Abstract
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 \((\Lambda_{\alpha,0}(Y,d),\Vert \cdot|_{Y,d}^{\alpha})\) the asymmetric normed cone of real-valued \(d\)-semi-H\”{o}lder functions defined on \(Y\) of exponent \(\alpha \in(0,1]\), vanishing in \(y_{0}\), and by \((\Lambda_{\alpha,0}(Y,\bar {d}),\Vert \cdot|_{Y,\bar{d}}^{\alpha})\) the similar cone if \(d\) is replaced by conjugate \(\bar{d}\) of \(d\).
Authors
Costică Mustăţa
Tiberiu Popoviciu Institute of Numerical Analysis, Romania
Keywords
Extensions, semi-Lipschitz functions, semi-Holder functions, best approximation, quasi-metric spaces.
Paper coordinates
C. Mustăţa, On the existence and uniqueness of extensions of semi-Holder real-valued functions, Rev. Anal. Numer. Theor. Approx., 39 (2010) no. 2, 134-140.
About this paper
Journal
Revue Analysis Numer Theor. Approx.
Publisher Name
Publishing House of the Romanian Academy
Print ISSN
2502-059X
Online ISSN
2457-6794
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[1] S. Cobzas, Phelps type duality reuslts in best approximation,Rev. Anal. Numer. Theor.Approx.,31, no. 1., pp. 29–43, 2002.
[2] J. Collins and J. Zimmer, An asymmetric Arzela-Ascoli Theorem, Topology Appl.,154, no. 11, pp. 2312–2322, 2007.
[3] P. Flectherand W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York,1982.
[4] M.G. Kreinand 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. Matouskova, 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. Mustata, Best approximation and unique extension of Lipschitz functions, J. Approx. Theory,19, no. 3, pp. 222–230, 1977.
[9]C. Mustata,Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal.Numer. Theor. Approx., 30, no. 1, pp. 61–67, 2001.
[10]C. Mustata, A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint. Univ. Baia Mare, Ser. B. Matematica-Informatica,18, pp. 275–280, 2002.
[11]C. Mustata, Extensions of semi-Holder real valued functions on a quasi-metric space, Rev. Anal. Numer. Theor. Approx., 38, no. 2, pp. 164–169, 2009.
[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. Received by the editors: April 13, 2010.
Paper (preprint) in HTML form
ON THE EXISTENCE AND UNIQUENESS OF EXTENSIONS OF SEMI-HÖLDER REAL-VALUED FUNCTIONS
Abstract
Let ( X,dX, d ) be a quasi-metric space, y_(0)in Xy_{0} \in X a fixed element and YY a subset of XX such that y_(0)in Yy_{0} \in Y. Denote by (Lambda_(alpha,0)(Y,d),||*|_(Y,d)^(alpha))\left(\Lambda_{\alpha, 0}(Y, d), \|\left.\cdot\right|_{Y, d} ^{\alpha}\right) the asymmetric normed cone of real-valued dd-semi-Hölder functions defined on YY of exponent alpha in(0,1]\alpha \in(0,1], vanishing in y_(0)y_{0}, and by ( Lambda_(alpha,0)(Y, bar(d)),||*|_(Y, bar(d))^(alpha)\Lambda_{\alpha, 0}(Y, \bar{d}), \|\left.\cdot\right|_{Y, \bar{d}} ^{\alpha} ) the similar cone if dd is replaced by conjugate bar(d)\bar{d} of dd.
One considers the following claims: (a) For every ff in the linear space Lambda_(alpha,0)(Y)=Lambda_(alpha,0)(Y,d)nnLambda_(alpha,0)(Y, bar(d))\Lambda_{\alpha, 0}(Y)=\Lambda_{\alpha, 0}(Y, d) \cap \Lambda_{\alpha, 0}(Y, \bar{d}) there exist F inLambda_(alpha,0)(X,d)F \in \Lambda_{\alpha, 0}(X, d) such that F|_(Y)=f\left.F\right|_{Y}=f and ||F|_(Y,d)^(alpha)=||f|_(Y,d)^(alpha)\left.\left\|\left.F\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha}; (b) For every f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y) there exists bar(F)inLambda_(alpha,0)(X, bar(d))\bar{F} \in \Lambda_{\alpha, 0}(X, \bar{d}) such that ( bar(F))|_(Y)=f\left.\bar{F}\right|_{Y}=f and ||( bar(F))|_(Y, bar(d))^(alpha)=||f|_(Y, bar(d))^(alpha);\left.\left\|\left.\bar{F}\right|_{Y, \bar{d}} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha} ; (c) The extension FF in (a) is unique; (d) The extension bar(F)\bar{F} in (b) is unique; (e) The annihilator Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp} of YY in Lambda_(alpha,0)(X, bar(d))\Lambda_{\alpha, 0}(X, \bar{d}) is proximinal for the elements of Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X) with respect to the distance generated by ||*|_(Y,d)^(alpha)\|\left.\cdot\right|_{Y, d} ^{\alpha}; (f) The annihilator Y_(d)^(_|_)Y_{d}^{\perp} of YY in Lambda_(alpha,0)(X,d)\Lambda_{\alpha, 0}(X, d) is proximinal for the elements of Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X) with respect to the distance generated by ||*|_(Y, bar(d))^(alpha)\|\left.\cdot\right|_{Y, \bar{d}} ^{\alpha}; (g) Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp} in the claim (e) is Chebyshevian; (h) Y_(d)^(_|_)Y_{d}^{\perp} in the claim ( ff ) is Chebyshevian.
