Abstract
The aim of this note is to prove an extension theorem for semi-Lipschitz real functions defined on quasi-metric spaces, similar to McShane extension theorem for real-valued Lipschitz functions defined on a metric space ([2], [4]).
Authors
Costica Mustata
“Tibeiru Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Paper coordinates
C. Mustăţa, Extension of semi Lipschitz function on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx. 30 (2001) nr. 1, 61-67, https://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art8
About this paper
Journal
Revue d’Analyse Numer. Theor. Approx.
Publisher Name
Publishing Romanian Academy
Print ISSN
2457-6794
Online ISSN
2501-059X
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[1] S. Cobzas and C. Mustata, Norm preserving extension of convex Lipschitz functions,J. Approx. Theory,29(1978), 555–569.
[2] J. Czipserand L. Geher, Extension of functions satisfying a Lipschitz condition, ActaMath. Sci. Hungar.,6(1955), 213–220.
[3] P. Fletcherand W. F. Lindgren, Quasi-Uniform Spaces, Dekker, New York, 1982.
[4] J. A. McShane, Extension of range of functions, Bull. Amer. Math. Soc.,40(1939),837–842.
[5] C. Mustata, Best approximation and unique extension of Lipschitz functions, J. Ap-prox. Theory,19(1977), 222–230.
[6] S. Romaguera and M. Sanchis,Semi-Lipschitz functions and best approximation inquasi-metric spaces, J. Approx. Theory,103(2000), 292–301.
[7] J. H. Wellsand L. R. Williams, Embeddings and Extensions in Analysis, Springer-Verlag, Berlin, 1975
[3] P. Fletcherand W. F. Lindgren, Quasi-Uniform Spaces, Dekker, New York, 1982.
[4] J. A. McShane, Extension of range of functions, Bull. Amer. Math. Soc.,40(1939),837–842.
[5] C. Mustata, Best approximation and unique extension of Lipschitz functions, J. Ap-prox. Theory,19(1977), 222–230.
[6] S. Romaguera and M. Sanchis,Semi-Lipschitz functions and best approximation inquasi-metric spaces, J. Approx. Theory,103(2000), 292–301.
[7] J. H. Wellsand L. R. Williams, Embeddings and Extensions in Analysis, Springer-Verlag, Berlin, 1975
Paper (preprint) in HTML form
EXTENSIONS OF SEMI-LIPSCHITZ FUNCTIONS ON QUASI-METRIC SPACES
Abstract
The aim of this note is to prove an extension theorem for semiLipschitz real functions defined on quasi-metric spaces, similar to McShane extension theorem for real-valued Lipschitz functions defined on a metric space ([2], [4]).
MSC 2000. 46A22, 26A16, 26A48.
1. INTRODUCTION
Let XX be a nonvoid set. A quasi-metric on XX is a function d:X xx X rarr[0,oo)d: X \times X \rightarrow [0, \infty) satisfying the conditions
{:(i)d(x","y)=d(y","x)=0Longleftrightarrow x=y;quad x","y in X",":}\begin{equation*}
d(x, y)=d(y, x)=0 \Longleftrightarrow x=y ; \quad x, y \in X, \tag{i}
\end{equation*}
(ii)
d(x,y) <= d(x,z)+d(z,y),quad x,y,z in X.d(x, y) \leq d(x, z)+d(z, y), \quad x, y, z \in X .
If dd is a quasi-metric on XX, then the pair ( X,dX, d ) is called a quasi-metric space.
The conjugate of quasi-metric dd, denoted by d^(-1)d^{-1} is defined by d^(-1)(x,y)=d(y,x),x,y in Xd^{-1}(x, y)= d(y, x), x, y \in X.
Obviously the function d^(s):X xx X rarr[0,oo)d^{s}: X \times X \rightarrow[0, \infty) defined by
d^(s)(x,y)=max{d(x,y),d^(-1)(x,y)};quad x,y in Xd^{s}(x, y)=\max \left\{d(x, y), d^{-1}(x, y)\right\} ; \quad x, y \in X
is a metric on XX.
If the quasi-metric dd can take the value +oo+\infty, then it is called an extended quasi-metric.
