In this paper we consider the problem of best uniform approximation of a real valued semi-Lipschitz function \(F\) defined on an asymmetric metric space \((X,d)\), by the elements of the set \(E_{d}(F|_{Y})\) of all extensions of \(F|_{Y}(Y\subset X)\), preserving the smallest semi-Lipschitz constant. It is proved that, this problem has always at least a solution, if \((X,d)\) is \((d,\overline{d})\)-sequentially compact, or of finite diameter.
Authors
Costică Mustăţa
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academi, Romania
Keywords
Semi-Lipschitz functions; uniform approximation; extensions of semi-Lipschitz functions.
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[3] Cobzas, S. and Mustata, C., Best approximation in spaces with asymmetric norm,. Rev. Anal. Numer. Theor. Approx., 33 (1), pp. 17–31, 2006.
[4] Cobzas, S. and Mustata, C., Extension of bounded linear functionals and best approximation in spaces with asymmetric norm, Rev. Anal. Numer. Theor. Approx., 31, 1, pp. 35–50, 2004.
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[6] Garcia-Raffi, L. M., Romaguera, S. and Sanchez-Perez, E. A., The dual space of an asymmetric linear space, Quaest. Math., 26, pp. 83–96, 2003.
[7] Kunzi, H. P. A., Nonsymmetric distances and their associated topologies: about the origin of basic ideas in the area of asymmetric topologies, in: Handbook of the History of General Topology, ed. by C.E. Aull and R. Lower, 3, Hist. Topol. 3, Kluwer Acad. Publ. Dordrecht, pp. 853–968, 2001.
[8] Mc.Shane, E. T., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837–842, 1934.
[9] Menucci, A., On asymmetric distances, Technical Report, Scuola Normale Superiore, Pisa, 2004.
[10] Mustata, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx, 30, 1, pp. 61–67, 2001.
[11] Mustata, C., On the extremal semi-Lipschitz functions, Rev. Anal. Numer. Theor. Approx, 31, 1, pp. 103–108, 2002.
[12] Mustata, C., On the approximation of the global extremum of a semi-Lipschitz function, IJMMS (to appear).
[13] Reilly, I.L., Subrahmanyam, P. V. and Vamanamurthy, M. K., Cauchy sequences in quasi-pseudo-metric spaces, Mh. Math., 93 , pp. 127–140, 1982.
[14] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292–301, 2000.
[15] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitz functions, Acta Math. Hungar., 108(1-2), pp. 55–70, 2005.
[16] Romaguera, S., Sanchez-Alvarez, J.M. and Sanchis, M., El espacio de funciones semi-Lipschitz, VI Jornadas de Matematica Aplicada, Universiadad Politecnica de Valencia, pp. 1–15, 2005.
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Paper (preprint) in HTML form
2007-Mustata-Best uniform approximation of semi-Lipschitz-Jnaat
BEST UNIFORM APPROXIMATION OF SEMI-LIPSCHITZ FUNCTIONS BY EXTENSIONS*
COSTICĂ MUSTĂŢA ^(†){ }^{\dagger}
Abstract
In this paper we consider the problem of best uniform approximation of a real valued semi-Lipschitz function FF defined on an asymmetric metric space (X,d)(X, d), by the elements of the set E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right) of all extensions of F|_(Y)(Y sub X)\left.F\right|_{Y}(Y \subset X), preserving the smallest semi-Lipschitz constant. It is proved that, this problem has always at least a solution, if ( X,dX, d ) is ( d, bar(d)d, \bar{d} )-sequentially compact, or of finite diameter.
Let XX be a non-empty set. A function d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty) is called a quasi-metric on X 14X 14 if the following conditions hold:
d(x,z) <= d(x,y)+d(y,z)d(x, z) \leq d(x, y)+d(y, z), for all x,y,z in Xx, y, z \in X.
The function bar(d):X xx X rarr[0,oo)\bar{d}: X \times X \rightarrow[0, \infty) defined by bar(d)(x,y)=d(y,x)\bar{d}(x, y)=d(y, x), for all x,y in Xx, y \in X is also a quasi-metric on XX, called the conjugate quasi-metric of dd.
A pair ( X,dX, d ) where XX is a non-empty set and dd a quasi-metric on XX, is called a quasi-metric space.
If dd can take the value +oo+\infty, then it is called a quasi-distance on XX.
Each quasi-metric dd on XX induces a topology tau(d)\tau(d) which has as a basis the family of balls (forward open balls [5])
{:(1)B^(+)(x","epsi):={y in X:d(x","y) < epsi}","x in X","epsi > 0.:}\begin{equation*}
B^{+}(x, \varepsilon):=\{y \in X: d(x, y)<\varepsilon\}, x \in X, \varepsilon>0 . \tag{1}
\end{equation*}
This topology is called the forward topology of XX ([5], [9]), and is denoted also by tau_(+)\tau_{+}.
Observe that the topology tau_(+)\tau_{+}is a T_(0)T_{0}-topology. If the condition 1) is replaced by 1^(')1^{\prime} ) d(x,y)=0d(x, y)=0 iff x=yx=y, then the topology tau_(+)\tau_{+}is a T_(1)T_{1}-topology (see [14, [15]).
Analogously, the quasi-metric bar(d)\bar{d} induces the topology tau( bar(d))\tau(\bar{d}) on XX, which has as a basis the family of backward open balls (5)
{:(2)B^(-)(x","epsi):={y in X:d(y","x) < epsi}","x in X","epsi > 0:}\begin{equation*}
B^{-}(x, \varepsilon):=\{y \in X: d(y, x)<\varepsilon\}, x \in X, \varepsilon>0 \tag{2}
\end{equation*}
This topology is called the backward topology of XX ([5], [9]) and is denoted also by tau_(-)\tau_{-}.