Then the following equivalences hold:
(a)<=>(e);(b)<=>(f);(c)<=>(g);(d)<=>(h)(a) \Leftrightarrow(e) ;(b) \Leftrightarrow(f) ;(c) \Leftrightarrow(g) ;(d) \Leftrightarrow(h).
(a)<=>(e);(b)<=>(f);(c)<=>(g);(d)<=>(h)(a) \Leftrightarrow(e) ;(b) \Leftrightarrow(f) ;(c) \Leftrightarrow(g) ;(d) \Leftrightarrow(h).
MSC 2000. 46A22, 41A50, 41A52.
Keywords. Extensions, semi-Lipschitz functions, semi-Hölder functions, best approximation, quasi-metric spaces.
Keywords. Extensions, semi-Lipschitz functions, semi-Hölder functions, best approximation, quasi-metric spaces.
1. INTRODUCTION
Let XX be a nonempty set and d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty) a function with the properties:
(QM_(1))d(x,y)=d(y,x)=0\left(\mathrm{QM}_{1}\right) d(x, y)=d(y, x)=0 iff x=yx=y,
(QM_(2))d(x,y) <= d(x,z)+d(z,y)\left(\mathrm{QM}_{2}\right) d(x, y) \leq d(x, z)+d(z, y),
for all x,y,z in Xx, y, z \in X.
Then the function dd is called a quasi-metric on XX and the pair ( X,dX, d ) is called quasi-metric space ([13]).
(QM_(1))d(x,y)=d(y,x)=0\left(\mathrm{QM}_{1}\right) d(x, y)=d(y, x)=0 iff x=yx=y,
(QM_(2))d(x,y) <= d(x,z)+d(z,y)\left(\mathrm{QM}_{2}\right) d(x, y) \leq d(x, z)+d(z, y),
for all x,y,z in Xx, y, z \in X.
Then the function dd is called a quasi-metric on XX and the pair ( X,dX, d ) is called quasi-metric space ([13]).
Because, in general, d(x,y)!=d(y,x)d(x, y) \neq d(y, x), for x,y in Xx, y \in X one defines the conjugate quasi-metric bar(d)\bar{d} of dd, by the equality bar(d)(x,y)=d(y,x)\bar{d}(x, y)=d(y, x), for all x,y in Xx, y \in X.
Let YY be a nonvoid subset of ( X,dX, d ) and alpha in(0,1]\alpha \in(0,1] a fixed number.
Definition 1. a) AA function f:Y rarrRf: Y \rightarrow \mathbb{R} is called dd-semi-Hölder (of exponent alpha\alpha ) if there exists a constant K_(Y)(f) >= 0K_{Y}(f) \geq 0 such that
Definition 1. a) AA function f:Y rarrRf: Y \rightarrow \mathbb{R} is called dd-semi-Hölder (of exponent alpha\alpha ) if there exists a constant K_(Y)(f) >= 0K_{Y}(f) \geq 0 such that
{:(1)f(x)-f(y) <= K_(Y)(f)d^(alpha)(x","y)",":}\begin{equation*}
f(x)-f(y) \leq K_{Y}(f) d^{\alpha}(x, y), \tag{1}
\end{equation*}
for all x,y in Yx, y \in Y.
b) f:Y rarrRf: Y \rightarrow \mathbb{R} is called bar(d)\bar{d}-semi-Hölder (of exponent alpha\alpha ) if there exists a constant bar(K)_(Y)(f) >= 0\bar{K}_{Y}(f) \geq 0 such that
b) f:Y rarrRf: Y \rightarrow \mathbb{R} is called bar(d)\bar{d}-semi-Hölder (of exponent alpha\alpha ) if there exists a constant bar(K)_(Y)(f) >= 0\bar{K}_{Y}(f) \geq 0 such that
{:(2)f(x)-f(y) <= bar(K)_(Y)(f)*d^(alpha)(y","x)",":}\begin{equation*}
f(x)-f(y) \leq \bar{K}_{Y}(f) \cdot d^{\alpha}(y, x), \tag{2}
\end{equation*}
for all x,y in Yx, y \in Y.
The smallest constant K_(Y)(f)K_{Y}(f) in (1) is denoted by ||f|_(Y,d)^(alpha)\|\left. f\right|_{Y, d} ^{\alpha} and one shows that
The smallest constant K_(Y)(f)K_{Y}(f) in (1) is denoted by ||f|_(Y,d)^(alpha)\|\left. f\right|_{Y, d} ^{\alpha} and one shows that
{:(3)||f|_(Y,d)^(alpha):=s u p{((f(x)-f(y))vv0)/(d^(alpha)(x,y)):d(x,y) > 0;x,y in Y}.:}\begin{equation*}
\|\left. f\right|_{Y, d} ^{\alpha}:=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d^{\alpha}(x, y)}: d(x, y)>0 ; x, y \in Y\right\} . \tag{3}
\end{equation*}
Analogously one defines ||f|_(Y, bar(d))^(alpha)\|\left. f\right|_{Y, \bar{d}} ^{\alpha}.