Let ( X,dX, d ) be a quasi-metric space. A function f:X rarrRf: X \rightarrow \mathbb{R} is called semiLipschitz if there exists a constant K >= 0K \geq 0 so that
{:(1)f(x)-f(y) <= K*d(x","y):}\begin{equation*}
f(x)-f(y) \leq K \cdot d(x, y) \tag{1}
\end{equation*}
for all x,y in Xx, y \in X. The number K >= 0K \geq 0 in (1) is called a semi-Lipschitz constant for ff.
For a quasi-metric space ( X,dX, d ) the real-valued function f:X rarrRf: X \rightarrow \mathbb{R} is said to be <= _(d)\leq_{d}-increasing if
{:(2)d(x","y)=0quad" implies "quad f(x)-f(y) <= 0","quad x","y in X:}\begin{equation*}
d(x, y)=0 \quad \text { implies } \quad f(x)-f(y) \leq 0, \quad x, y \in X \tag{2}
\end{equation*}
or equivalently,
{:(3)f(x)-f(y) > 0quad" implies "quad d(x","y) > 0","quad x","y in X.:}\begin{equation*}
f(x)-f(y)>0 \quad \text { implies } \quad d(x, y)>0, \quad x, y \in X . \tag{3}
\end{equation*}
Note that every semi-Lipschitz function on quasi-metric space (X,d)(X, d) is <= _(d^(-))\leq_{d^{-}} increasing (see (1)).
For a semi-Lipschitz function f:X rarrRf: X \rightarrow \mathbb{R}, where ( X,dX, d ) is a quasi-metric space, denote by ||f||_(d)\|f\|_{d} the constant:
{:(4)||f||_(d)=s u p{((f(x)-f(y))vv0)/(d(x,y)):d(x,y) > 0,quad x,y in X}:}\begin{equation*}
\|f\|_{d}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: d(x, y)>0, \quad x, y \in X\right\} \tag{4}
\end{equation*}
Theorem 1. Let ( X,dX, d ) a quasi-metric space and f:X rarrRf: X \rightarrow \mathbb{R} a semiLipschitz function. Then ||f||_(d)\|f\|_{d} defined by (4) is the smallest semi-Lipschitz constant for ff.
Proof. If f:X rarrRf: X \rightarrow \mathbb{R} is semi-Lipschitz, then ff is <= _(d)\leq_{d}-increasing, and then f(x)-f(y) > 0f(x)-f(y)>0 implies d(x,y) > 0d(x, y)>0. It follows that
((f(x)-f(y))vv0)/(d(x,y))=(f(x)-f(y))/(d(x,y)) > 0\frac{(f(x)-f(y)) \vee 0}{d(x, y)}=\frac{f(x)-f(y)}{d(x, y)}>0
The inequalities f(x)-f(y) <= 0f(x)-f(y) \leq 0 and d(x,y) > 0d(x, y)>0 imply
((f(x)-f(y))vv0)/(d(x,y))=0\frac{(f(x)-f(y)) \vee 0}{d(x, y)}=0
Consequently ||f||_(d) >= 0\|f\|_{d} \geq 0.
For f(x)-f(y) < 0f(x)-f(y)<0 it follows (f(x)-f(y))//d(x,y) <= ||f||_(d)(f(x)-f(y)) / d(x, y) \leq\|f\|_{d} and obviously for f(x)-f(y) <= 0f(x)-f(y) \leq 0 we have f(x)-f(y) <= 0 <= ||f||_(d)*d(x,y)f(x)-f(y) \leq 0 \leq\|f\|_{d} \cdot d(x, y).
For f(x)-f(y) < 0f(x)-f(y)<0 it follows (f(x)-f(y))//d(x,y) <= ||f||_(d)(f(x)-f(y)) / d(x, y) \leq\|f\|_{d} and obviously for f(x)-f(y) <= 0f(x)-f(y) \leq 0 we have f(x)-f(y) <= 0 <= ||f||_(d)*d(x,y)f(x)-f(y) \leq 0 \leq\|f\|_{d} \cdot d(x, y).
Consequently
f(x)-f(y) <= ||f||_(d)*d(x,y)f(x)-f(y) \leq\|f\|_{d} \cdot d(x, y)
for all x,y in Xx, y \in X.