For more information about quasi-metric spaces and their applications see, for example, the papers [5, 66, 7, 9], 14) and the references quoted therein.
Let ( X,dX, d ) be a quasi-metric space. A sequence (x_(k))_(k >= 1)sub X\left(x_{k}\right)_{k \geq 1} \subset X is called dd-convergent (forward convergent) to x_(0)in Xx_{0} \in X, respectively bar(d)\bar{d}-convergent (backward convergent) to x_(0)in Xx_{0} \in X iff
(see 5], Definition 2.4)
A subset KK of XX is called dd-compact (forward compact) if every open cover of KK with respect to the forward topology tau_(+)\tau_{+}has a finite subcover. We say that a subset KK of XX is dd-sequentially compact (forward-sequentially compact) if every sequence in KK has a dd-convergent (forward convergent) subsequence with limit in KK ([5], Definition 4.1).
The bar(d)\bar{d}-compact (backward compact) and bar(d)\bar{d}-sequentially compact (backward -sequentially compact) subset of XX - are defined in a similar way.
Finally, a subset YY of ( X,dX, d ) is called ( d, bar(d)d, \bar{d} )-sequentially compact if every sequence (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1} in YY has a subsequence (y_(n_(k)))_(k >= 1),d\left(y_{n_{k}}\right)_{k \geq 1}, d-convergent to some u in Yu \in Y and bar(d)\bar{d}-convergent to some v in Yv \in Y. By Lemma 3.1 in [5] if follows that we can take u=vu=v in the definition of ( d, bar(d)d, \bar{d} )-sequentially compactness, if ( X,dX, d ) is a T_(1)T_{1} quasi-metric space. A subset YY of ( X,dX, d ) is called dd-bounded (forward bounded in [5]) if there exist x in Xx \in X and r > 0r>0, such that Y subB^(+)(x,r).YY \subset B^{+}(x, r) . Y is called dd-totally bounded if for every epsi > 0\varepsilon>0, there exists n inNn \in \mathbb{N}, and the forward balls B^(+)(y_(1),epsi),B^(+)(y_(2),epsi),dots,B_(n)(y_(n),epsi),y_(i)in Y,i= bar(1,n)B^{+}\left(y_{1}, \varepsilon\right), B^{+}\left(y_{2}, \varepsilon\right), \ldots, B_{n}\left(y_{n}, \varepsilon\right), y_{i} \in Y, i=\overline{1, n} such that Y subuuu_(i=1)^(n)B^(+)(y_(i),epsi)Y \subset \bigcup_{i=1}^{n} B^{+}\left(y_{i}, \varepsilon\right).
Similar definitions are given for bar(d)\bar{d}-boundedness and bar(d)\bar{d}-total boundedness of a subset YY of ( X,dX, d ).
2. THE CONE OF SEMI-LIPSCHITZ FUNCTIONS
Definition 1. [15] Let YY be a non-empty subset of a quasi-metric space (X,d)(X, d). A function f:Y rarrRf: Y \rightarrow \mathbb{R} is called dd-semi-Lipschitz if there exists a number L >= 0L \geq 0 (named a dd-semi-Lipschitz constant for ff ) such that
{:(4)f(x)-f(y) <= Ld(x","y):}\begin{equation*}
f(x)-f(y) \leq L d(x, y) \tag{4}
\end{equation*}
for all x,y in Yx, y \in Y.
A function f:Y rarrRf: Y \rightarrow \mathbb{R}, is called <= _(d)\leq_{d}-increasing if f(x) <= f(y)f(x) \leq f(y), whenever d(x,y)=0d(x, y)=0.
Denote by R_( <= d)^(Y)\mathbb{R}_{\leq d}^{Y} the set of all <= _(d)\leq_{d}-increasing functions on YY. This set is a cone in the linear space R^(Y)\mathbb{R}^{Y} of real valued functions defined on YY, i.e. for each f,g inR_( <= d)^(Y)f, g \in \mathbb{R}_{\leq d}^{Y} and lambda >= 0\lambda \geq 0 it follows that f+g inR_( <= d)^(Y)f+g \in \mathbb{R}_{\leq d}^{Y} and lambda f inR_( <= d)^(Y)\lambda f \in \mathbb{R}_{\leq d}^{Y}.
For a dd-semi-Lipschitz function ff on YY, put [14]:
{:(5)||f|_(d)=s u p{((f(x)-f(y))vv0)/(d(x,y)):d(x,y) > 0;x,y in Y}:}\begin{equation*}
\|\left. f\right|_{d}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: d(x, y)>0 ; x, y \in Y\right\} \tag{5}
\end{equation*}
Then ||f|_(d)\|\left. f\right|_{d} is the smallest dd-semi-Lipschitz constant of ff (see also [10, [15]).
For a fixed element theta in Y\theta \in Y denote
the set of all dd-semi-Lipschitz real valued functions defined on YY vanishing at the fixed element theta in Y\theta \in Y.
Observe that if ( X,dX, d ) is a T_(1)T_{1} quasi-metric space, then every real-valued function on XX is <= _(d)\leq_{d}-increasing [14].
The set d-SLip_(0)Yd-\operatorname{SLip}_{0} Y is a cone (a subcone of R_( <= d)^(Y)\mathbb{R}_{\leq d}^{Y} ) and the functional ||*|_(d):d-SLip_(0)Y rarr[0,oo)\|\left.\cdot\right|_{d}: d- \operatorname{SLip}_{0} Y \rightarrow[0, \infty) defined by (5) is subadditive and positive homogeneous on dd-SLip _(0)Y{ }_{0} Y. Moreover ||f|_(d)=0\|\left. f\right|_{d}=0 iff f=0f=0, and consequently ||*|_(d)\|\left.\cdot\right|_{d} is a quasi-norm (asymmetric norm) on the cone d-SLip_(0)Yd-\operatorname{SLip}_{0} Y.
