Best uniform approximation of semi-Lipschitz function by extension

Costica Mustata
(original), Approximation Theory, best approximation, paper, uniform approximation

Abstract

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.

Paper coordinates

C. Mustăţa, Best uniform approximation of semi-Lipschitz function by extension, Rev. Anal. Numér. Théor. Approx. 36 (2007) 2, pp. 161-171.

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https://ictp.acad.ro/jnaat/journal/article/view/2007-vol36-no2-art4/Mustata-pdf

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Journal

Revue d’Analyse Numer. Theor. Approx.

Publisher Name

Publishing House of the Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2007-vol36-no2-art4

Print ISSN

2501-059X

Online ISSN

2457-6794

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[2] Cobzas, S., Asymmetric locally convex spaces, Int. J. Math. Math. Sci., 16, pp. 2585–2608, 2005.
[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.
[5] Collins, J. and Zimmer, J., An asymmetric Arzela-Ascoli theorem, http://bath.ac.uk/math-sci/BICS, Preprint, 16, 12 pp, 2005.
[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.
[17] Sanchez-Alvarez, J. M., On semi-Lipschitz functions with values in a quasi-normed linear space, Applied General Topology, 6, 2, pp. 216–228, 2005.

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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 F F FFF defined on an asymmetric metric space ( X , d ) ( X , d ) (X,d)(X, d)(X,d), by the elements of the set E d ( F | Y ) E d F Y E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right)Ed(F|Y) of all extensions of F | Y ( Y ⊂ X ) F Y ( Y ⊂ X ) F|_(Y)(Y sub X)\left.F\right|_{Y}(Y \subset X)F|Y(Y⊂X), preserving the smallest semi-Lipschitz constant. It is proved that, this problem has always at least a solution, if ( X , d X , d X,dX, dX,d ) is ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compact, or of finite diameter.

MSC 2000. 41A65, 41A30.
Keywords. Semi-Lipschitz functions, uniform approximation, extensions of semi-Lipschitz functions.