Now let K >= 0K \geq 0 such that
Now let K >= 0K \geq 0 such that
f(x)-f(y) <= K*d(x,y),quad" for all "x,y in X.f(x)-f(y) \leq K \cdot d(x, y), \quad \text { for all } x, y \in X .
The function ff is <= _(d)\leq_{d}-increasing, and then
((f(x)-f(y))vv0)/(d(x,y))={[(f(x)-f(y))/(d(x,y)) <= K","," if "f(x)-f(y) > 0","],[0 <= K","," if "f(x)-f(y) <= 0","]:}\frac{(f(x)-f(y)) \vee 0}{d(x, y)}= \begin{cases}\frac{f(x)-f(y)}{d(x, y)} \leq K, & \text { if } f(x)-f(y)>0, \\ 0 \leq K, & \text { if } f(x)-f(y) \leq 0,\end{cases}
Consequently ||f||_(d) <= K\|f\|_{d} \leq K.
For a quasi-metric ( X,dX, d ) let us consider the set:
{:(5)S Lip X={f:X rarrR∣f" is " <= _(d)"-increasing, "s u p_(d(x,y)!=0)((f(x)-f(y))vv0)/(d(x,y)) < oo}.:}\begin{equation*}
S \operatorname{Lip} X=\left\{f: X \rightarrow \mathbb{R} \mid f \text { is } \leq_{d} \text {-increasing, } \sup _{d(x, y) \neq 0} \frac{(f(x)-f(y)) \vee 0}{d(x, y)}<\infty\right\} . \tag{5}
\end{equation*}
It is straightforward to see that SS Lip XX is exactly the set of all semiLipschitz functions on (X,d)(X, d) (see [6]).
2. EXTENSIONS OF SEMI-LIPSCHITZ FUNCTIONS
Let Y sub XY \subset X where ( X,dX, d ) is a quasi-metric space. Then ( Y,dY, d ) is a quasimetric space with the quasi-metric induced by dd (denoted by dd too). Let us denote by SS Lip YY the set of all semi-Lipschitz functions defined on YY and let
{:(6)||f||_(d)=s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in Y,d(x,y)!=0}:}\begin{equation*}
\|f\|_{d}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in Y, d(x, y) \neq 0\right\} \tag{6}
\end{equation*}
be the smallest semi-Lipschitz constant for f in Sf \in S Lip YY.
If f in Sf \in S Lip YY, a function F in SF \in S Lip XX is called an extension (preserving the smallest semi-Lipschitz constant) of ff if:
If f in Sf \in S Lip YY, a function F in SF \in S Lip XX is called an extension (preserving the smallest semi-Lipschitz constant) of ff if:
{:(7)F|_(Y)=f quad" and "quad||F||_(d)=||f||_(d).:}\begin{equation*}
\left.F\right|_{Y}=f \quad \text { and } \quad\|F\|_{d}=\|f\|_{d} . \tag{7}
\end{equation*}
Denote by E_(Y)(f)E_{Y}(f) the set of all extensions of the function f in Sf \in S Lip YY, i.e.
{:(8)E_(Y)(f)={F in S Lip X:F|_(Y)=f" and "||F||_(d)=||f||_(d)}:}\begin{equation*}
E_{Y}(f)=\left\{F \in S \operatorname{Lip} X:\left.F\right|_{Y}=f \text { and }\|F\|_{d}=\|f\|_{d}\right\} \tag{8}
\end{equation*}
Theorem 2. Let ( X,dX, d ) be a quasi-metric space and YY a nonvoid subset of XX. Then for every f in Sf \in S Lip YY the set E_(Y)(f)E_{Y}(f) is nonvoid.
Proof. Let f in Sf \in S Lip YY and the constant ||f||_(d)\|f\|_{d} defined by (6).
Consider the function
Consider the function
{:(9)F(x)=i n f_(y in Y){f(y)+||f||_(d)d(x,y)}","x in X:}\begin{equation*}
F(x)=\inf _{y \in Y}\left\{f(y)+\|f\|_{d} d(x, y)\right\}, x \in X \tag{9}
\end{equation*}
a) First we show that FF is well defined.