In [15] some properties of the "normed cone" ( d-SLip_(0)Y,||*|_(d)d-\operatorname{SLip}_{0} Y, \|\left.\cdot\right|_{d} ) are presented. Similar properties in the case of dd-semi-Lipschitz functions on a quasi-metric space with values in a quasi-normed space (space with asymmetric norm) are discussed in [16], [17]. For more information concerning other properties of quasi-metric spaces, see also [7], [13].
Now, let ( X,dX, d ) be a quasi-metric space and let YY be a non-empty subset of XX. A real valued function ff defined on YY is called tau_(+)\tau_{+}-lower semi-continuous ( tau_(+)\tau_{+}-l.s.c in short) (respectively tau_(-)\tau_{-}-upper semi-continuous ( tau_(-)-\tau_{-}-u.s.c. ))) at x_(0)in Yx_{0} \in Y, if for every epsi > 0\varepsilon>0 there exists r > 0r>0 such that for every x inB^(+)(x_(0),r)x \in B^{+}\left(x_{0}, r\right) (respectively, for every {:x inB^(-)(x_(0),r)),f(x) > f(x_(0))-epsi\left.x \in B^{-}\left(x_{0}, r\right)\right), f(x)>f\left(x_{0}\right)-\varepsilon (respectively f(x) < {:f(x_(0))+epsi)f(x)< \left.f\left(x_{0}\right)+\varepsilon\right).
Proposition 2. Let ( X,dX, d ) be a quasi-metric space, theta in X\theta \in X a fixed element, and Y sube XY \subseteq X with theta in Y\theta \in Y. Then every f in d-SLip_(0)Yf \in d-\operatorname{SLip}_{0} Y is tau_(-)-\tau_{-}-u.s.c and tau_(+)\tau_{+}-l.s.c., and every f in bar(d)-SLip_(0)Yf \in \bar{d}-\operatorname{SLip}_{0} Y is tau_(+)-\tau_{+}-u.s.c. and tau_(-)-\tau_{-}-l.s.c. on YY.
Proof. Let f in d-SLip_(0)Yf \in d-\operatorname{SLip}_{0} Y such that ||f|_(d)=0\|\left. f\right|_{d}=0. Then f-=0f \equiv 0 and ff is tau_(-)-\tau_{-}-u.s.c. and tau_(+)\tau_{+}-l.s.c at every y in Yy \in Y.
Now, let ||f|_(d) > 0\|\left. f\right|_{d}>0 and y_(0)in Yy_{0} \in Y. The inequality
f(y)-f(y_(0)) <= ||f|_(d)d(y,y_(0)),y in Yf(y)-f\left(y_{0}\right) \leq \|\left. f\right|_{d} d\left(y, y_{0}\right), y \in Y
implies
f(y) <= f(y_(0))+||f|_(d)d(y,y_(0)),y in Y.f(y) \leq f\left(y_{0}\right)+\|\left. f\right|_{d} d\left(y, y_{0}\right), y \in Y .
for every epsi > 0\varepsilon>0 and every y inB^(-)(y_(0),(epsi)/(||f|_(d)))y \in B^{-}\left(y_{0}, \frac{\varepsilon}{\|\left. f\right|_{d}}\right), showing that ff is tau_(-)-\tau_{-}-u.s.c at y_(0)in Yy_{0} \in Y.
Similarly,
f(y_(0))-f(y) <= ||f|_(d)*d(y_(0),y),y in Y,f\left(y_{0}\right)-f(y) \leq \|\left. f\right|_{d} \cdot d\left(y_{0}, y\right), y \in Y,
for every y inB^(+)(Y_(0),(epsi)/(||f|_(d)))y \in B^{+}\left(Y_{0}, \frac{\varepsilon}{\|\left. f\right|_{d}}\right), showing that ff is tau_(+)\tau_{+}-l.s.c. in y_(0)in Yy_{0} \in Y.
Similarly one prove that every f in bar(d)-SLip_(0)Yf \in \bar{d}-\operatorname{SLip}_{0} Y is tau_(+)-\tau_{+}-u.s.c. and tau_(-)-\tau_{-}-l.s.c. on YY.
Observe that if ff is in d-SLip_(0)Yd-\operatorname{SLip}_{0} Y, then -f in bar(d)-SLip_(0)Y-f \in \bar{d}-\operatorname{SLip}_{0} Y, and -f-f is tau_(+)-\tau_{+}-u.s.c, and tau_(-)-\tau_{-}-l.s.c. on YY, i.e. if y_(0)in Yy_{0} \in Y then
AA epsi > 0,EE r > 0\forall \varepsilon>0, \exists r>0 such that (-f)(y) < (-f)(y_(0))+epsi(-f)(y)<(-f)\left(y_{0}\right)+\varepsilon, for all y inB^(+)(y_(0),r)y \in B^{+}\left(y_{0}, r\right), and respectively
AA epsi > 0,EE r > 0\forall \varepsilon>0, \exists r>0 such that (-f)(y) > (-f)(y_(0))-epsi(-f)(y)>(-f)\left(y_{0}\right)-\varepsilon, for all y inB^(-)(y_(0),r)y \in B^{-}\left(y_{0}, r\right).
Proposition 3. Let ( X,dX, d ) be a quasi-metric space, theta in X\theta \in X a fixed element, and Y sub XY \subset X, with theta in Y\theta \in Y.
(a) If YY is bar(d)\bar{d}-sequentially compact, then each f in df \in d-SLip _(0)Y{ }_{0} Y attains its maximum value on YY;
(b) If YY is dd - sequentially compact, then each f in df \in d-SLip _(0)Y{ }_{0} Y attains its minimum value on YY.