Observe that the function ff is dd-semi-Hölder on YY iff -f-f is bar(d)\bar{d}-semi-Hölder on YY. Moreover
Observe that the function ff is dd-semi-Hölder on YY iff -f-f is bar(d)\bar{d}-semi-Hölder on YY. Moreover
{:(4)||f|_(Y,d)^(alpha)=||-f|_(Y, bar(d))^(alpha):}\begin{equation*}
\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, \bar{d}} ^{\alpha} \tag{4}
\end{equation*}
Definition 2. ([14). Let ( X,dX, d ) be a quasi-metric space and Y sube XY \subseteq X a nonempty set. The function f:Y rarrRf: Y \rightarrow \mathbb{R} is called <= _(d)\leq_{d}-increasing on YY if f(x) <= f(y)f(x) \leq f(y) whenever d(x,y)=0,x,y in Yd(x, y)=0, x, y \in Y.
The set of all <= _(d)\leq_{d}-increasing functions on YY is denoted by R_( <= _(d))^(Y)\mathbb{R}_{\leq_{d}}^{Y} and it is a cone in the linear space R^(Y)\mathbb{R}^{Y} of all real-valued functions on YY.
The set
{:(5)Lambda_(alpha)(Y","d):={f inR_( <= d)^(Y);f" is "d"-semi-Hölder and "||f|_(Y,d)^(alpha) < oo}:}\begin{equation*}
\Lambda_{\alpha}(Y, d):=\left\{f \in \mathbb{R}_{\leq d}^{Y} ; f \text { is } d \text {-semi-Hölder and } \|\left. f\right|_{Y, d} ^{\alpha}<\infty\right\} \tag{5}
\end{equation*}ö
is also a cone, called the cone of dd-semi-Hölder functions on YY.
If y_(0)in Yy_{0} \in Y is arbitrary, but fixed, one considers the cone
If y_(0)in Yy_{0} \in Y is arbitrary, but fixed, one considers the cone
{:(6)Lambda_(alpha,0)(Y","d):={f inLambda_(alpha)(Y,d):f(y_(0))=0}.:}\begin{equation*}
\Lambda_{\alpha, 0}(Y, d):=\left\{f \in \Lambda_{\alpha}(Y, d): f\left(y_{0}\right)=0\right\} . \tag{6}
\end{equation*}
Then the functional |||_(Y,d)^(alpha):Lambda_(alpha,0)(Y,d)rarr[0,oo)\|\left.\right|_{Y, d} ^{\alpha}: \Lambda_{\alpha, 0}(Y, d) \rightarrow[0, \infty) is subadditive, positively homogeneous and the equality ||f|_(Y,d)^(alpha)=||-f|_(Y,d)^(alpha)=0\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, d} ^{\alpha}=0 implies f-=0f \equiv 0. This means that ||*|_(Y,d)^(alpha)\|\left.\cdot\right|_{Y, d} ^{\alpha} is an asymmetric norm (see [13, [14]), on the cone Lambda_(alpha,0)(Y,d)\Lambda_{\alpha, 0}(Y, d).
The pair ( Lambda_(alpha,0)(Y,d),|||_(Y,d)^(alpha)\Lambda_{\alpha, 0}(Y, d), \|\left.\right|_{Y, d} ^{\alpha} ) is called the asymmetric normed cone of dd -semi-Hölder real-valued function on YY (compare with [14]).
Analogously, one defines the asymmetric normed cone ( Lambda_(alpha,0)(Y, bar(d)),||*|_(Y, bar(d))^(alpha)\Lambda_{\alpha, 0}(Y, \bar{d}), \|\left.\cdot\right|_{Y, \bar{d}} ^{\alpha} ). of all bar(d)\bar{d}-semi-Hölder real-valued functions on YY, vanishing at the fixed point y_(0)in Yy_{0} \in Y.
By the above definitions it follows that
f in(Lambda_(alpha,0)(Y,d),|||_(Y,d)^(alpha))" iff "-f in(Lambda_(alpha,0)(Y,( bar(d))),|||_(Y, bar(d))^(alpha))f \in\left(\Lambda_{\alpha, 0}(Y, d), \|\left.\right|_{Y, d} ^{\alpha}\right) \text { iff }-f \in\left(\Lambda_{\alpha, 0}(Y, \bar{d}), \|\left.\right|_{Y, \bar{d}} ^{\alpha}\right)
and, moreover, ||f|_(Y,d)^(alpha)=||-f|_(Y, bar(d))^(alpha)\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, \bar{d}} ^{\alpha}.
Defining Lambda_(alpha,0)(Y)\Lambda_{\alpha, 0}(Y) by
Defining Lambda_(alpha,0)(Y)\Lambda_{\alpha, 0}(Y) by
{:(7)Lambda_(alpha,0)(Y)=Lambda_(alpha,0)(Y","d)nnLambda_(alpha,0)(Y"," bar(d)):}\begin{equation*}
\Lambda_{\alpha, 0}(Y)=\Lambda_{\alpha, 0}(Y, d) \cap \Lambda_{\alpha, 0}(Y, \bar{d}) \tag{7}
\end{equation*}
It follows that Lambda_(alpha,0)(Y)\Lambda_{\alpha, 0}(Y) is a linear subspace. The following, theorem holds.