1. INTRODUCTION

Let X X XXX be a non-empty set. A function d : X × X → [ 0 , ∞ ) d : X × X → [ 0 , ∞ ) d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty)d:X×X→[0,∞) is called a quasi-metric on X 14 X 14 X 14X 14X14 if the following conditions hold:
  1. d ( x , y ) = d ( y , x ) = 0 d ( x , y ) = d ( y , x ) = 0 d(x,y)=d(y,x)=0quadd(x, y)=d(y, x)=0 \quadd(x,y)=d(y,x)=0 iff x = y x = y quad x=y\quad x=yx=y,
  2. d ( x , z ) ≤ d ( x , y ) + d ( y , z ) d ( x , z ) ≤ d ( x , y ) + d ( y , z ) d(x,z) <= d(x,y)+d(y,z)d(x, z) \leq d(x, y)+d(y, z)d(x,z)≤d(x,y)+d(y,z), for all x , y , z ∈ X x , y , z ∈ X x,y,z in Xx, y, z \in Xx,y,z∈X.
The function d ¯ : X × X → [ 0 , ∞ ) d ¯ : X × X → [ 0 , ∞ ) bar(d):X xx X rarr[0,oo)\bar{d}: X \times X \rightarrow[0, \infty)d¯:X×X→[0,∞) defined by d ¯ ( x , y ) = d ( y , x ) d ¯ ( x , y ) = d ( y , x ) bar(d)(x,y)=d(y,x)\bar{d}(x, y)=d(y, x)d¯(x,y)=d(y,x), for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X is also a quasi-metric on X X XXX, called the conjugate quasi-metric of d d ddd.
A pair ( X , d X , d X,dX, dX,d ) where X X XXX is a non-empty set and d d ddd a quasi-metric on X X XXX, is called a quasi-metric space.
If d d ddd can take the value + ∞ + ∞ +oo+\infty+∞, then it is called a quasi-distance on X X XXX.
Each quasi-metric d d ddd on X X XXX induces a topology τ ( d ) τ ( d ) tau(d)\tau(d)τ(d) which has as a basis the family of balls (forward open balls [5])
(1) B + ( x , ε ) := { y ∈ X : d ( x , y ) < ε } , x ∈ X , ε > 0 . (1) B + ( x , ε ) := { y ∈ X : d ( x , y ) < ε } , x ∈ X , ε > 0 . {:(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*}(1)B+(x,ε):={y∈X:d(x,y)<ε},x∈X,ε>0.
This topology is called the forward topology of X X XXX ([5], [9]), and is denoted also by τ + τ + tau_(+)\tau_{+}τ+.
Observe that the topology τ + τ + tau_(+)\tau_{+}τ+is a T 0 T 0 T_(0)T_{0}T0-topology. If the condition 1) is replaced by 1 ′ 1 ′ 1^(')1^{\prime}1′ ) d ( x , y ) = 0 d ( x , y ) = 0 d(x,y)=0d(x, y)=0d(x,y)=0 iff x = y x = y x=yx=yx=y, then the topology τ + τ + tau_(+)\tau_{+}τ+is a T 1 T 1 T_(1)T_{1}T1-topology (see [14, [15]).
Analogously, the quasi-metric d ¯ d ¯ bar(d)\bar{d}d¯ induces the topology τ ( d ¯ ) τ ( d ¯ ) tau( bar(d))\tau(\bar{d})τ(d¯) on X X XXX, which has as a basis the family of backward open balls (5)
(2) B − ( x , ε ) := { y ∈ X : d ( y , x ) < ε } , x ∈ X , ε > 0 (2) B − ( x , ε ) := { y ∈ X : d ( y , x ) < ε } , x ∈ X , ε > 0 {:(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*}(2)B−(x,ε):={y∈X:d(y,x)<ε},x∈X,ε>0
This topology is called the backward topology of X X XXX ([5], [9]) and is denoted also by τ − τ − tau_(-)\tau_{-}τ−.
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 , d X , d X,dX, dX,d ) be a quasi-metric space. A sequence ( x k ) k ≥ 1 ⊂ X x k k ≥ 1 ⊂ X (x_(k))_(k >= 1)sub X\left(x_{k}\right)_{k \geq 1} \subset X(xk)k≥1⊂X is called d d ddd-convergent (forward convergent) to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X, respectively d ¯ d ¯ bar(d)\bar{d}d¯-convergent (backward convergent) to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X iff
(3) lim k → ∞ d ( x 0 , x k ) = 0 , respectively lim k → 0 d ( x k , x 0 ) = lim k → ∞ d ¯ ( x 0 , x k ) = 0 (3) lim k → ∞   d x 0 , x k = 0 ,  respectively  lim k → 0   d x k , x 0 = lim k → ∞   d ¯ x 0 , x k = 0 {:(3)lim_(k rarr oo)d(x_(0),x_(k))=0","" respectively "lim_(k rarr0)d(x_(k),x_(0))=lim_(k rarr oo) bar(d)(x_(0),x_(k))=0:}\begin{equation*} \lim _{k \rightarrow \infty} d\left(x_{0}, x_{k}\right)=0, \text { respectively } \lim _{k \rightarrow 0} d\left(x_{k}, x_{0}\right)=\lim _{k \rightarrow \infty} \bar{d}\left(x_{0}, x_{k}\right)=0 \tag{3} \end{equation*}(3)limk→∞d(x0,xk)=0, respectively limk→0d(xk,x0)=limk→∞d¯(x0,xk)=0
(see 5], Definition 2.4)
A subset K K KKK of X X XXX is called d d ddd-compact (forward compact) if every open cover of K K KKK with respect to the forward topology τ + τ + tau_(+)\tau_{+}τ+has a finite subcover. We say that a subset K K KKK of X X XXX is d d ddd-sequentially compact (forward-sequentially compact) if every sequence in K K KKK has a d d ddd-convergent (forward convergent) subsequence with limit in K K KKK ([5], Definition 4.1).
The d ¯ d ¯ bar(d)\bar{d}d¯-compact (backward compact) and d ¯ d ¯ bar(d)\bar{d}d¯-sequentially compact (backward -sequentially compact) subset of X X XXX - are defined in a similar way.
Finally, a subset Y Y YYY of ( X , d X , d X,dX, dX,d ) is called ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compact if every sequence ( y n ) n ≥ 1 y n n ≥ 1 (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1}(yn)n≥1 in Y Y YYY has a subsequence ( y n k ) k ≥ 1 , d y n k k ≥ 1 , d (y_(n_(k)))_(k >= 1),d\left(y_{n_{k}}\right)_{k \geq 1}, d(ynk)k≥1,d-convergent to some u ∈ Y u ∈ Y u in Yu \in Yu∈Y and d ¯ d ¯ bar(d)\bar{d}d¯-convergent to some v ∈ Y v ∈ Y v in Yv \in Yv∈Y. By Lemma 3.1 in [5] if follows that we can take u = v u = v u=vu=vu=v in the definition of ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compactness, if ( X , d X , d X,dX, dX,d ) is a T 1 T 1 T_(1)T_{1}T1 quasi-metric space. A subset Y Y YYY of ( X , d X , d X,dX, dX,d ) is called d d ddd-bounded (forward bounded in [5]) if there exist x ∈ X x ∈ X x in Xx \in Xx∈X and r > 0 r > 0 r > 0r>0r>0, such that Y ⊂ B + ( x , r ) . Y Y ⊂ B + ( x , r ) . Y Y subB^(+)(x,r).YY \subset B^{+}(x, r) . YY⊂B+(x,r).Y is called d d ddd-totally bounded if for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, there exists n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N, and the forward balls B + ( y 1 , ε ) , B + ( y 2 , ε ) , … , B n ( y n , ε ) , y i ∈ Y , i = 1 , n ― B + y 1 , ε , B + y 2 , ε , … , B n y n , ε , y i ∈ Y , i = 1 , n ¯ 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}B+(y1,ε),B+(y2,ε),…,Bn(yn,ε),yi∈Y,i=1,n― such that Y ⊂ ⋃ i = 1 n B + ( y i , ε ) Y ⊂ ⋃ i = 1 n   B + y i , ε Y subuuu_(i=1)^(n)B^(+)(y_(i),epsi)Y \subset \bigcup_{i=1}^{n} B^{+}\left(y_{i}, \varepsilon\right)Y⊂⋃i=1nB+(yi,ε).
Similar definitions are given for d ¯ d ¯ bar(d)\bar{d}d¯-boundedness and d ¯ d ¯ bar(d)\bar{d}d¯-total boundedness of a subset Y Y YYY of ( X , d X , d X,dX, dX,d ).