Let z in Yz \in Y and x in Xx \in X. For any y in Yy \in Y we have
{:[f(y)+||f||_(d)d(x","y)=f(z)+||f||_(d)d(x","y)-(f(z)-f(y))],[ >= f(z)+||f||_(d)d(x","y)-||f||_(d)d(z","y)],[=f(z)-||f||_(d)(d(z","y)-d(x","y))]:}\begin{aligned}
f(y)+\|f\|_{d} d(x, y) & =f(z)+\|f\|_{d} d(x, y)-(f(z)-f(y)) \\
& \geq f(z)+\|f\|_{d} d(x, y)-\|f\|_{d} d(z, y) \\
& =f(z)-\|f\|_{d}(d(z, y)-d(x, y))
\end{aligned}
The inequality d(z,y)-d(x,y) <= d(z,x)=d^(-1)(x,z)d(z, y)-d(x, y) \leq d(z, x)=d^{-1}(x, z) implies
{:(10)f(y)+||f||_(d)d(x","y) >= f(z)-||f||_(d)*d^(-1)(x","z):}\begin{equation*}
f(y)+\|f\|_{d} d(x, y) \geq f(z)-\|f\|_{d} \cdot d^{-1}(x, z) \tag{10}
\end{equation*}
showing that for every x in Xx \in X the set {f(y)+||f||_(d)d(x,y):y in Y}\left\{f(y)+\|f\|_{d} d(x, y): y \in Y\right\} is bounded from above by f(z)-||f||_(d)d^(-1)(x,z)f(z)-\|f\|_{d} d^{-1}(x, z), and the infimum (9) is finite.
b) We show now that F(y)=f(y)F(y)=f(y) for all y in Yy \in Y.
b) We show now that F(y)=f(y)F(y)=f(y) for all y in Yy \in Y.
Let y in Yy \in Y. Then
F(y) <= f(y)+||f||_(d)d(y,y)=f(y).F(y) \leq f(y)+\|f\|_{d} d(y, y)=f(y) .
For any v in Yv \in Y we have
f(y)-f(v) <= ||f||_(d)*d(y,v)f(y)-f(v) \leq\|f\|_{d} \cdot d(y, v)
so that
f(v)+||f||_(d)*d(y,v) >= f(y)f(v)+\|f\|_{d} \cdot d(y, v) \geq f(y)
and
F(y)=i n f{f(v)+||f||_(d)d(y,v):v in Y} >= f(y)F(y)=\inf \left\{f(v)+\|f\|_{d} d(y, v): v \in Y\right\} \geq f(y)
It follows F(y)=f(y)F(y)=f(y).
c) We prove that ||F||_(d)=||f||_(d)\|F\|_{d}=\|f\|_{d}.
c) We prove that ||F||_(d)=||f||_(d)\|F\|_{d}=\|f\|_{d}.
Since F|_(Y)=f\left.F\right|_{Y}=f, the definitions of ||F||_(d)\|F\|_{d} and ||f||_(d)\|f\|_{d} yield ||F||_(d) >= ||f||_(d)\|F\|_{d} \geq\|f\|_{d}.
Let x_(1),x_(2)in Xx_{1}, x_{2} \in X and epsi > 0\varepsilon>0. Choosing y in Yy \in Y such that
Let x_(1),x_(2)in Xx_{1}, x_{2} \in X and epsi > 0\varepsilon>0. Choosing y in Yy \in Y such that
F(x_(1)) >= f(y)+||f||_(d)d(x_(1),y)-epsiF\left(x_{1}\right) \geq f(y)+\|f\|_{d} d\left(x_{1}, y\right)-\varepsilon
we obtain
{:[F(x_(2))-F(x_(1)) <= f(y)+||f||_(d)d(x_(2),y)-(f(y)+||f||_(d)*d(x_(1),y)-epsi)],[=||f||_(d)[d(x_(2),y)-d(x_(1),y)]+epsi],[ <= ||f||_(d)*d(x_(2),x_(1))+epsi.]:}\begin{aligned}
F\left(x_{2}\right)-F\left(x_{1}\right) & \leq f(y)+\|f\|_{d} d\left(x_{2}, y\right)-\left(f(y)+\|f\|_{d} \cdot d\left(x_{1}, y\right)-\varepsilon\right) \\
& =\|f\|_{d}\left[d\left(x_{2}, y\right)-d\left(x_{1}, y\right)\right]+\varepsilon \\
& \leq\|f\|_{d} \cdot d\left(x_{2}, x_{1}\right)+\varepsilon .