Theorem 3. For every f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y) there exist at least one function F inLambda_(alpha,0)(Y,d)F \in \Lambda_{\alpha, 0}(Y, d) and at least one function bar(F)inLambda_(alpha,0)(Y, bar(d))\bar{F} \in \Lambda_{\alpha, 0}(Y, \bar{d}) such that
a) F|_(Y)=( bar(F))|_(Y)=f\left.F\right|_{Y}=\left.\bar{F}\right|_{Y}=f.
b) ||F|_(Y,d)^(alpha)=||f|_(Y,d)^(alpha)\left.\left\|\left.F\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha} and ||( bar(F))|_(Y, bar(d))^(alpha)=||f|_(Y, bar(d))^(alpha)\left.\left\|\left.\bar{F}\right|_{Y, \bar{d}} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha}.
Theorem 3. For every f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y) there exist at least one function F inLambda_(alpha,0)(Y,d)F \in \Lambda_{\alpha, 0}(Y, d) and at least one function bar(F)inLambda_(alpha,0)(Y, bar(d))\bar{F} \in \Lambda_{\alpha, 0}(Y, \bar{d}) such that
a) F|_(Y)=( bar(F))|_(Y)=f\left.F\right|_{Y}=\left.\bar{F}\right|_{Y}=f.
b) ||F|_(Y,d)^(alpha)=||f|_(Y,d)^(alpha)\left.\left\|\left.F\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha} and ||( bar(F))|_(Y, bar(d))^(alpha)=||f|_(Y, bar(d))^(alpha)\left.\left\|\left.\bar{F}\right|_{Y, \bar{d}} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha}.
Proof. By Theorem 2 and Remark 3 in [11] it follows that the functions defined by the formulae:
{:[(8)F(f)(x)=i n f_(y in Y){f(y)+||f|_(Y,d)^(alpha)d^(alpha)(x,y)}","x in X],[G(f)(x)=s u p_(y in Y){f(y)-||f|_(Y,d)^(alpha)d^(alpha)(y,x},x in X:}]:}\begin{align*}
& F(f)(x)=\inf _{y \in Y}\left\{f(y)+\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(x, y)\right\}, x \in X \tag{8}\\
& G(f)(x)=\sup _{y \in Y}\left\{f(y)-\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(y, x\}, x \in X\right.
\end{align*}
are elements of Lambda_(alpha,0)(X,d)\Lambda_{\alpha, 0}(X, d) and, respectively, the functions given by
{:[(9) bar(F)(f)(x)=i n f_(y in Y){f(y)+||f|_(Y, bar(d))^(alpha)d^(alpha)(y,x)}","x in X],[ bar(G)(f)(x)=s u p_(y in Y){f(y)-||f|_(Y, bar(d))^(alpha)d^(alpha)(x,y)}","x in X]:}\begin{align*}
& \bar{F}(f)(x)=\inf _{y \in Y}\left\{f(y)+\|\left. f\right|_{Y, \bar{d}} ^{\alpha} d^{\alpha}(y, x)\right\}, x \in X \tag{9}\\
& \bar{G}(f)(x)=\sup _{y \in Y}\left\{f(y)-\|\left. f\right|_{Y, \bar{d}} ^{\alpha} d^{\alpha}(x, y)\right\}, x \in X
\end{align*}
are elements of Lambda_(alpha,0)(X, bar(d))\Lambda_{\alpha, 0}(X, \bar{d}) such that
{:(10)F(f)|_(Y)=G(f)|_(Y)=f" and "||F(f)|_(Y,d)^(alpha)=||G(f)|_(Y,d)^(alpha)=||f|_(Y,d)^(alpha):}\begin{equation*}
\left.F(f)\right|_{Y}=\left.G(f)\right|_{Y}=f \text { and }\left.\left\|\left.F(f)\right|_{Y, d} ^{\alpha}=\right\| G(f)\right|_{Y, d} ^{\alpha}=\|\left. f\right|_{Y, d} ^{\alpha} \tag{10}
\end{equation*}
respectively
{:(11)( bar(F))(f)|_(Y)=( bar(G))(f)|_(Y)=f" and "||( bar(F))(f)|_(Y, bar(d))^(alpha)=||( bar(G))(f)|_(Y, bar(d))^(alpha)=||f|_(Y, bar(d))^(alpha):}\begin{equation*}
\left.\bar{F}(f)\right|_{Y}=\left.\bar{G}(f)\right|_{Y}=f \text { and }\left.\left\|\left.\bar{F}(f)\right|_{Y, \bar{d}} ^{\alpha}=\right\| \bar{G}(f)\right|_{Y, \bar{d}} ^{\alpha}=\|\left. f\right|_{Y, \bar{d}} ^{\alpha} \tag{11}
\end{equation*}
For f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y) let us consider the following (nonempty) sets of extensions:
{:(12)E_(d)(f):={H inLambda_(alpha,0)(X,d):H|_(Y)=f" and "||H|_(Y,d)^(alpha)=||f|_(Y,d)^(alpha)}:}\begin{equation*}
\mathcal{E}_{d}(f):=\left\{H \in \Lambda_{\alpha, 0}(X, d):\left.H\right|_{Y}=f \text { and }\left.\left\|\left.H\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha}\right\} \tag{12}
\end{equation*}
and
{:(13)E_( bar(d))(f):={( bar(H))in^^_(alpha,0)(X,( bar(d))):( bar(H))|_(Y)=f" and "||( bar(H))|_(Y, bar(d))^(alpha)=||f|_(Y, bar(d))^(alpha)}:}\begin{equation*}
\mathcal{E}_{\bar{d}}(f):=\left\{\bar{H} \in \wedge_{\alpha, 0}(X, \bar{d}):\left.\bar{H}\right|_{Y}=f \text { and }\left.\left\|\left.\bar{H}\right|_{Y, \bar{d}} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha}\right\} \tag{13}
\end{equation*}
The sets E_(d)(f)\mathcal{E}_{d}(f) and E_( bar(d))(f)\mathcal{E}_{\bar{d}}(f) are convex and
{:(14)F(f)(x) >= H(x) >= G(f)(x)","x in X:}\begin{equation*}
F(f)(x) \geq H(x) \geq G(f)(x), x \in X \tag{14}
\end{equation*}
for all H inE_(d)(f)H \in \mathcal{E}_{d}(f);
{:(15) bar(F)(f)(x) >= bar(H)(x) >= bar(G)(f)(x)","x in H",":}\begin{equation*}
\bar{F}(f)(x) \geq \bar{H}(x) \geq \bar{G}(f)(x), x \in H, \tag{15}
\end{equation*}
for all bar(H)inE_( bar(d))(f)\bar{H} \in \mathcal{E}_{\bar{d}}(f).