2. THE CONE OF SEMI-LIPSCHITZ FUNCTIONS

Definition 1. [15] Let Y Y YYY be a non-empty subset of a quasi-metric space ( X , d ) ( X , d ) (X,d)(X, d)(X,d). A function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is called d d ddd-semi-Lipschitz if there exists a number L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 (named a d d ddd-semi-Lipschitz constant for f f fff ) such that
(4) f ( x ) − f ( y ) ≤ L d ( x , y ) (4) f ( x ) − f ( y ) ≤ L d ( x , y ) {:(4)f(x)-f(y) <= Ld(x","y):}\begin{equation*} f(x)-f(y) \leq L d(x, y) \tag{4} \end{equation*}(4)f(x)−f(y)≤Ld(x,y)
for all x , y ∈ Y x , y ∈ Y x,y in Yx, y \in Yx,y∈Y.
A function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R, is called ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing if f ( x ) ≤ f ( y ) f ( x ) ≤ f ( y ) f(x) <= f(y)f(x) \leq f(y)f(x)≤f(y), whenever d ( x , y ) = 0 d ( x , y ) = 0 d(x,y)=0d(x, y)=0d(x,y)=0.
Denote by R ≤ d Y R ≤ d Y R_( <= d)^(Y)\mathbb{R}_{\leq d}^{Y}R≤dY the set of all ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing functions on Y Y YYY. This set is a cone in the linear space R Y R Y R^(Y)\mathbb{R}^{Y}RY of real valued functions defined on Y Y YYY, i.e. for each f , g ∈ R ≤ d Y f , g ∈ R ≤ d Y f,g inR_( <= d)^(Y)f, g \in \mathbb{R}_{\leq d}^{Y}f,g∈R≤dY and λ ≥ 0 λ ≥ 0 lambda >= 0\lambda \geq 0λ≥0 it follows that f + g ∈ R ≤ d Y f + g ∈ R ≤ d Y f+g inR_( <= d)^(Y)f+g \in \mathbb{R}_{\leq d}^{Y}f+g∈R≤dY and λ f ∈ R ≤ d Y λ f ∈ R ≤ d Y lambda f inR_( <= d)^(Y)\lambda f \in \mathbb{R}_{\leq d}^{Y}λf∈R≤dY.
For a d d ddd-semi-Lipschitz function f f fff on Y Y YYY, put [14]:
(5) ‖ f | d = sup { ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : d ( x , y ) > 0 ; x , y ∈ Y } (5) ‖ f d = sup ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : d ( x , y ) > 0 ; x , y ∈ Y {:(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*}(5)‖f|d=sup{(f(x)−f(y))∨0d(x,y):d(x,y)>0;x,y∈Y}
Then ‖ f | d ‖ f d ||f|_(d)\|\left. f\right|_{d}‖f|d is the smallest d d ddd-semi-Lipschitz constant of f f fff (see also [10, [15]).
For a fixed element θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y denote
(6) d − SLip 0 Y := { f ∈ R ≤ d Y : ‖ f | d < ∞ and f ( θ ) = 0 } (6) d − SLip 0 Y := f ∈ R ≤ d Y : ‖ f d < ∞  and  f ( θ ) = 0 {:(6)d-SLip_(0)Y:={f inR_( <= d)^(Y):||f|_(d) < oo" and "f(theta)=0}:}\begin{equation*} d-\operatorname{SLip}_{0} Y:=\left\{f \in \mathbb{R}_{\leq d}^{Y}: \|\left. f\right|_{d}<\infty \text { and } f(\theta)=0\right\} \tag{6} \end{equation*}(6)d−SLip0Y:={f∈R≤dY:‖f|d<∞ and f(θ)=0}
the set of all d d ddd-semi-Lipschitz real valued functions defined on Y Y YYY vanishing at the fixed element θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y.
Observe that if ( X , d X , d X,dX, dX,d ) is a T 1 T 1 T_(1)T_{1}T1 quasi-metric space, then every real-valued function on X X XXX is ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing [14].
The set d − SLip 0 Y d − SLip 0 Y d-SLip_(0)Yd-\operatorname{SLip}_{0} Yd−SLip0Y is a cone (a subcone of R ≤ d Y R ≤ d Y R_( <= d)^(Y)\mathbb{R}_{\leq d}^{Y}R≤dY ) and the functional ‖ ⋅ | d : d − SLip 0 Y → [ 0 , ∞ ) ‖ ⋅ d : d − SLip 0 Y → [ 0 , ∞ ) ||*|_(d):d-SLip_(0)Y rarr[0,oo)\|\left.\cdot\right|_{d}: d- \operatorname{SLip}_{0} Y \rightarrow[0, \infty)‖⋅|d:d−SLip0Y→[0,∞) defined by (5) is subadditive and positive homogeneous on d d ddd-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y. Moreover ‖ f | d = 0 ‖ f d = 0 ||f|_(d)=0\|\left. f\right|_{d}=0‖f|d=0 iff f = 0 f = 0 f=0f=0f=0, and consequently ‖ ⋅ | d ‖ ⋅ d ||*|_(d)\|\left.\cdot\right|_{d}‖⋅|d is a quasi-norm (asymmetric norm) on the cone d − SLip 0 Y d − SLip 0 Y d-SLip_(0)Yd-\operatorname{SLip}_{0} Yd−SLip0Y.