Proof. (a) Let YY be bar(d)\bar{d}-sequentially compact and M:=s u p f(Y)M:=\sup f(Y), where M inRuu{+oo}M \in \mathbb{R} \cup\{+\infty\}. Then there exists a sequence (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1} in YY such that lim_(n rarr oo)f(y_(n))=M\lim _{n \rightarrow \infty} f\left(y_{n}\right)= M. Because YY is bar(d)\bar{d}-sequentially compact, there exists y_(0)in Yy_{0} \in Y and a subsequence (y_(n_(k)))_(k >= 1)\left(y_{n_{k}}\right)_{k \geq 1} of (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1} such that lim_(n rarr oo)d(y_(n,k),y_(0))=0\lim _{n \rightarrow \infty} d\left(y_{n, k}, y_{0}\right)=0. By the tau_(-)-\tau_{-}-u.s.c. of ff at y_(0)y_{0} it follows:
M=lim_(k rarr oo)f(y_(n_(k)))=l i m s u p_(k)f(y_(n_(k))) <= f(y_(0))=M,M=\lim _{k \rightarrow \infty} f\left(y_{n_{k}}\right)=\limsup _{k} f\left(y_{n_{k}}\right) \leq f\left(y_{0}\right)=M,
implying M < ooM<\infty and f(y_(0))=Mf\left(y_{0}\right)=M.
(b) If f in df \in d-SLip _(0)Y{ }_{0} Y, it follows- f in bar(d)f \in \bar{d}-SLip _(0)Y{ }_{0} Y, and because YY is dd-sequentially compact, by (a), it follows that -f-f attains its maximum value on YY, i.e. ff attains its minimum value on YY.
Proposition 4. Let ( X,dX, d ) be a quasi-metric space, theta in X\theta \in X a fixed element, and Y sube XY \subseteq X with theta in Y\theta \in Y.
(a) If YY is bar(d)\bar{d}-sequentially compact, then the functional ||*|_(oo)^( bar(d)):d-SLip_(0)Y rarr[0,oo)\|\left.\cdot\right|_{\infty} ^{\bar{d}}: d-\operatorname{SLip}_{0} Y \rightarrow [0, \infty) defined by
{:(7)||f|_(oo)^( bar(d))=max{f(y):y in Y}:}\begin{equation*}
\|\left. f\right|_{\infty} ^{\bar{d}}=\max \{f(y): y \in Y\} \tag{7}
\end{equation*}
is an asymmetric norm on d-SLip_(0)Yd-\operatorname{SLip}_{0} Y.
(b) If YY is dd-sequentially compact, then the functional ||*|_(oo)^(d):d-SLip_(0)Y rarr[0,oo)\|\left.\cdot\right|_{\infty} ^{d}: d-\operatorname{SLip}_{0} Y \rightarrow [0, \infty) defined by
{:(8)||f|_(oo)^(d)=max{-f(y):y in Y}","f in d-SLip_(0)Y:}\begin{equation*}
\|\left. f\right|_{\infty} ^{d}=\max \{-f(y): y \in Y\}, f \in d-\operatorname{SLip}_{0} Y \tag{8}
\end{equation*}
is an asymmetric norm on d-SLip_(0)Yd-\operatorname{SLip}_{0} Y;
(c) If YY is (d, bar(d))(d, \bar{d})-sequentially compact, then the functional ||*|_(oo):d-SLip_(0)Y rarr[0,oo)\|\left.\cdot\right|_{\infty}: d-\operatorname{SLip}_{0} Y \rightarrow [0, \infty) defined by
{:(9)||f|_(oo)=||f|_(oo)^(d)vv||f|_(oo)^( bar(d))","f in d-SLip_(0)Y:}\begin{equation*}
\left.\left\|\left.f\right|_{\infty}=\right\| f\right|_{\infty} ^{d} \vee \|\left. f\right|_{\infty} ^{\bar{d}}, f \in d-\operatorname{SLip}_{0} Y \tag{9}
\end{equation*}
is the uniform norm on the cone d-SLip_(0)Yd-\operatorname{SLip}_{0} Y.
Proof. (a) By Proposition 3 (a), the functional (7) is well defined. For every f in df \in d - SLip_(0)Y\operatorname{SLip}_{0} Y, we have ||f|_(oo)^( bar(d)) >= f(theta)=0\|\left. f\right|_{\infty} ^{\bar{d}} \geq f(\theta)=0. If f in d-SLip_(0)Yf \in d-\operatorname{SLip}_{0} Y and ||f|_(oo)^(d) > 0\|\left. f\right|_{\infty} ^{d}>0 then there exists y_(0)in Yy_{0} \in Y such that f(y_(0))=||f|_(oo)^( bar(d)) > 0f\left(y_{0}\right)=\|\left. f\right|_{\infty} ^{\bar{d}}>0. It follows f!=0f \neq 0.
for all f,g in d-SLip_(0)Yf, g \in d-\operatorname{SLip}_{0} Y and lambda >= 0\lambda \geq 0.
(b) For every f in df \in d - SLip_(0)Y\operatorname{SLip}_{0} Y it follows that -f in bar(d)-SLip_(0)Y-f \in \bar{d}-\operatorname{SLip}_{0} Y, and because YY is dd-sequentially compact, then -f-f attains its maximum value on YY, and
||f|_(oo)^(d)=max{-f(y):y in Y}\|\left. f\right|_{\infty} ^{d}=\max \{-f(y): y \in Y\}
is an asymmetric norm on d-SLip_(0)Yd-\operatorname{SLip}_{0} Y.
(c) By Proposition 3, if YY is (d, bar(d))(d, \bar{d})-sequentially compact, then every f in df \in dSLip_(0)Y\operatorname{SLip}_{0} Y, attains its maximum and minimum value on YY.
