\left\vert _{Y,d}^{\alpha }\right. =\left\Vert H\right\vert _{X,d}^{\alpha }=\left\Vert F-(F-H)\right\vert _{X,d}^{\alpha }. \]

Because $F-H\in Y_{\overline{d}}^{\perp },$ it follows $\left\Vert F\right\vert _{Y}| _{Y,d}^{\alpha } \geq \rho _{d}(F,Y_{d}^{\perp }).$

Consequently, $Y_{\overline{d}}^{\perp }$ is proximinal with respect to the distance $\rho _{d}$ ($\rho _{d}$-proximinal in short) and

\[ \rho _{d}(F,Y_{\overline{d}}^{\perp })=\left\Vert F\right\vert _{Y}\left\vert _{Y,d}^{\alpha }\right. . \]

Now, let $\overline{G}_{0}\in P_{Y_{\overline{d}}^{\perp }}(F).$ Then $(F-\overline{G}_{0})\left\vert _{Y}\right. =\left. F\right\vert _{Y}$ and $\left\Vert F-\overline{G}_{0}\right\vert _{X,d}^{\alpha }=\left\Vert F\right\vert _{Y}| _{Y,d}^{\alpha }.$ This means that $F-\overline{G}_{0}\in \mathcal{E}_{d}(\left. F\right\vert _{Y}),$ i.e., $\overline{G}_{0}\in F-\mathcal{E}_{d}(\left. F\right\vert _{Y}).$ Consequently, $\overline{G}_{0}\in P_{Y_{\overline{d}}^{\perp }}(F)$ implies $\overline{G}_{0}\in F-\mathcal{E}_{d}(\left. F\right\vert _{Y}).$

Taking into account the first part of the proof it follows $P_{Y_{\overline{d}}^{\perp }}(F)=F-\mathcal{E}_{d}(\left. F\right\vert _{Y}).$

Analogously, one obtains $\rho _{\overline{d}}(F,Y_{d}^{\perp })=\left\Vert F\right\vert _{Y}| _{Y,\overline{d}}^{\alpha }$ and $P_{Y_{d}^{\perp }}(F)=F-\mathcal{E}_{\overline{d}}(\left. F\right\vert _{Y}). $

By the equalities (22) and (23) it follows.

Corollary 5

Let $F\in \Lambda _{\alpha ,0}(X)$ and $Y\subset X$ such that $y_{0}\in Y.$ Then

card $\mathcal{E}_{d}(\left. F\right\vert _{Y})=1$ iff $Y_{\overline{d}}^{\perp }$ is $\rho _{d}$-Chebyshevian;
card $\mathcal{E}_{\overline{d}}(\left. F\right\vert _{Y})=1$ iff $Y_{d}^{\perp }$ is $\rho _{\overline{d}}$-Chebyshevian.

Remark 6

Observe that the linear space $\Lambda _{\alpha ,0}(X)=\Lambda _{\alpha ,0}(X,d)\cap \Lambda _{\alpha ,0}(X,\overline{d})$ is a Banach space with respect to the norm

\begin{equation} \left\Vert F\right\vert _{X}^{\alpha }=\max \{ \left\Vert F\right\vert _{X,d}^{\alpha },\left\Vert F\right\vert _{X,\overline{d}}^{\alpha }\} .\label{f.1.28}\end{equation}

In fact this space in the space of all real-valued Lipschitz functions defined on the quasi-metric space $(X,d^{\alpha }),$ vanishing at a fixed point $y_{0}\in X.$ Obviously,

\begin{equation} \left\Vert F\right\Vert _{X}^{\alpha }=\sup \left\{ \tfrac {\left\vert F(x)-F(y)\right\vert }{d^{\alpha }(x,y)}:d(x,y)>0;\ x,y\in X\right\} \label{f.1.29}\end{equation}

is a norm on $\Lambda _{\alpha ,0}(X).$â–¡

Corollary 7

For every element $f$ in the space $\Lambda _{\alpha ,0}(Y)=\Lambda _{\alpha ,0}(Y,d)\cap \Lambda _{\alpha ,0}(Y,\overline{d})$ there exists $F\in \Lambda _{\alpha ,0}(X)$ such that

\[ \left. F\right\vert _{Y}=f\ \text{and }\left\Vert F\right\Vert _{X}^{\alpha }=\left\Vert f\right\Vert _{Y}^{\alpha }. \]

The set of all extensions of $f\in \Lambda _{\alpha ,0}(Y)$ preserving the norm $\left\Vert f\right\Vert _{Y}^{\alpha }$ (of the form (29), is denoted by $\mathcal{E}(f),$ i.e.,

\begin{equation} \mathcal{E(}f\mathcal{)}:\mathcal{=\{ }F\mathcal{\in }\Lambda _{\alpha ,0}(X):\left. F\right\vert _{Y}=f\ \text{and }\left\Vert F\right\Vert _{X}^{\alpha }=\left\Vert f\right\Vert _{Y}^{\alpha }\} .\label{f.1.30}\end{equation}