In [15] some properties of the "normed cone" ( d − SLip 0 Y , ‖ ⋅ | d d − SLip 0 Y , ‖ ⋅ d d-SLip_(0)Y,||*|_(d)d-\operatorname{SLip}_{0} Y, \|\left.\cdot\right|_{d}d−SLip0Y,‖⋅|d ) are presented. Similar properties in the case of d d ddd-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 , d X , d X,dX, dX,d ) be a quasi-metric space and let Y Y YYY be a non-empty subset of X X XXX. A real valued function f f fff defined on Y Y YYY is called τ + τ + tau_(+)\tau_{+}τ+-lower semi-continuous ( τ + τ + tau_(+)\tau_{+}τ+-l.s.c in short) (respectively τ − τ − tau_(-)\tau_{-}τ−-upper semi-continuous ( τ − − τ − − tau_(-)-\tau_{-}-τ−−u.s.c. ) ) )))) at x 0 ∈ Y x 0 ∈ Y x_(0)in Yx_{0} \in Yx0∈Y, if for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 there exists r > 0 r > 0 r > 0r>0r>0 such that for every x ∈ B + ( x 0 , r ) x ∈ B + x 0 , r x inB^(+)(x_(0),r)x \in B^{+}\left(x_{0}, r\right)x∈B+(x0,r) (respectively, for every x ∈ B − ( x 0 , r ) ) , f ( x ) > f ( x 0 ) − ε x ∈ B − x 0 , r , f ( x ) > f x 0 − ε {: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)-\varepsilonx∈B−(x0,r)),f(x)>f(x0)−ε (respectively f ( x ) < f ( x 0 ) + ε ) f ( x ) < f x 0 + ε f(x) < {:f(x_(0))+epsi)f(x)< \left.f\left(x_{0}\right)+\varepsilon\right)f(x)<f(x0)+ε).
Proposition 2. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X a fixed element, and Y ⊆ X Y ⊆ X Y sube XY \subseteq XY⊆X with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y. Then every f ∈ d − SLip 0 Y f ∈ d − SLip 0 Y f in d-SLip_(0)Yf \in d-\operatorname{SLip}_{0} Yf∈d−SLip0Y is τ − − τ − − tau_(-)-\tau_{-}-τ−−u.s.c and τ + τ + tau_(+)\tau_{+}τ+-l.s.c., and every f ∈ d ¯ − SLip 0 Y f ∈ d ¯ − SLip 0 Y f in bar(d)-SLip_(0)Yf \in \bar{d}-\operatorname{SLip}_{0} Yf∈d¯−SLip0Y is τ + − τ + − tau_(+)-\tau_{+}-τ+−u.s.c. and τ − − τ − − tau_(-)-\tau_{-}-τ−−l.s.c. on Y Y YYY.
Proof. Let f ∈ d − SLip 0 Y f ∈ d − SLip 0 Y f in d-SLip_(0)Yf \in d-\operatorname{SLip}_{0} Yf∈d−SLip0Y such that ‖ f | d = 0 ‖ f d = 0 ||f|_(d)=0\|\left. f\right|_{d}=0‖f|d=0. Then f ≡ 0 f ≡ 0 f-=0f \equiv 0f≡0 and f f fff is τ − − τ − − tau_(-)-\tau_{-}-τ−−u.s.c. and τ + τ + tau_(+)\tau_{+}τ+-l.s.c at every y ∈ Y y ∈ Y y in Yy \in Yy∈Y.
Now, let ‖ f | d > 0 ‖ f d > 0 ||f|_(d) > 0\|\left. f\right|_{d}>0‖f|d>0 and y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y. The inequality
f ( y ) − f ( y 0 ) ≤ ‖ f | d d ( y , y 0 ) , y ∈ Y f ( y ) − f y 0 ≤ ‖ f d d y , y 0 , y ∈ Y 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 Yf(y)−f(y0)≤‖f|dd(y,y0),y∈Y
implies
f ( y ) ≤ f ( y 0 ) + ‖ f | d d ( y , y 0 ) , y ∈ Y . f ( y ) ≤ f y 0 + ‖ f d d y , y 0 , y ∈ Y . 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 .f(y)≤f(y0)+‖f|dd(y,y0),y∈Y.
So that
f ( y ) < f ( y 0 ) + ε , f ( y ) < f y 0 + ε , f(y) < f(y_(0))+epsi,f(y)<f\left(y_{0}\right)+\varepsilon,f(y)<f(y0)+ε,
for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 and every y ∈ B − ( y 0 , ε ‖ f | d ) y ∈ B − y 0 , ε ‖ f d y inB^(-)(y_(0),(epsi)/(||f|_(d)))y \in B^{-}\left(y_{0}, \frac{\varepsilon}{\|\left. f\right|_{d}}\right)y∈B−(y0,ε‖f|d), showing that f f fff is τ − − τ − − tau_(-)-\tau_{-}-τ−−u.s.c at y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y.
Similarly,
f ( y 0 ) − f ( y ) ≤ ‖ f | d ⋅ d ( y 0 , y ) , y ∈ Y , f y 0 − f ( y ) ≤ ‖ f d ⋅ d y 0 , y , y ∈ Y , 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,f(y0)−f(y)≤‖f|d⋅d(y0,y),y∈Y,
implies
f ( y ) ≥ f ( y 0 ) − ‖ f | d d ( y 0 , y ) , f ( y ) ≥ f y 0 − ‖ f d d y 0 , y , f(y) >= f(y_(0))-||f|_(d)d(y_(0),y),f(y) \geq f\left(y_{0}\right)-\|\left. f\right|_{d} d\left(y_{0}, y\right),f(y)≥f(y0)−‖f|dd(y0,y),
so that
f ( y ) > f ( y 0 ) − ε , f ( y ) > f y 0 − ε , f(y) > f(y_(0))-epsi,f(y)>f\left(y_{0}\right)-\varepsilon,f(y)>f(y0)−ε,
for every y ∈ B + ( Y 0 , ε ‖ f | d ) y ∈ B + Y 0 , ε ‖ f d y inB^(+)(Y_(0),(epsi)/(||f|_(d)))y \in B^{+}\left(Y_{0}, \frac{\varepsilon}{\|\left. f\right|_{d}}\right)y∈B+(Y0,ε‖f|d), showing that f f fff is τ + τ + tau_(+)\tau_{+}τ+-l.s.c. in y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y.
Similarly one prove that every f ∈ d ¯ − SLip 0 Y f ∈ d ¯ − SLip 0 Y f in bar(d)-SLip_(0)Yf \in \bar{d}-\operatorname{SLip}_{0} Yf∈d¯−SLip0Y is τ + − τ + − tau_(+)-\tau_{+}-τ+−u.s.c. and τ − − τ − − tau_(-)-\tau_{-}-τ−−l.s.c. on Y Y YYY.