\end{aligned}
Since epsi > 0\varepsilon>0 is arbitrary, it follows
F(x_(2))-F(x_(1)) <= ||f||_(d)*d(x_(2),x_(1))F\left(x_{2}\right)-F\left(x_{1}\right) \leq\|f\|_{d} \cdot d\left(x_{2}, x_{1}\right)
for any x_(1),x_(2)in Xx_{1}, x_{2} \in X and ||F||_(d) <= ||f||_(d)\|F\|_{d} \leq\|f\|_{d}.
d) The function FF is <= _(d)\leq_{d}-increasing.
d) The function FF is <= _(d)\leq_{d}-increasing.
Indeed, let be u,v in Xu, v \in X and d(u,v)=0d(u, v)=0. We have d(u,y) <= d(u,v)+d(v,y)d(u, y) \leq d(u, v)+d(v, y). Consequently
d(u,y) <= d(v,y)d(u, y) \leq d(v, y)
Then
f(y)+||f||_(d)d(u,y) <= f(y)+||f||_(d)d(v,y).f(y)+\|f\|_{d} d(u, y) \leq f(y)+\|f\|_{d} d(v, y) .
It follows that
F(u) <= F(v),F(u) \leq F(v),
and consequently d(u,v)=0d(u, v)=0 implies F(u) <= F(v)F(u) \leq F(v).
It follows that F inE_(Y)^(d)(f)F \in E_{Y}^{d}(f) so that E_(Y)^(d)(t)!=O/E_{Y}^{d}(t) \neq \emptyset.
Remarks 1. 1^(0)1^{0} Similarly, the function
It follows that F inE_(Y)^(d)(f)F \in E_{Y}^{d}(f) so that E_(Y)^(d)(t)!=O/E_{Y}^{d}(t) \neq \emptyset.
Remarks 1. 1^(0)1^{0} Similarly, the function
{:(11)G(x)=s u p_(y in Y){f(y)-||f||_(d)d^(-1)(x,y)}:}\begin{equation*}
G(x)=\sup _{y \in Y}\left\{f(y)-\|f\|_{d} d^{-1}(x, y)\right\} \tag{11}
\end{equation*}
is <= _(d)\leq_{d}-increasing, and GG belongs to E_(Y)^(d)(f)E_{Y}^{d}(f) too.
2^(0)2^{0} The inequality
2^(0)2^{0} The inequality
{:(12)G(x) <= F(x)",":}\begin{equation*}
G(x) \leq F(x), \tag{12}
\end{equation*}
holds for every x in Xx \in X.
Indeed, taking the infimum with respect to z in Yz \in Y and then the supremum with respect to y in Yy \in Y in (10) we find
Indeed, taking the infimum with respect to z in Yz \in Y and then the supremum with respect to y in Yy \in Y in (10) we find
G(x)=s u p_(y in Y){f(y)-||f||_(d)d^(-1)(x,y)} <= i n f_(z in Y){f(z)+||f||_(d)d(x,z)}=F(x)G(x)=\sup _{y \in Y}\left\{f(y)-\|f\|_{d} d^{-1}(x, y)\right\} \leq \inf _{z \in Y}\left\{f(z)+\|f\|_{d} d(x, z)\right\}=F(x)
In fact, the following theorem holds:
Theorem 3. Let ( X,dX, d ) be a quasi-metric space, YY a nonvoid subset of XX and f in Sf \in S Lip YY.
Theorem 3. Let ( X,dX, d ) be a quasi-metric space, YY a nonvoid subset of XX and f in Sf \in S Lip YY.