Denote by

\begin{equation} Y^{\perp }:=\{ G\in \Lambda _{\alpha ,0}(X):\left. G\right\vert _{Y}=0\} .\label{f.1.31}\end{equation}

the annihilator of the set $Y$ in Banach space $\Lambda _{\alpha ,0}(X),$ and one considers the following problem of best approximation:

P. For $F\in \Lambda _{\alpha ,0}(X)$ find $G_{0}\in Y^{\perp }$ such that

\[ \left\Vert F-G_{0}\right\Vert _{X}^{\alpha }=\inf \{ \left\Vert F-G\right\Vert _{X}^{\alpha }:G\in Y^{\perp }\} =\rho (F,Y^{\perp }). \]

Corollary 8

The subspace $Y^{\perp }is$ proximinal in $\Lambda _{\alpha ,0}(X)$ and the set of elements of best approximation for $F\in \Lambda _{\alpha ,0}(X)$ is

\[ P_{Y^{\perp }}(F)=F-\mathcal{E(}\left. F\right\vert _{Y}). \]

The distance of $F$ to $Y^{\perp }$ is given by

\[ \rho (F,Y^{\perp })=\left\Vert F\right\vert _{Y}\left\Vert _{Y}^{\alpha }\right. . \]

The subspace $Y^{\perp }$ is Chebyshevian for $F$ iff card $\mathcal{E(}\left. F\right\vert _{Y})=1.$

For $f$ in the linear space $\Lambda _{\alpha ,0}(Y),$ the equalities $F(f)(x)=\overline{F}(f)(x),$ $x\in X$ and $G(f)(x)=\overline{G}(f)(x),$ $x\in X$ are verified iff $\left\Vert f\right\vert _{Y,d}^{\alpha }=\left\Vert f\right\vert _{Y,\overline{d}}^{\alpha }.$ This means that $\left\Vert f\right\vert _{Y,d}^{\alpha }=\left\Vert -f\right\vert _{Y,d}^{\alpha }$ and, consequently,

\[ \left\Vert f\right\Vert _{Y}^{\alpha }=\max \{ \left\Vert f\right\vert _{Y,d}^{\alpha };\left\Vert f\right\vert _{Y,\overline{d}}^{\alpha }\} =\left\Vert f\right\vert _{Y,d}^{\alpha }. \]

By Theorem 3 in [ 14 ] , it follows that $\Lambda _{a,0}(Y)$ is a Banach space and $(Y,d^{\alpha })$ is a metric space.

Bibliography

1: S. Cobzaş, Phelps type duality reuslts in best approximation, Rev. Anal. Numér. Théor. Approx., 31, no. 1., pp. 29–43, 2002. $\includegraphics[scale=0.1]{ext-link.png}$
2: J. Collins and J. Zimmer, An asymmetric Arzelā-Ascoli Theorem, Topology Appl., 154, no. 11, pp. 2312–2322, 2007.
3: P. Flecther and W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982.
4: M.G. Krein and A.A. Nudel’man, The Markov Moment Problem and Extremum Problems, Nauka, Moscow 1973 (in Russian), English translation: American Mathematical Society, Providence, R.I., 1977.
5: E. Matouškova, Extensions of continuous and Lipschitz functions, Canad. Math. Bull., 43, no. 2, pp. 208–217, 2000.
6: E.T. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837–842, 1934.
7: A. Mennucci, On asymmetric distances, Tehnical report, Scuola Normale Superiore, Pisa, 2004.
8: C. Mustăţa, Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19, no. 3, pp. 222–230, 1977.
9: C. Mustăţa, Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Théor. Approx., 30, no. 1, pp. 61–67, 2001. $\includegraphics[scale=0.1]{ext-link.png}$
10: C. Mustăţa, A Phelps type theorem for spaces with asymmetric norms, Bul. Ştiinţ. Univ. Baia Mare, Ser. B. Matematică-Informatică, 18, pp. 275–280, 2002.
11: C. Mustăţa, Extensions of semi-Hölder real valued functions on a quasi-metric space, Rev. Anal. Numér. Théor. Approx., 38, no. 2, pp. 164–169, 2009. $\includegraphics[scale=0.1]{ext-link.png}$
12: R.R. Phelps, Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Numer. Math. Soc., 95, pp. 238–255, 1960.
13: S. Romaguera and M. Sanchis, Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292–301, 2000.
14: S. Romaguera and M. Sanchis, Properties of the normed cone of semi-Lipschitz functions, Acta Math. Hungar, 108, nos. 1–2, pp. 55–70, 2005.
15: J.H. Wells and L.R. Williams, Embeddings and Extensions in Analysis, Springer-Verlag, Berlin, 1975.

This work is licensed under a Creative Commons Attribution 4.0 International License.