Observe that if f f fff is in d − SLip 0 Y d − SLip 0 Y d-SLip_(0)Yd-\operatorname{SLip}_{0} Yd−SLip0Y, then − f ∈ d ¯ − SLip 0 Y − f ∈ d ¯ − SLip 0 Y -f in bar(d)-SLip_(0)Y-f \in \bar{d}-\operatorname{SLip}_{0} Y−f∈d¯−SLip0Y, and − f − f -f-f−f is τ + − τ + − tau_(+)-\tau_{+}-τ+−u.s.c, and τ − − τ − − tau_(-)-\tau_{-}-τ−−l.s.c. on Y Y YYY, i.e. if y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y then
  • ∀ ε > 0 , ∃ r > 0 ∀ ε > 0 , ∃ r > 0 AA epsi > 0,EE r > 0\forall \varepsilon>0, \exists r>0∀ε>0,∃r>0 such that ( − f ) ( y ) < ( − f ) ( y 0 ) + ε ( − f ) ( y ) < ( − f ) y 0 + ε (-f)(y) < (-f)(y_(0))+epsi(-f)(y)<(-f)\left(y_{0}\right)+\varepsilon(−f)(y)<(−f)(y0)+ε, for all y ∈ B + ( y 0 , r ) y ∈ B + y 0 , r y inB^(+)(y_(0),r)y \in B^{+}\left(y_{0}, r\right)y∈B+(y0,r), and respectively
  • ∀ ε > 0 , ∃ r > 0 ∀ ε > 0 , ∃ r > 0 AA epsi > 0,EE r > 0\forall \varepsilon>0, \exists r>0∀ε>0,∃r>0 such that ( − f ) ( y ) > ( − f ) ( y 0 ) − ε ( − f ) ( y ) > ( − f ) y 0 − ε (-f)(y) > (-f)(y_(0))-epsi(-f)(y)>(-f)\left(y_{0}\right)-\varepsilon(−f)(y)>(−f)(y0)−ε, for all y ∈ B − ( y 0 , r ) y ∈ B − y 0 , r y inB^(-)(y_(0),r)y \in B^{-}\left(y_{0}, r\right)y∈B−(y0,r).
Proposition 3. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X a fixed element, and Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X, with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y.
(a) If Y Y YYY is d ¯ d ¯ bar(d)\bar{d}d¯-sequentially compact, then each f ∈ d f ∈ d f in df \in df∈d-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y attains its maximum value on Y Y YYY;
(b) If Y Y YYY is d d ddd - sequentially compact, then each f ∈ d f ∈ d f in df \in df∈d-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y attains its minimum value on Y Y YYY.
Proof. (a) Let Y Y YYY be d ¯ d ¯ bar(d)\bar{d}d¯-sequentially compact and M := sup f ( Y ) M := sup f ( Y ) M:=s u p f(Y)M:=\sup f(Y)M:=supf(Y), where M ∈ R ∪ { + ∞ } M ∈ R ∪ { + ∞ } M inRuu{+oo}M \in \mathbb{R} \cup\{+\infty\}M∈R∪{+∞}. Then there exists a sequence ( y n ) n ≥ 1 y n n ≥ 1 (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1}(yn)n≥1 in Y Y YYY such that lim n → ∞ f ( y n ) = M lim n → ∞   f y n = M lim_(n rarr oo)f(y_(n))=M\lim _{n \rightarrow \infty} f\left(y_{n}\right)= Mlimn→∞f(yn)=M. Because Y Y YYY is d ¯ d ¯ bar(d)\bar{d}d¯-sequentially compact, there exists y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y and a subsequence ( y n k ) k ≥ 1 y n k k ≥ 1 (y_(n_(k)))_(k >= 1)\left(y_{n_{k}}\right)_{k \geq 1}(ynk)k≥1 of ( y n ) n ≥ 1 y n n ≥ 1 (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1}(yn)n≥1 such that lim n → ∞ d ( y n , k , y 0 ) = 0 lim n → ∞   d y n , k , y 0 = 0 lim_(n rarr oo)d(y_(n,k),y_(0))=0\lim _{n \rightarrow \infty} d\left(y_{n, k}, y_{0}\right)=0limn→∞d(yn,k,y0)=0. By the τ − − τ − − tau_(-)-\tau_{-}-τ−−u.s.c. of f f fff at y 0 y 0 y_(0)y_{0}y0 it follows:
M = lim k → ∞ f ( y n k ) = lim sup k f ( y n k ) ≤ f ( y 0 ) = M , M = lim k → ∞   f y n k = lim sup k   f y n k ≤ f y 0 = M , 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,M=limk→∞f(ynk)=lim supkf(ynk)≤f(y0)=M,
implying M < ∞ M < ∞ M < ooM<\inftyM<∞ and f ( y 0 ) = M f y 0 = M f(y_(0))=Mf\left(y_{0}\right)=Mf(y0)=M.
(b) If f ∈ d f ∈ d f in df \in df∈d-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y, it follows- f ∈ d ¯ f ∈ d ¯ f in bar(d)f \in \bar{d}f∈d¯-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y, and because Y Y YYY is d d ddd-sequentially compact, by (a), it follows that − f − f -f-f−f attains its maximum value on Y Y YYY, i.e. f f fff attains its minimum value on Y Y YYY.
Proposition 4. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X a fixed element, and Y ⊆ X Y ⊆ X Y sube XY \subseteq XY⊆X with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y.
(a) If Y Y YYY is d ¯ d ¯ bar(d)\bar{d}d¯-sequentially compact, then the functional ‖ ⋅ | ∞ d ¯ : d − SLip 0 Y → [ 0 , ∞ ) ‖ ⋅ ∞ d ¯ : d − SLip 0 Y → [ 0 , ∞ ) ||*|_(oo)^( bar(d)):d-SLip_(0)Y rarr[0,oo)\|\left.\cdot\right|_{\infty} ^{\bar{d}}: d-\operatorname{SLip}_{0} Y \rightarrow [0, \infty)‖⋅|∞d¯:d−SLip0Y→[0,∞) defined by
(7) ‖ f | ∞ d ¯ = max { f ( y ) : y ∈ Y } (7) ‖ f ∞ d ¯ = max { f ( y ) : y ∈ Y } {:(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*}(7)‖f|∞d¯=max{f(y):y∈Y}