We have
{:[||f||_(oo)=max{|f(y)|:y in Y}=],[=(max{f(y):y in Y})vv(max{-f(y):y in Y})],[=||f|_(oo)^(d)vv||f|_(oo)^( bar(d))]:}\begin{aligned}
\|f\|_{\infty} & =\max \{|f(y)|: y \in Y\}= \\
& =(\max \{f(y): y \in Y\}) \vee(\max \{-f(y): y \in Y\}) \\
& =\left.\left\|\left.f\right|_{\infty} ^{d} \vee\right\| f\right|_{\infty} ^{\bar{d}}
\end{aligned}
3. BEST UNIFORM APPROXIMATION BY EXTENSIONS
In the following the quasi-metric space (X,d)(X, d) is supposed (d, bar(d))(d, \bar{d})-sequentially compact. Let theta in X\theta \in X be a fixed element, and Y sube XY \subseteq X with theta in Y\theta \in Y. Consider also the normed cones ( d-SLip_(0)Y,||*|_(d)d-\operatorname{SLip}_{0} Y, \|\left.\cdot\right|_{d} ) and ( bar(d)-SLip_(0)X,||*|_( bar(d))\bar{d}-\operatorname{SLip}_{0} X, \|\left.\cdot\right|_{\bar{d}} ), where ||*|_( bar(d))\|\left.\cdot\right|_{\bar{d}} is the asymmetric norm defined as in (5), where dd is replaced by bar(d)\bar{d}.
An extension results for semi-Lipschitz functions, analogous to Mc Shane's Extension Theorem [8] for real-valued Lipschitz functions defined on a subset of a metric space was proved in [10] (see also [12]).
Proposition 5. 10] For every f in df \in d - SLip_(0)Y\operatorname{SLip}_{0} Y there exists at least one function F in d-SLip_(0)XF \in d-\mathrm{SLip}_{0} X, such that
{:(10)F|_(Y)=f" and "||F|_(d)=||f|_(d).:}\begin{equation*}
\left.F\right|_{Y}=f \text { and }\left.\left\|\left.F\right|_{d}=\right\| f\right|_{d} . \tag{10}
\end{equation*}
A function FF with the properties included in Proposition 5, is called an extension, preserving the asymmetric norm of ff (or an extension preserving the smallest semi-Lipschitz constant of ff ).
Denote the set of all extensions of ff preserving asymmetric norm, by
{:(11)E_(d)(f)={F in d-SLip_(0)X:F|_(Y)=f" and "||F|_(d)=||f|_(d)}:}\begin{equation*}
\mathcal{E}_{d}(f)=\left\{F \in d-\operatorname{SLip}_{0} X:\left.F\right|_{Y}=f \text { and }\left.\left\|\left.F\right|_{d}=\right\| f\right|_{d}\right\} \tag{11}
\end{equation*}
The set E_(d)(f)\mathcal{E}_{d}(f) is convex in dd-SLip _(0)X{ }_{0} X, the functions
{:(12)F_(d)(f)(x)=i n f{f(y)+||f|_(d)d(x,y):y in Y}","x in X:}\begin{equation*}
F_{d}(f)(x)=\inf \left\{f(y)+\|\left. f\right|_{d} d(x, y): y \in Y\right\}, x \in X \tag{12}
\end{equation*}
and
{:(13)G_(d)(f)(x)=s u p{f(y)-||f|_(d)*d(y,x):y in Y}","x in X:}\begin{equation*}
G_{d}(f)(x)=\sup \left\{f(y)-\|\left. f\right|_{d} \cdot d(y, x): y \in Y\right\}, x \in X \tag{13}
\end{equation*}
are extremal elements of E_(d)(f)\mathcal{E}_{d}(f), and
for all F inE_(d)(f)F \in \mathcal{E}_{d}(f) (see [10, [1]).
Now let R^(X)\mathbb{R}^{X} be the linear space of all real valued functions defined on (X,d)(X, d). One considers the quasi-distance ( [15, p.67)
{:(15)D_(d)(f","g)=s u p{(f(x)-g(x))vv0:x in X}:}\begin{equation*}
D_{d}(f, g)=\sup \{(f(x)-g(x)) \vee 0: x \in X\} \tag{15}
\end{equation*}
Obviously, dd-SLip _(0)X subR_( <= d)^(X)subR^(X){ }_{0} X \subset \mathbb{R}_{\leq d}^{X} \subset \mathbb{R}^{X}, and the quasi-distance D_(d)D_{d} may be restricted to d-SLip_(0)Xd-\operatorname{SLip}_{0} X.
The quasi-distance D_(d)D_{d} generates the topology tau(D_(d))\tau\left(D_{d}\right), named the topology of quasi-uniform convergence. In [15] (Corollary 4, p.67), it is proved that the unit ball U_(0)U_{0} of dd - SLip_(0)X\mathrm{SLip}_{0} X is compact with respect to the topology of quasiuniform convergence tau(D_(d))\tau\left(D_{d}\right), (and tau( bar(D)_(d))\tau\left(\bar{D}_{d}\right) too, where bar(D)_(d)(f,g)=D_(d)(g,f),f,g in{:d-SLip_(0)X)\bar{D}_{d}(f, g)=D_{d}(g, f), f, g \in \left.d-\operatorname{SLip}_{0} X\right).
We have
Proposition 6. For every f in d-SLip_(0)Yf \in d-\operatorname{SLip}_{0} Y, the set E_(d)(f)\mathcal{E}_{d}(f) is compact with respect to the topology tau(D_(d))\tau\left(D_{d}\right), (and tau( bar(D)_(d))\tau\left(\bar{D}_{d}\right), too).
Proof. Because F_(d)(f)F_{d}(f) defined in (12) and G_(d)(f)G_{d}(f) defined in (13) are in E_(d)(f)\mathcal{E}_{d}(f), and they satisfy the inequalities (14), it follows
D_(d)(F,F_(d)(f))=0," and " bar(D)_(d)(F,G_(d)(f)=D_(d)(G_(d)(f),F)=0:}D_{d}\left(F, F_{d}(f)\right)=0, \text { and } \bar{D}_{d}\left(F, G_{d}(f)=D_{d}\left(G_{d}(f), F\right)=0\right.
for every F inE_(d)(f)F \in \mathcal{E}_{d}(f). It follows that E_(d)(f)\mathcal{E}_{d}(f) is D_(d^(-))D_{d^{-}}-totally bounded (and bar(D)_(d^(-))\bar{D}_{d^{-}} totally bounded too).