Also, for F inLambda_(alpha,0)(X),F|_(Y)inLambda_(alpha,0)(Y)F \in \Lambda_{\alpha, 0}(X),\left.F\right|_{Y} \in \Lambda_{\alpha, 0}(Y) and
{:[F-H inLambda_(alpha,0)(X"," bar(d))","" for all "H inE_(d)(F|_(Y))","],[F- bar(H)inLambda_(alpha,0)(X","d)" for all " bar(H)inE_( bar(d))(F|_(Y)).]:}\begin{aligned}
& F-H \in \Lambda_{\alpha, 0}(X, \bar{d}), \text { for all } H \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right), \\
& F-\bar{H} \in \Lambda_{\alpha, 0}(X, d) \text { for all } \bar{H} \in \mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right) .
\end{aligned}
Let ( X,dX, d ) be a quasi-metric space, y_(0)in Xy_{0} \in X fixed and Y sube XY \subseteq X such that y_(0)in Yy_{0} \in Y.
Let
Let
{:(16)Y_(d)^(_|_):={G inLambda_(alpha,0)(X,d):G|_(Y)=0}:}\begin{equation*}
Y_{d}^{\perp}:=\left\{G \in \Lambda_{\alpha, 0}(X, d):\left.G\right|_{Y}=0\right\} \tag{16}
\end{equation*}
and
{:(17)Y_( bar(d))^(_|_):={( bar(G))in^^_(alpha,d)(X,( bar(d))):( bar(G))|_(Y)=0}.:}\begin{equation*}
Y_{\bar{d}}^{\perp}:=\left\{\bar{G} \in \wedge_{\alpha, d}(X, \bar{d}):\left.\bar{G}\right|_{Y}=0\right\} . \tag{17}
\end{equation*}
Obviously, for F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X)
{:(18)F-E_(d)(F|_(Y))subLambda_(alpha,0)(X"," bar(d)):}\begin{equation*}
F-\mathcal{E}_{d}\left(\left.F\right|_{Y}\right) \subset \Lambda_{\alpha, 0}(X, \bar{d}) \tag{18}
\end{equation*}
and
{:(19)F-E_( bar(d))(F|_(Y))subLambda_(alpha,0)(X","d).:}\begin{equation*}
F-\mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right) \subset \Lambda_{\alpha, 0}(X, d) . \tag{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)^(_|_)Y_{d}^{\perp} and Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp}, respectively.
Let (X,||||)(X,\| \|) be an asymmetric norm (see [13, [14]) and let MM be a nonempty set of XX. The set MM is called proximinal for x in Xx \in X iff there exists at least one element m_(0)in Mm_{0} \in M such that
||x-m_(0)|=i n f{||x-m∣:m in M}=rho(x,M).:}\left|\left|x-m_{0}\right|=\inf \{| | x-m \mid: m \in M\}=\rho(x, M) .\right.
If MM is proximinal for xx, then the set P_(M)(x)={m_(0)in M:||x-m_(0)∣=:}rho(x,M)}P_{M}(x)=\left\{m_{0} \in M: \| x-m_{0} \mid=\right. \rho(x, M)\} is called the set of elements of best approximations for xx in MM. If card P_(M)(x)=1P_{M}(x)=1 then the set MM is called Chebyshevian for xx.
The set MM is called proximinal if MM is proximinal for every x in Xx \in X, and Chebyshevian if MM is Chebyshevian for every x in Xx \in X.