is an asymmetric norm on d − SLip 0 Y d − SLip 0 Y d-SLip_(0)Yd-\operatorname{SLip}_{0} Yd−SLip0Y.
(b) If Y Y YYY is d d ddd-sequentially compact, then the functional ‖ ⋅ | ∞ d : d − SLip 0 Y → [ 0 , ∞ ) ‖ ⋅ ∞ d : d − SLip 0 Y → [ 0 , ∞ ) ||*|_(oo)^(d):d-SLip_(0)Y rarr[0,oo)\|\left.\cdot\right|_{\infty} ^{d}: d-\operatorname{SLip}_{0} Y \rightarrow [0, \infty)‖⋅|∞d:d−SLip0Y→[0,∞) defined by
(8) ‖ f | ∞ d = max { − f ( y ) : y ∈ Y } , f ∈ d − SLip 0 Y (8) ‖ f ∞ d = max { − f ( y ) : y ∈ Y } , f ∈ d − SLip 0 Y {:(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*}(8)‖f|∞d=max{−f(y):y∈Y},f∈d−SLip0Y
is an asymmetric norm on d − SLip 0 Y d − SLip 0 Y d-SLip_(0)Yd-\operatorname{SLip}_{0} Yd−SLip0Y;
(c) If Y Y YYY is ( d , d ¯ ) ( d , d ¯ ) (d, bar(d))(d, \bar{d})(d,d¯)-sequentially compact, then the functional ‖ ⋅ | ∞ : d − SLip 0 Y → [ 0 , ∞ ) ‖ ⋅ ∞ : d − SLip 0 Y → [ 0 , ∞ ) ||*|_(oo):d-SLip_(0)Y rarr[0,oo)\|\left.\cdot\right|_{\infty}: d-\operatorname{SLip}_{0} Y \rightarrow [0, \infty)‖⋅|∞:d−SLip0Y→[0,∞) defined by
(9) ‖ f | ∞ = ‖ f | ∞ d ∨ ‖ f | ∞ d ¯ , f ∈ d − SLip 0 Y (9) f ∞ = f ∞ d ∨ ‖ f ∞ d ¯ , f ∈ d − SLip 0 Y {:(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*}(9)‖f|∞=‖f|∞d∨‖f|∞d¯,f∈d−SLip0Y
is the uniform norm on the cone d − SLip 0 Y d − SLip 0 Y d-SLip_(0)Yd-\operatorname{SLip}_{0} Yd−SLip0Y.
Proof. (a) By Proposition 3 (a), the functional (7) is well defined. For every f ∈ d f ∈ d f in df \in df∈d - SLip 0 Y SLip 0 Y SLip_(0)Y\operatorname{SLip}_{0} YSLip0Y, we have ‖ f | ∞ d ¯ ≥ f ( θ ) = 0 ‖ f ∞ d ¯ ≥ f ( θ ) = 0 ||f|_(oo)^( bar(d)) >= f(theta)=0\|\left. f\right|_{\infty} ^{\bar{d}} \geq f(\theta)=0‖f|∞d¯≥f(θ)=0. If f ∈ d − SLip 0 Y f ∈ d − SLip 0 Y f in d-SLip_(0)Yf \in d-\operatorname{SLip}_{0} Yf∈d−SLip0Y and ‖ f | ∞ d > 0 ‖ f ∞ d > 0 ||f|_(oo)^(d) > 0\|\left. f\right|_{\infty} ^{d}>0‖f|∞d>0 then there exists y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y such that f ( y 0 ) = ‖ f | ∞ d ¯ > 0 f y 0 = ‖ f ∞ d ¯ > 0 f(y_(0))=||f|_(oo)^( bar(d)) > 0f\left(y_{0}\right)=\|\left. f\right|_{\infty} ^{\bar{d}}>0f(y0)=‖f|∞d¯>0. It follows f ≠ 0 f ≠ 0 f!=0f \neq 0f≠0.
  • Obviously,
‖ f + g | ∞ d ¯ ≤ ‖ f | ∞ d ¯ + ‖ g | ∞ d ¯ f + g ∞ d ¯ ≤ f ∞ d ¯ + ‖ g ∞ d ¯ ||f+g|_(oo)^( bar(d)) <= ||f|_(oo)^( bar(d))+||g|_(oo)^( bar(d))\left.\left\|f+\left.g\right|_{\infty} ^{\bar{d}} \leq\right\| f\right|_{\infty} ^{\bar{d}}+\|\left. g\right|_{\infty} ^{\bar{d}}‖f+g|∞d¯≤‖f|∞d¯+‖g|∞d¯
and
‖ λ f | ∞ d ¯ = λ ‖ f | ∞ d ¯ λ f ∞ d ¯ = λ f ∞ d ¯ || lambda f|_(oo)^( bar(d))=lambda||f|_(oo)^( bar(d))\left.\left\|\left.\lambda f\right|_{\infty} ^{\bar{d}}=\lambda\right\| f\right|_{\infty} ^{\bar{d}}‖λf|∞d¯=λ‖f|∞d¯
for all f , g ∈ d − SLip 0 Y f , g ∈ d − SLip 0 Y f,g in d-SLip_(0)Yf, g \in d-\operatorname{SLip}_{0} Yf,g∈d−SLip0Y and λ ≥ 0 λ ≥ 0 lambda >= 0\lambda \geq 0λ≥0.
(b) For every f ∈ d f ∈ d f in df \in df∈d - SLip 0 Y SLip 0 Y SLip_(0)Y\operatorname{SLip}_{0} YSLip0Y it follows that − f ∈ d ¯ − SLip 0 Y − f ∈ d ¯ − SLip 0 Y -f in bar(d)-SLip_(0)Y-f \in \bar{d}-\operatorname{SLip}_{0} Y−f∈d¯−SLip0Y, and because Y Y YYY is d d ddd-sequentially compact, then − f − f -f-f−f attains its maximum value on Y Y YYY, and
‖ f | ∞ d = max { − f ( y ) : y ∈ Y } ‖ f ∞ d = max { − f ( y ) : y ∈ Y } ||f|_(oo)^(d)=max{-f(y):y in Y}\|\left. f\right|_{\infty} ^{d}=\max \{-f(y): y \in Y\}‖f|∞d=max{−f(y):y∈Y}
is an asymmetric norm on d − SLip 0 Y d − SLip 0 Y d-SLip_(0)Yd-\operatorname{SLip}_{0} Yd−SLip0Y.
(c) By Proposition 3, if Y Y YYY is ( d , d ¯ ) ( d , d ¯ ) (d, bar(d))(d, \bar{d})(d,d¯)-sequentially compact, then every f ∈ d f ∈ d f in df \in df∈d SLip 0 Y SLip 0 Y SLip_(0)Y\operatorname{SLip}_{0} YSLip0Y, attains its maximum and minimum value on Y Y YYY.
  • We have
‖ f ‖ ∞ = max { | f ( y ) | : y ∈ Y } = = ( max { f ( y ) : y ∈ Y } ) ∨ ( max { − f ( y ) : y ∈ Y } ) = ‖ f | ∞ d ∨ ‖ f | ∞ d ¯ ‖ f ‖ ∞ = max { | f ( y ) | : y ∈ Y } = = ( max { f ( y ) : y ∈ Y } ) ∨ ( max { − f ( y ) : y ∈ Y } ) = f ∞ d ∨ f ∞ d ¯ {:[||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}‖f‖∞=max{|f(y)|:y∈Y}==(max{f(y):y∈Y})∨(max{−f(y):y∈Y})=‖f|∞d∨‖f|∞d¯