Let (F_(n))_(n >= 1)\left(F_{n}\right)_{n \geq 1} be a sequence in E_(d)(f)\mathcal{E}_{d}(f). Because F_(n)(x) <= F_(d)(f)(x)F_{n}(x) \leq F_{d}(f)(x), for all x in Xx \in X, it follows that D_(d)(F_(n),F_(d)(f))=0,n=1,2,dotsD_{d}\left(F_{n}, F_{d}(f)\right)=0, n=1,2, \ldots, i.e. (F_(n))_(n >= 1)\left(F_{n}\right)_{n \geq 1} is D_(d^(-))D_{d^{-}} convergent to F_(d)(f)F_{d}(f). It follows that E_(d)(f)\mathcal{E}_{d}(f) is D_(d)D_{d}-sequentially compact. By Proposition 4.6 in [5], because E_(d)(f)\mathcal{E}_{d}(f) is totally D_(d)D_{d}-bounded an D_(d)D_{d}-sequentially compact it follows that the set E_(d)(f)\mathcal{E}_{d}(f) is D_(d)D_{d}-compact (i.e. compact with respect to the topology tau(D_(d))\tau\left(D_{d}\right) ).
Because G_(d)(f)(x) <= F(x)G_{d}(f)(x) \leq F(x), for all x in Xx \in X and every F inE_(d)(f)F \in \mathcal{E}_{d}(f), it follows that D_(d)(G_(d)(f),F)= bar(D)_(d)(F,G_(d)(f))=0D_{d}\left(G_{d}(f), F\right)=\bar{D}_{d}\left(F, G_{d}(f)\right)=0. Consequently, E_(d)(f)\mathcal{E}_{d}(f) is bar(D)_(d)\bar{D}_{d}-compact too. (i.e. with respect to the topology {: tau( bar(D)_(d)))\left.\tau\left(\bar{D}_{d}\right)\right).
Obviously, for every F in d-SLip_(0)X,F|_(Y)in d-SLip_(0)YF \in d-\operatorname{SLip}_{0} X,\left.F\right|_{Y} \in d-\operatorname{SLip}_{0} Y and the set E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right) is a (D_(d), bar(D)_(d))\left(D_{d}, \bar{D}_{d}\right)-compact subset of d-SLip_(0)Xd-\operatorname{SLip}_{0} X, by Proposition 6 .
Now, we consider the following optimization problem:
For F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} X, find G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right) such that
{:(16)D_(d)(F,G_(0))=i n f{D_(d)(F,G):G inE_(d)(F|_(Y))}:}\begin{equation*}
D_{d}\left(F, G_{0}\right)=\inf \left\{D_{d}(F, G): G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)\right\} \tag{16}
\end{equation*}
This problem (of best approximation) has always at least one solution, because E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right) is D_(d)D_{d}-compact. Analogously, the problem of existence of an element bar(G)_(0)inE_(d)(F|_(Y))\bar{G}_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right) such that
{:(17) bar(D)_(d)(F, bar(G)_(0))=i n f{ bar(D)_(d)(F,G):G inE_(d)(F|_(Y))}:}\begin{equation*}
\bar{D}_{d}\left(F, \bar{G}_{0}\right)=\inf \left\{\bar{D}_{d}(F, G): G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)\right\} \tag{17}
\end{equation*}
is also assured, because E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right) is bar(D)_(d)\bar{D}_{d}-compact too.
Now, because ( X,dX, d ) is supposed ( d, bar(d)d, \bar{d} )-sequentially compact, every F in dF \in dSLip_(0)X\operatorname{SLip}_{0} X is bounded, and the uniform norm
{:(18)||F||_(oo)=max{F(x):x in X}vv max{-F(x):x in X}:}\begin{equation*}
\|F\|_{\infty}=\max \{F(x): x \in X\} \vee \max \{-F(x): x \in X\} \tag{18}
\end{equation*}
is well defined, by Proposition 4, (c).
Moreover, for every G inE_(d)(F|_(Y))G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right), we have
Now, we consider the following problem of uniform best approximation:
For F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} X, find G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right), such that
{:(20)||F-G_(0)||_(oo)=i n f{||F-G||_(oo):G inE_(d)(F|_(Y))}:}\begin{equation*}
\left\|F-G_{0}\right\|_{\infty}=\inf \left\{\|F-G\|_{\infty}: G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)\right\} \tag{20}
\end{equation*}
Proposition 7. Let (X,d)(X, d) be a (d, bar(d))(d, \bar{d})-sequentially compact quasi-metric space, theta in X\theta \in X a fixed element, and Y sub XY \subset X with theta in Y\theta \in Y. Then for every F in dF \in dSLip_(0)X\operatorname{SLip}_{0} X, there exists at least one element G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right), such that
||F-G_(0)||_(oo)=i n f{||F-G||_(oo):G inE_(d)(F|_(Y))}\left\|F-G_{0}\right\|_{\infty}=\inf \left\{\|F-G\|_{\infty}: G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)\right\}
Proof. For every G inE_(d)(F|_(Y))G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right), using the equality (18), one obtains
{:[i n f{||F-G||_(oo):}{::G inE_(d)(F|_(Y))}=],[=i n f{D_(d)(F,G)vvD_(d)(G,F):G inE_(d)(F|_(Y))}]:}\begin{aligned}
\inf \left\{\|F-G\|_{\infty}\right. & \left.: G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)\right\}= \\
& =\inf \left\{D_{d}(F, G) \vee D_{d}(G, F): G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)\right\}
\end{aligned}
Because E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right) is ( D_(d), bar(D)_(d)D_{d}, \bar{D}_{d} )-compact, the conclusion of Proposition follows.
