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

1 Introduction

Let $X$ be a nonempty set and $d:X\times X\to [0,\mathrm{\infty })$ a function with the properties:

• $d(x,y)=d(y,x)=0$iff $x=y,$

• $d(x,y)\le d(x,z)+d(z,y),$

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

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

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

Analogously one defines ${\Vert f|}_{Y,\bar{d}}^{\alpha }.$

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

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

The set

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

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

The pair $({\mathrm{\Lambda }}_{\alpha ,0}(Y,d),{\Vert |}_{Y,d}^{\alpha })$ 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 $({\mathrm{\Lambda }}_{\alpha ,0}(Y,\bar{d}),\Vert \cdot {|}_{Y,\bar{d}}^{\alpha }).$ of all $\bar{d}$-semi-Hölder real-valued functions on $Y$, vanishing at the fixed point $y_0\in Y.$

By the above definitions it follows that

and, moreover, ${\Vert f|}_{Y,d}^{\alpha }={\Vert -f|}_{Y,\bar{d}}^{\alpha }.$

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

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

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

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

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

respectively

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

and

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

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

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

Also, for $F\in {\mathrm{\Lambda }}_{\alpha ,0}(X),$ ${F|}_Y\in {\mathrm{\Lambda }}_{\alpha ,0}(Y)$ and

Let $(X,d)$ be a quasi-metric space, $y_0\in X$ fixed and $Y\subseteq X$ such that $y_0\in Y.$ Let

and

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

and

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 $Y_{\bar{d}}^{\perp },$ respectively.

Let $(X,\Vert |)$ 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

If $M$ is proximinal for $x,$ then the set $P_M(x)=\{m_0\in M:\Vert x-m_0|=\rho (x,M)\}$ 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${}_{\bar{\text{d}}}$(F). For $F\in {\mathrm{\Lambda }}_{\alpha ,0}(X)$ find $G_0\in Y_d^{\perp }$ such that

and

P${}_{\text{d}}$(F). For $F\in {\mathrm{\Lambda }}_{\alpha ,0}(X)$ find ${\bar{G}}_0\in Y_{\bar{d}}^{\perp }$ such that

Let

and

The following theorem holds.

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

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

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

Because $F-H\in Y_{\bar{d}}^{\perp },$ it follows ${\Vert F|}_Y{|}_{Y,d}^{\alpha }\ge {\rho }_d(F,Y_d^{\perp }).$

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

    Now, let ${\bar{G}}_0\in P_{Y_{\bar{d}}^{\perp }}(F).$ Then $(F-{\bar{G}}_0)|{}_Y={F|}_Y$ and ${\Vert F-{\bar{G}}_0|}_{X,d}^{\alpha }={\Vert F|}_Y{|}_{Y,d}^{\alpha }.$ This means that $F-{\bar{G}}_0\in {\mathcal{E}}_d({F|}_Y),$ i.e., ${\bar{G}}_0\in F-{\mathcal{E}}_d({F|}_Y).$ Consequently, ${\bar{G}}_0\in P_{Y_{\bar{d}}^{\perp }}(F)$ implies ${\bar{G}}_0\in F-{\mathcal{E}}_d({F|}_Y).$

Taking into account the first part of the proof it follows $P_{Y_{\bar{d}}^{\perp }}(F)=F-{\mathcal{E}}_d({F|}_Y).$

Analogously, one obtains ${\rho }_{\bar{d}}(F,Y_d^{\perp })={\Vert F|}_Y{|}_{Y,\bar{d}}^{\alpha }$ and $P_{Y_d^{\perp }}(F)=F-{\mathcal{E}}_{\bar{d}}({F|}_Y).$

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

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

Denote by

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

P. For $F\in {\mathrm{\Lambda }}_{\alpha ,0}(X)$ find $G_0\in Y^{\perp }$ such that

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

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