3. BEST UNIFORM APPROXIMATION BY EXTENSIONS

In the following the quasi-metric space ( X , d ) ( X , d ) (X,d)(X, d)(X,d) is supposed ( d , d ¯ ) ( d , d ¯ ) (d, bar(d))(d, \bar{d})(d,d¯)-sequentially compact. Let θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X be a fixed element, and Y ⊆ X Y ⊆ X Y sube XY \subseteq XY⊆X with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y. Consider also the normed cones ( d − SLip 0 Y , ‖ ⋅ | d d − SLip 0 Y , ‖ ⋅ d d-SLip_(0)Y,||*|_(d)d-\operatorname{SLip}_{0} Y, \|\left.\cdot\right|_{d}d−SLip0Y,‖⋅|d ) and ( d ¯ − SLip 0 X , ‖ ⋅ | d ¯ d ¯ − SLip 0 X , ‖ ⋅ d ¯ bar(d)-SLip_(0)X,||*|_( bar(d))\bar{d}-\operatorname{SLip}_{0} X, \|\left.\cdot\right|_{\bar{d}}d¯−SLip0X,‖⋅|d¯ ), where ‖ ⋅ | d ¯ ‖ ⋅ d ¯ ||*|_( bar(d))\|\left.\cdot\right|_{\bar{d}}‖⋅|d¯ is the asymmetric norm defined as in (5), where d d ddd is replaced by d ¯ d ¯ bar(d)\bar{d}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 ∈ d f ∈ d f in df \in df∈d - SLip 0 Y SLip 0 Y SLip_(0)Y\operatorname{SLip}_{0} YSLip0Y there exists at least one function F ∈ d − SLip 0 X F ∈ d − SLip 0 X F in d-SLip_(0)XF \in d-\mathrm{SLip}_{0} XF∈d−SLip0X, such that
(10) F | Y = f and ‖ F | d = ‖ f | d . (10) F Y = f  and  F d = f d . {:(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*}(10)F|Y=f and ‖F|d=‖f|d.
A function F F FFF with the properties included in Proposition 5, is called an extension, preserving the asymmetric norm of f f fff (or an extension preserving the smallest semi-Lipschitz constant of f f fff ).
Denote the set of all extensions of f f fff preserving asymmetric norm, by
(11) E d ( f ) = { F ∈ d − SLip 0 X : F | Y = f and ‖ F | d = ‖ f | d } (11) E d ( f ) = F ∈ d − SLip 0 X : F Y = f  and  F d = f d {:(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*}(11)Ed(f)={F∈d−SLip0X:F|Y=f and ‖F|d=‖f|d}
The set E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) is convex in d d ddd-SLip 0 X 0 X _(0)X{ }_{0} X0X, the functions
(12) F d ( f ) ( x ) = inf { f ( y ) + ‖ f | d d ( x , y ) : y ∈ Y } , x ∈ X (12) F d ( f ) ( x ) = inf f ( y ) + ‖ f d d ( x , y ) : y ∈ Y , x ∈ X {:(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*}(12)Fd(f)(x)=inf{f(y)+‖f|dd(x,y):y∈Y},x∈X
and
(13) G d ( f ) ( x ) = sup { f ( y ) − ‖ f | d ⋅ d ( y , x ) : y ∈ Y } , x ∈ X (13) G d ( f ) ( x ) = sup f ( y ) − ‖ f d ⋅ d ( y , x ) : y ∈ Y , x ∈ X {:(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*}(13)Gd(f)(x)=sup{f(y)−‖f|d⋅d(y,x):y∈Y},x∈X
are extremal elements of E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f), and
(14) G d ( f ) ( x ) ≤ F ( x ) ≤ F d ( f ) ( x ) (14) G d ( f ) ( x ) ≤ F ( x ) ≤ F d ( f ) ( x ) {:(14)G_(d)(f)(x) <= F(x) <= F_(d)(f)(x):}\begin{equation*} G_{d}(f)(x) \leq F(x) \leq F_{d}(f)(x) \tag{14} \end{equation*}(14)Gd(f)(x)≤F(x)≤Fd(f)(x)
for all F ∈ E d ( f ) F ∈ E d ( f ) F inE_(d)(f)F \in \mathcal{E}_{d}(f)F∈Ed(f) (see [10, [1]).
Now let R X R X R^(X)\mathbb{R}^{X}RX be the linear space of all real valued functions defined on ( X , d ) ( X , d ) (X,d)(X, d)(X,d). One considers the quasi-distance ( [15, p.67)