Then for any H inE_(Y)^(d)(f)H \in E_{Y}^{d}(f) we have
{:(13)G(x) <= H(x) <= F(x)","quad x in X.:}\begin{equation*}
G(x) \leq H(x) \leq F(x), \quad x \in X . \tag{13}
\end{equation*}
Proof. Let H inE_(Y)^(d)(f)H \in E_{Y}^{d}(f). For arbitrary x in Xx \in X and y in Yy \in Y we have
H(x)-H(y) <= ||f||_(d)d(x,y)H(x)-H(y) \leq\|f\|_{d} d(x, y)
implying
H(x) <= H(y)+||f||_(d)d(x,y)=f(y)+||f||_(d)(x,y)H(x) \leq H(y)+\|f\|_{d} d(x, y)=f(y)+\|f\|_{d}(x, y)
Taking the imfimum with respect to y in Yy \in Y we get
H(x) <= i n f_(y in Y){f(y)+||f||_(d)d(x,y)}=F(x)H(x) \leq \inf _{y \in Y}\left\{f(y)+\|f\|_{d} d(x, y)\right\}=F(x)
The inequality H(x) >= G(x),x in XH(x) \geq G(x), x \in X can be proved similarly.
Corollary 4. A function f in Sf \in S Lip YY has a unique extension in SS Lip XX if and only if the following relation
Corollary 4. A function f in Sf \in S Lip YY has a unique extension in SS Lip XX if and only if the following relation
{:(14)i n f_(y in Y){f(y)+||f||_(d)d(x,y)}=s u p_(y in Y){f(y)-||f||d(y","x)}:}\begin{equation*}
\inf _{y \in Y}\left\{f(y)+\|f\|_{d} d(x, y)\right\}=\sup _{y \in Y}\{f(y)-\|f\| d(y, x)\} \tag{14}
\end{equation*}
holds for every x in Xx \in X.
Example.
Let R\mathbb{R} be the real axis and d:RxxRrarr[0,oo)d: \mathbb{R} \times \mathbb{R} \rightarrow[0, \infty) the quasi-metric defined by
Example.
Let R\mathbb{R} be the real axis and d:RxxRrarr[0,oo)d: \mathbb{R} \times \mathbb{R} \rightarrow[0, \infty) the quasi-metric defined by
d(x,y)={[x-y","," if "quad x >= y],[1","," if "quad x < y]:}d(x, y)=\left\{\begin{array}{cc}
x-y, & \text { if } \quad x \geq y \\
1, & \text { if } \quad x<y
\end{array}\right.
Let YY be given by Y=[0,1]subRY=[0,1] \subset \mathbb{R} and f:Y rarrR,f(y)=2yf: Y \rightarrow \mathbb{R}, f(y)=2 y. Then ff is semi-Lipschitz on YY and ||f||_(d)=2\|f\|_{d}=2. The extension FF defined by (9) is
F(x)={[2","," if ",x < 0],[2x","," if ",x >= 0]:}F(x)=\left\{\begin{array}{cll}
2, & \text { if } & x<0 \\
2 x, & \text { if } & x \geq 0
\end{array}\right.
and the extension GG defined by (11) is
G(x)={[2x",",x <= 1],[0",",x > 1]:}G(x)=\left\{\begin{array}{cl}
2 x, & x \leq 1 \\
0, & x>1
\end{array}\right.
Obviously, G(x) <= F(x),x inRG(x) \leq F(x), x \in \mathbb{R}.
REFERENCES
[1] S. Cobzas and C. Mustăţa, Norm preserving extension of convex Lipschitz functions, J. Approx. Theory, 29 (1978), 555-569.
[2] J. Czipser and L. Gehér, Extension of functions satisfying a Lipschitz condition, Acta Math. Sci. Hungar., 6 (1955), 213-220.
[3] P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Dekker, New York, 1982.
[4] J. A. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1939), 837-842.
[5] C. Mustăţa, Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19 (1977), 222-230.
[6] S. Romaguera and M. Sanchis, Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103 (2000), 292-301.
[7] J. H. WellS and L. R. Williams, Embeddings and Extensions in Analysis, SpringerVerlag, Berlin, 1975.
Received: August 8, 2000.
[2] J. Czipser and L. Gehér, Extension of functions satisfying a Lipschitz condition, Acta Math. Sci. Hungar., 6 (1955), 213-220.
[3] P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Dekker, New York, 1982.
[4] J. A. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1939), 837-842.
[5] C. Mustăţa, Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19 (1977), 222-230.
[6] S. Romaguera and M. Sanchis, Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103 (2000), 292-301.
[7] J. H. WellS and L. R. Williams, Embeddings and Extensions in Analysis, SpringerVerlag, Berlin, 1975.
Received: August 8, 2000.
- "T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania, e-mail: cmustata@ictp-acad.math.ubbcluj.ro.