Any solution G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right) of problem (20) is called an element of best uniform approximation of FF by elements of bar(E_(d))(F|_(Y))\overline{\mathcal{E}_{d}}\left(\left.F\right|_{Y}\right).
Using (19), one obtains:
If FF is such that
F(x) >= F_(d)(F|_(Y))(x),x in XF(x) \geq F_{d}\left(\left.F\right|_{Y}\right)(x), x \in X
then G_(0)=F_(d)(F|_(Y))G_{0}=F_{d}\left(\left.F\right|_{Y}\right) is the unique solution of (20), where F_(d)(F|_(Y))F_{d}\left(\left.F\right|_{Y}\right) is defined as in (12);
If FF is such that
F(x) <= G_(d)(F|_(Y))(x),x in X,F(x) \leq G_{d}\left(\left.F\right|_{Y}\right)(x), x \in X,
then G_(0)=G_(d)(F|_(Y))G_{0}=G_{d}\left(\left.F\right|_{Y}\right) is the unique solution of (20), where G_(d)(F|_(Y))G_{d}\left(\left.F\right|_{Y}\right) is defined as in (13);
Finally, if F inE_(d)(F|_(Y))F \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right) i.e. ||F|_(d)=||F|_(Y)|_(d)\left.\left.\left\|\left.F\right|_{d}=\right\| F\right|_{Y}\right|_{d}, then G_(0)=FG_{0}=F.
In the following we consider another situation where a uniform best approximation problem by extensions may be posed and solved.
This is the case when the quasi-metric space ( X,dX, d ) is of finite diameter, i.e. such that s u p{d(x,y):x,y in X}=diam X < oo\sup \{d(x, y): x, y \in X\}=\operatorname{diam} X<\infty.
For theta in(X,d)\theta \in(X, d) denote cl_(tau(d)){theta}={x in X:d(theta,x)=0}c l_{\tau(d)}\{\theta\}=\{x \in X: d(\theta, x)=0\} and cl_(tau( bar(d))){theta}={x in X:d(x,theta)=0}c l_{\tau(\bar{d})}\{\theta\}= \{x \in X: d(x, \theta)=0\} (see 15, p.68). Let also cl{theta}=cl_(tau(d)){theta}uu cl_(tau( bar(d))){theta}c l\{\theta\}=c l_{\tau(d)}\{\theta\} \cup c l_{\tau(\bar{d})}\{\theta\}.
The following proposition holds:
Proposition 8. Let ( X,dX, d ) be a quasi-metric space of finite diameter, and theta in X\theta \in X a fixed element. Then every f in d-SLip_(0)Xf \in d-\operatorname{SLip}_{0} X is bounded on X\\cl{theta}X \backslash \operatorname{cl}\{\theta\}.
Proof. Let ff be in dd-SLip _(0)X{ }_{0} X. By definition, we have f(theta)=0f(\theta)=0, and for x in cl_(tau( bar(d))){theta}={x in X:d(x,theta)=0}x \in c l_{\tau(\bar{d})}\{\theta\}=\{x \in X: d(x, \theta)=0\}-it follows f(x) <= 0f(x) \leq 0, because d(x,theta)=0d(x, \theta)=0 implies f(x) <= f(theta)=0f(x) \leq f(\theta)=0.
Analogously, for x in cl_(tau(d)){theta}={x in X:d(theta,x)=0}x \in c l_{\tau(d)}\{\theta\}=\{x \in X: d(\theta, x)=0\} it follows 0=f(theta) <= f(x)0=f(\theta) \leq f(x).
For every x in X\\cl_(tau( bar(d))){theta}x \in X \backslash c l_{\tau(\bar{d})}\{\theta\}, we have
and consequently f(x) <= ||f|_(d)diam X < oof(x) \leq \|\left. f\right|_{d} \operatorname{diam} X<\infty.
It follows, f(x) <= ||f|_(d)diam X < oof(x) \leq \|\left. f\right|_{d} \operatorname{diam} X<\infty for all x in X\\cl_(tau( bar(d))){theta}x \in X \backslash c l_{\tau(\bar{d})}\{\theta\}.
For every x in X\\cl_(tau(d)){theta}x \in X \backslash c l_{\tau(d)}\{\theta\} it follows
Then f(x) >= -||f|_(d)diam X > -oof(x) \geq-\|\left. f\right|_{d} \operatorname{diam} X>-\infty, for all x in X\\cl_(tau(d)){theta}x \in X \backslash c l_{\tau(d)}\{\theta\}. Consequently -||f|_(d)diam X <= f(x) <= ||f|_(d)diam X,x in X\\cl{theta}-\left.\left\|\left.f\right|_{d} \operatorname{diam} X \leq f(x) \leq\right\| f\right|_{d} \operatorname{diam} X, x \in X \backslash c l\{\theta\}.
Now, let ( X,dX, d ) be a quasi-metric space of finite diameter, theta in X\theta \in X a fixed element, and Y sub XY \subset X with theta in Y\theta \in Y. Then, for every F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} X, it follows F|_(Y)in d-SLip_(0)Y\left.F\right|_{Y} \in d-\operatorname{SLip}_{0} Y, and the set
E_(d)(F|_(Y))={G in d-SLip_(0)X:G|_(Y)=F|_(Y),||G|_(d)=||F|_(Y)|_(d)}\mathcal{E}_{d}\left(\left.F\right|_{Y}\right)=\left\{G \in d-\operatorname{SLip}_{0} X:\left.G\right|_{Y}=\left.F\right|_{Y},\left.\left.\left\|\left.G\right|_{d}=\right\| F\right|_{Y}\right|_{d}\right\}
is non empty.