Now, consider the following two problems of best approximation:
P_( bar(d))(F)\mathbf{P}_{\overline{\mathrm{d}}}(\mathbf{F}). For F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) find G_(0)inY_(d)^(_|_)G_{0} \in Y_{d}^{\perp} such that
P_( bar(d))(F)\mathbf{P}_{\overline{\mathrm{d}}}(\mathbf{F}). For F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) find G_(0)inY_(d)^(_|_)G_{0} \in Y_{d}^{\perp} such that
{:(20)||F-G_(0)|_(Y, bar(d))^(alpha)=i n f{||F-G|_(Y, bar(d))^(alpha):G inY_(d)^(_|_)}=rho_( bar(d))(F,Y_(d)^(_|_)):}\begin{equation*}
\| F-\left.G_{0}\right|_{Y, \bar{d}} ^{\alpha}=\inf \left\{\| F-\left.G\right|_{Y, \bar{d}} ^{\alpha}: G \in Y_{d}^{\perp}\right\}=\rho_{\bar{d}}\left(F, Y_{d}^{\perp}\right) \tag{20}
\end{equation*}
and
P_(d)(F)\mathbf{P}_{\mathrm{d}}(\mathbf{F}). For F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) find bar(G)_(0)inY_( bar(d))^(_|_)\bar{G}_{0} \in Y_{\bar{d}}^{\perp} such that
P_(d)(F)\mathbf{P}_{\mathrm{d}}(\mathbf{F}). For F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) find bar(G)_(0)inY_( bar(d))^(_|_)\bar{G}_{0} \in Y_{\bar{d}}^{\perp} such that
{:(21)||F- bar(G)_(0)|_(Y,d)^(alpha)=i n f{||F-( bar(G))|_(X,d)^(alpha):( bar(G))inY_( bar(d))^(_|_)}=rho_(d)(F,Y_( bar(d))^(_|_)).:}\begin{equation*}
\| F-\left.\bar{G}_{0}\right|_{Y, d} ^{\alpha}=\inf \left\{\| F-\left.\bar{G}\right|_{X, d} ^{\alpha}: \bar{G} \in Y_{\bar{d}}^{\perp}\right\}=\rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right) . \tag{21}
\end{equation*}
Let
{:(22)P_(Y_( bar(d))^(_|_))(F):={ bar(G)_(0)inY_( bar(d))^(_|_):||F- bar(G)_(0)|_(X,d)^(alpha)=rho_(d)(F,Y_( bar(d))^(_|_))}:}\begin{equation*}
P_{Y_{\bar{d}}^{\perp}}(F):=\left\{\bar{G}_{0} \in Y_{\bar{d}}^{\perp}: \| F-\left.\bar{G}_{0}\right|_{X, d} ^{\alpha}=\rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right)\right\} \tag{22}
\end{equation*}
and
{:(23)P_(Y_(d)^(_|_))(F):={G_(0)inY_(d)^(_|_):||F-G_(0)|_(X, bar(d))^(alpha)=rho_( bar(d))(F,Y_(d)^(_|_))}.:}\begin{equation*}
P_{Y_{d}^{\perp}}(F):=\left\{G_{0} \in Y_{d}^{\perp}: \| F-\left.G_{0}\right|_{X, \bar{d}} ^{\alpha}=\rho_{\bar{d}}\left(F, Y_{d}^{\perp}\right)\right\} . \tag{23}
\end{equation*}
The following theorem holds.
Theorem 4. If F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) then
and
Theorem 4. If F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) then
and
{:[(24)P_(Y_( bar(d))^(_|_))(F)=F-E_(d)(F|_(Y))],[(25)P_(Y_( bar(d))^(_|_))(F)=F-E_( bar(d))(F|_(Y))]:}\begin{align*}
& P_{Y_{\bar{d}}^{\perp}}(F)=F-\mathcal{E}_{d}\left(\left.F\right|_{Y}\right) \tag{24}\\
& P_{Y_{\bar{d}}^{\perp}}(F)=F-\mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right) \tag{25}
\end{align*}
and
{:[(26)rho_(d)(F,Y_( bar(d))^(_|_))=||F|_(Y)|_(Y,d)^(alpha)],[(27)rho_( bar(d))(F,Y_(d)^(_|_))=||F|_(Y)|_(Y, bar(d))^(alpha)]:}\begin{align*}
& \rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha} \tag{26}\\
& \rho_{\bar{d}}\left(F, Y_{d}^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|_{Y, \bar{d}} ^{\alpha} \tag{27}
\end{align*}
Proof. Let F inLambda_(alpha,0)(X)(=Lambda_(alpha,0)(X,d)nnLambda_(alpha,0)(X,( bar(d))))F \in \Lambda_{\alpha, 0}(X)\left(=\Lambda_{\alpha, 0}(X, d) \cap \Lambda_{\alpha, 0}(X, \bar{d})\right)
Then, for every bar(G)inY_( bar(d))^(_|_)\bar{G} \in Y_{\bar{d}}^{\perp},
Then, for every bar(G)inY_( bar(d))^(_|_)\bar{G} \in Y_{\bar{d}}^{\perp},
||F|_(Y)|_(Y,d)^(alpha)=||F|_(Y)-( bar(G))|_(Y)|_(Y,d)^(alpha) <= ||F-( bar(G))|_(X,d)^(alpha)\left.\left\|\left.\left.F\right|_{Y}\right|_{Y, d} ^{\alpha}=\right\| F\right|_{Y}-\left.\left.\bar{G}\right|_{Y}\right|_{Y, d} ^{\alpha} \leq \| F-\left.\bar{G}\right|_{X, d} ^{\alpha}
Taking the infimum with respect to bar(G)inY_( bar(d))^(_|_)\bar{G} \in Y_{\bar{d}}^{\perp}, one obtains ||F|_(Y)|_(Y,d)^(alpha) <= rho_(d)(F,Y_( bar(d))^(_|_))\|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha} \leq \rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right). On the other hand, for every H inE_(d)(F|_(Y))H \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right),
||F|_(Y)|_(Y,d)^(alpha)=||H|_(X,d)^(alpha)=||F-(F-H)|_(X,d)^(alpha)\left.\left\|\left.\left.F\right|_{Y}\right|_{Y, d} ^{\alpha}=\right\| H\right|_{X, d} ^{\alpha}=\| F-\left.(F-H)\right|_{X, d} ^{\alpha}
Because F-H inY_( bar(d))^(_|_)F-H \in Y_{\bar{d}}^{\perp}, it follows ||F|_(Y)|_(Y,d)^(alpha) >= rho_(d)(F,Y_(d)^(_|_))\|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha} \geq \rho_{d}\left(F, Y_{d}^{\perp}\right).