This set is also ( D_(d), bar(D)_(d)D_{d}, \bar{D}_{d} )-compact and the following proposition holds:
Proposition 9. Let ( X,dX, d ) be a quasi-metric space of finite diameter, theta in X\theta \in X a fixed element, and Y sub XY \subset X with theta in Y\theta \in Y. Then for every F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} X, there exists at least one element G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right) such that
||(F-G_(0))|_(X\\cl{theta})||_(oo)=i n f{||(F-G)|_(X\\cl{theta})||_(oo):G inE_(d)(F|_(Y))}.\left\|\left.\left(F-G_{0}\right)\right|_{X \backslash c l\{\theta\}}\right\|_{\infty}=\inf \left\{\left\|\left.(F-G)\right|_{X \backslash c l\{\theta\}}\right\|_{\infty}: G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)\right\} .
The proof is immediate.
Example 10. Let X=[-10,10]X=[-10,10] and the quasi-metric d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty) defined by
d(x,y)={[y-x" if "x <= y],[2(x-y)" if "x > y]:}d(x, y)=\left\{\begin{array}{c}
y-x \text { if } x \leq y \\
2(x-y) \text { if } x>y
\end{array}\right.
Consider theta=0\theta=0 and Y={-1,0,1}Y=\{-1,0,1\}. Then the function f:Y rarrRf: Y \rightarrow \mathbb{R}
is in d-SLip_(0)Yd-\operatorname{SLip}_{0} Y and ||f|_(d)=3\|\left. f\right|_{d}=3.
The functions
{:[F_(d)(f)(x)=i n f_(y in Y){f(y)+3d(x","y)}],[={[-4-3x","quad x in[-10","-1]","],[6x+5","quad x in(-1,(-5)/(9)]","],[-3x","quad x in((-5)/(9),0]","],[6x","quad x in(0,(2)/(3)]","],[6-3x","quad x in((2)/(3),1]","],[6x-3","quad x in(1","10].]:}]:}\begin{aligned}
F_{d}(f)(x) & =\inf _{y \in Y}\{f(y)+3 d(x, y)\} \\
& =\left\{\begin{array}{l}
-4-3 x, \quad x \in[-10,-1], \\
6 x+5, \quad x \in\left(-1, \frac{-5}{9}\right], \\
-3 x, \quad x \in\left(\frac{-5}{9}, 0\right], \\
6 x, \quad x \in\left(0, \frac{2}{3}\right], \\
6-3 x, \quad x \in\left(\frac{2}{3}, 1\right], \\
6 x-3, \quad x \in(1,10] .
\end{array}\right.
\end{aligned}
and, respectively
{:[G_(d)(f)(x)=s u p_(y in Y){f(y)-3d(y","x)}=],[={[6x+5","quad x in[-10","-1]],[-3x+4","quad x in(-1,(-4)/(9)]],[6x","quad x in((-4)/(9),0]],[-3x","quad x in(0,(1)/(3)]],[6x-3","quad x in((1)/(3),1]],[-3x-6","quad x in(1","10]]:}]:}\begin{aligned}
G_{d}(f)(x)= & \sup _{y \in Y}\{f(y)-3 d(y, x)\}= \\
= & \left\{\begin{array}{l}
6 x+5, \quad x \in[-10,-1] \\
-3 x+4, \quad x \in\left(-1, \frac{-4}{9}\right] \\
6 x, \quad x \in\left(\frac{-4}{9}, 0\right] \\
-3 x, \quad x \in\left(0, \frac{1}{3}\right] \\
6 x-3, \quad x \in\left(\frac{1}{3}, 1\right] \\
-3 x-6, \quad x \in(1,10]
\end{array}\right.
\end{aligned}
where H inE_(d)(f)H \in \mathcal{E}_{d}(f) is an arbitrary extension of ff.
Obviously, ( X,dX, d ) is ( d, bar(d)d, \bar{d} )-sequentially compact and E_(d)(f)\mathcal{E}_{d}(f) is compact in the uniform topology.
Let F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} X such that F|_(Y)=f\left.F\right|_{Y}=f.
Then
F(x) >= F_(d)(f)(x),AA x in[-10,10]F(x) \geq F_{d}(f)(x), \forall x \in[-10,10]
then
||F-F_(d)(f)||_(oo)=i n f{||F-H||_(oo):H inE_(d)(F|_(Y))}\left\|F-F_{d}(f)\right\|_{\infty}=\inf \left\{\|F-H\|_{\infty}: H \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)\right\}
For example, let FF be the function
F(x)={[F_(d)(f)(x)",",x in[-1","1]","],[-4x-5",",x in[-10","-1)],[7x-4",",x in(1","10)]:}F(x)=\left\{\begin{array}{cc}
F_{d}(f)(x), & x \in[-1,1], \\
-4 x-5, & x \in[-10,-1) \\
7 x-4, & x \in(1,10)
\end{array}\right.
Similarly, if F(x) <= G_(d)(f)(x),AA x in[-10,10]F(x) \leq G_{d}(f)(x), \forall x \in[-10,10]
then
||F-G_(d)(f)||_(oo)=i n f{||F-H||_(oo):H inE_(d)(F|_(Y))}\left\|F-G_{d}(f)\right\|_{\infty}=\inf \left\{\|F-H\|_{\infty}: H \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)\right\}
^(**){ }^{*} This work has been supported by MEdC under Grant 2-CEx06-11-96/ 19.09.2006. ^(†){ }^{\dagger} "Tiberiu Popoviciu" Institute of Numerical Analysis, P.O. Box. 68-1, Cluj-Napoca, Romania, e-mail: cmustata2001@yahoo.com.
Abstract AuthorsCostică Mustăţa Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania Keywords? Paper coordinatesC. Mustăţa, Extension and approximation of…
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