Consequently, Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp} is proximinal with respect to the distance rho_(d)\rho_{d} ( rho_(d)\rho_{d}-proximinal in short) and
Consequently, Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp} is proximinal with respect to the distance rho_(d)\rho_{d} ( rho_(d)\rho_{d}-proximinal in short) and
rho_(d)(F,Y_( bar(d))^(_|_))=||F|_(Y)|_(Y,d)^(alpha)\rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha}
Now, let bar(G)_(0)inP_(Y_( bar(d))^(_|_))(F)\bar{G}_{0} \in P_{Y_{\bar{d}}^{\perp}}(F). Then (F- bar(G)_(0))|_(Y)=F|_(Y)\left.\left(F-\bar{G}_{0}\right)\right|_{Y}=\left.F\right|_{Y} and ||F- bar(G)_(0)|_(X,d)^(alpha)=||F|_(Y)|_(Y,d)^(alpha)\| F-\left.\bar{G}_{0}\right|_{X, d} ^{\alpha}= \|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha}. This means that F- bar(G)_(0)inE_(d)(F|_(Y))F-\bar{G}_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right), i.e., bar(G)_(0)in F-E_(d)(F|_(Y))\bar{G}_{0} \in F-\mathcal{E}_{d}\left(\left.F\right|_{Y}\right). Consequently, bar(G)_(0)inP_(Y_( bar(d))^(_|_))(F)\bar{G}_{0} \in P_{Y_{\bar{d}}^{\perp}}(F) implies bar(G)_(0)in F-E_(d)(F|_(Y))\bar{G}_{0} \in F-\mathcal{E}_{d}\left(\left.F\right|_{Y}\right).
Taking into account the first part of the proof it follows P_(Y_( bar(d))^(_|_))(F)=F-E_(d)(F|_(Y))P_{Y_{\bar{d}}^{\perp}}(F)=F- \mathcal{E}_{d}\left(\left.F\right|_{Y}\right).
Analogously, one obtains rho_( bar(d))(F,Y_(d)^(_|_))=||F|_(Y)|_(Y, bar(d))^(alpha)\rho_{\bar{d}}\left(F, Y_{d}^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|_{Y, \bar{d}} ^{\alpha} and P_(Y_(d)^(_|_))(F)=F-E_( bar(d))(F|_(Y))P_{Y_{d}^{\perp}}(F)=F-\mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right).
By the equalities (22) and (23) it follows.
Corollary 5. Let F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) and Y sub XY \subset X such that y_(0)in Yy_{0} \in Y. Then
a) cardE_(d)(F|_(Y))=1\operatorname{card} \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)=1 iff Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp} is rho_(d)\rho_{d}-Chebyshevian;
b) cardE_( bar(d))(F|_(Y))=1\operatorname{card} \mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right)=1 iff Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp} is rho_( bar(d))\rho_{\bar{d}}-Chebyshevian.
Corollary 5. Let F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) and Y sub XY \subset X such that y_(0)in Yy_{0} \in Y. Then
a) cardE_(d)(F|_(Y))=1\operatorname{card} \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)=1 iff Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp} is rho_(d)\rho_{d}-Chebyshevian;
b) cardE_( bar(d))(F|_(Y))=1\operatorname{card} \mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right)=1 iff Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp} is rho_( bar(d))\rho_{\bar{d}}-Chebyshevian.
Remark 6. Observe that the linear space Lambda_(alpha,0)(X)=Lambda_(alpha,0)(X,d)nnLambda_(alpha,0)(X, bar(d))\Lambda_{\alpha, 0}(X)=\Lambda_{\alpha, 0}(X, d) \cap \Lambda_{\alpha, 0}(X, \bar{d}) is a Banach space with respect to the norm
{:(28)||F|_(X)^(alpha)=max{||F|_(X,d)^(alpha),||F|_(X, bar(d))^(alpha)}:}\begin{equation*}
\|\left. F\right|_{X} ^{\alpha}=\max \left\{\left.\left\|\left.F\right|_{X, d} ^{\alpha},\right\| F\right|_{X, \bar{d}} ^{\alpha}\right\} \tag{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)X, d^{\alpha} ), vanishing at a fixed point y_(0)in Xy_{0} \in X. Obviously,
{:(29)||F||_(X)^(alpha)=s u p{(|F(x)-F(y)|)/(d^(alpha)(x,y)):d(x,y) > 0;x,y in X}:}\begin{equation*}
\|F\|_{X}^{\alpha}=\sup \left\{\frac{|F(x)-F(y)|}{d^{\alpha}(x, y)}: d(x, y)>0 ; x, y \in X\right\} \tag{29}
\end{equation*}
is a norm on Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X).
Corollary 7. For every element ff in the space Lambda_(alpha,0)(Y)=Lambda_(alpha,0)(Y,d)nnLambda_(alpha,0)(Y, bar(d))\Lambda_{\alpha, 0}(Y)=\Lambda_{\alpha, 0}(Y, d) \cap \Lambda_{\alpha, 0}(Y, \bar{d}) there exists F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) such that
Corollary 7. For every element ff in the space Lambda_(alpha,0)(Y)=Lambda_(alpha,0)(Y,d)nnLambda_(alpha,0)(Y, bar(d))\Lambda_{\alpha, 0}(Y)=\Lambda_{\alpha, 0}(Y, d) \cap \Lambda_{\alpha, 0}(Y, \bar{d}) there exists F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) such that
F|_(Y)=f" and "||F||_(X)^(alpha)=||f||_(Y)^(alpha)\left.F\right|_{Y}=f \text { and }\|F\|_{X}^{\alpha}=\|f\|_{Y}^{\alpha}
The set of all extensions of f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y) preserving the norm ||f||_(Y)^(alpha)\|f\|_{Y}^{\alpha} (of the form (29), is denoted by E(f)\mathcal{E}(f), i.e.,
{:(30)E(f):={F inLambda_(alpha,0)(X):F|_(Y)=f" and "||F||_(X)^(alpha)=||f||_(Y)^(alpha)}.:}\begin{equation*}
\mathcal{E}(f):=\left\{F \in \Lambda_{\alpha, 0}(X):\left.F\right|_{Y}=f \text { and }\|F\|_{X}^{\alpha}=\|f\|_{Y}^{\alpha}\right\} . \tag{30}
\end{equation*}
Denote by
{:(31)Y^(_|_):={G inLambda_(alpha,0)(X):G|_(Y)=0}.:}\begin{equation*}
Y^{\perp}:=\left\{G \in \Lambda_{\alpha, 0}(X):\left.G\right|_{Y}=0\right\} . \tag{31}
\end{equation*}
the annihilator of the set YY in Banach space Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X), and one considers the following problem of best approximation:
P. For F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) find G_(0)inY^(_|_)G_{0} \in Y^{\perp} such that
P. For F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) find G_(0)inY^(_|_)G_{0} \in Y^{\perp} such that
||F-G_(0)||_(X)^(alpha)=i n f{||F-G||_(X)^(alpha):G inY^(_|_)}=rho(F,Y^(_|_)).\left\|F-G_{0}\right\|_{X}^{\alpha}=\inf \left\{\|F-G\|_{X}^{\alpha}: G \in Y^{\perp}\right\}=\rho\left(F, Y^{\perp}\right) .
Corollary 8. The subspace Y^(_|_)Y^{\perp} is proximinal in Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X) and the set of elements of best approximation for F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X) is
P_(Y^(_|_))(F)=F-E(F|_(Y)).P_{Y^{\perp}}(F)=F-\mathcal{E}\left(\left.F\right|_{Y}\right) .
The distance of FF to Y^(_|_)Y^{\perp} is given by
rho(F,Y^(_|_))=||F|_(Y)||_(Y)^(alpha)\rho\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|_{Y}^{\alpha}
The subspace Y^(_|_)Y^{\perp} is Chebyshevian for FF iff cardE(F|_(Y))=1\operatorname{card} \mathcal{E}\left(\left.F\right|_{Y}\right)=1.
For ff in the linear space Lambda_(alpha,0)(Y)\Lambda_{\alpha, 0}(Y), the equalities F(f)(x)= bar(F)(f)(x),x in XF(f)(x)=\bar{F}(f)(x), x \in X and G(f)(x)= bar(G)(f)(x),x in XG(f)(x)=\bar{G}(f)(x), x \in X are verified iff ||f|_(Y,d)^(alpha)=||f|_(Y, bar(d))^(alpha)\left.\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha}. This means that ||f|_(Y,d)^(alpha)=||-f|_(Y,d)^(alpha)\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, d} ^{\alpha} and, consequently,
For ff in the linear space Lambda_(alpha,0)(Y)\Lambda_{\alpha, 0}(Y), the equalities F(f)(x)= bar(F)(f)(x),x in XF(f)(x)=\bar{F}(f)(x), x \in X and G(f)(x)= bar(G)(f)(x),x in XG(f)(x)=\bar{G}(f)(x), x \in X are verified iff ||f|_(Y,d)^(alpha)=||f|_(Y, bar(d))^(alpha)\left.\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha}. This means that ||f|_(Y,d)^(alpha)=||-f|_(Y,d)^(alpha)\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, d} ^{\alpha} and, consequently,
||f||_(Y)^(alpha)=max{||f|_(Y,d)^(alpha);||f|_(Y, bar(d))^(alpha)}=||f|_(Y,d)^(alpha)\|f\|_{Y}^{\alpha}=\max \left\{\left.\left\|\left.f\right|_{Y, d} ^{\alpha} ;\right\| f\right|_{Y, \bar{d}} ^{\alpha}\right\}=\|\left. f\right|_{Y, d} ^{\alpha}
By Theorem 3 in [14], it follows that Lambda_(a,0)(Y)\Lambda_{a, 0}(Y) is a Banach space and ( Y,d^(alpha)Y, d^{\alpha} ) is a metric space.
REFERENCES
[1] S. Cobzaş, Phelps type duality reuslts in best approximation, Rev. Anal. Numér. Théor. Approx., 31, no. 1., pp. 29-43, 2002. 짗
[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. 837842, 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. 짖
[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. 줄
[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, SpringerVerlag, Berlin, 1975.
[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. 837842, 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. 짖
[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. 줄
[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, SpringerVerlag, Berlin, 1975.
Received by the editors: April 13, 2010.
- *"T. Popoviciu" Institute of Numerical Analysis, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro, cmustata2001@yahoo.com.