D d : R X × R X → [ 0 , ∞ ) D d : R X × R X → [ 0 , ∞ ) D_(d):R^(X)xxR^(X)rarr[0,oo)D_{d}: \mathbb{R}^{X} \times \mathbb{R}^{X} \rightarrow[0, \infty)Dd:RX×RX→[0,∞)
defined by
(15) D d ( f , g ) = sup { ( f ( x ) − g ( x ) ) ∨ 0 : x ∈ X } (15) D d ( f , g ) = sup { ( f ( x ) − g ( x ) ) ∨ 0 : x ∈ X } {:(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*}(15)Dd(f,g)=sup{(f(x)−g(x))∨0:x∈X}
Obviously, d d ddd-SLip 0 X ⊂ R ≤ d X ⊂ R X 0 X ⊂ R ≤ d X ⊂ R X _(0)X subR_( <= d)^(X)subR^(X){ }_{0} X \subset \mathbb{R}_{\leq d}^{X} \subset \mathbb{R}^{X}0X⊂R≤dX⊂RX, and the quasi-distance D d D d D_(d)D_{d}Dd may be restricted to d − SLip 0 X d − SLip 0 X d-SLip_(0)Xd-\operatorname{SLip}_{0} Xd−SLip0X.
The quasi-distance D d D d D_(d)D_{d}Dd generates the topology τ ( D d ) τ D d tau(D_(d))\tau\left(D_{d}\right)τ(Dd), named the topology of quasi-uniform convergence. In [15] (Corollary 4, p.67), it is proved that the unit ball U 0 U 0 U_(0)U_{0}U0 of d d ddd - SLip 0 X SLip 0 X SLip_(0)X\mathrm{SLip}_{0} XSLip0X is compact with respect to the topology of quasiuniform convergence τ ( D d ) τ D d tau(D_(d))\tau\left(D_{d}\right)τ(Dd), (and τ ( D ¯ d ) τ D ¯ d tau( bar(D)_(d))\tau\left(\bar{D}_{d}\right)τ(D¯d) too, where D ¯ d ( f , g ) = D d ( g , f ) , f , g ∈ d − SLip 0 X ) D ¯ d ( f , g ) = D d ( g , f ) , f , g ∈ d − SLip 0 X 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)D¯d(f,g)=Dd(g,f),f,g∈d−SLip0X).
We have
Proposition 6. For every f ∈ d − SLip 0 Y f ∈ d − SLip 0 Y f in d-SLip_(0)Yf \in d-\operatorname{SLip}_{0} Yf∈d−SLip0Y, the set E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) is compact with respect to the topology τ ( D d ) τ D d tau(D_(d))\tau\left(D_{d}\right)τ(Dd), (and τ ( D ¯ d ) τ D ¯ d tau( bar(D)_(d))\tau\left(\bar{D}_{d}\right)τ(D¯d), too).
Proof. Because F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) defined in (12) and G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) defined in (13) are in E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f), and they satisfy the inequalities (14), it follows
D d ( F , F d ( f ) ) = 0 , and D ¯ d ( F , G d ( f ) = D d ( G d ( f ) , F ) = 0 D d F , F d ( f ) = 0 ,  and  D ¯ d F , G d ( f ) = D d G d ( f ) , F = 0 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.Dd(F,Fd(f))=0, and D¯d(F,Gd(f)=Dd(Gd(f),F)=0
for every F ∈ E d ( f ) F ∈ E d ( f ) F inE_(d)(f)F \in \mathcal{E}_{d}(f)F∈Ed(f). It follows that E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) is D d − D d − D_(d^(-))D_{d^{-}}Dd−-totally bounded (and D ¯ d − D ¯ d − bar(D)_(d^(-))\bar{D}_{d^{-}}D¯d− totally bounded too).
Let ( F n ) n ≥ 1 F n n ≥ 1 (F_(n))_(n >= 1)\left(F_{n}\right)_{n \geq 1}(Fn)n≥1 be a sequence in E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f). Because F n ( x ) ≤ F d ( f ) ( x ) F n ( x ) ≤ F d ( f ) ( x ) F_(n)(x) <= F_(d)(f)(x)F_{n}(x) \leq F_{d}(f)(x)Fn(x)≤Fd(f)(x), for all x ∈ X x ∈ X x in Xx \in Xx∈X, it follows that D d ( F n , F d ( f ) ) = 0 , n = 1 , 2 , … D d F n , F d ( f ) = 0 , n = 1 , 2 , … D_(d)(F_(n),F_(d)(f))=0,n=1,2,dotsD_{d}\left(F_{n}, F_{d}(f)\right)=0, n=1,2, \ldotsDd(Fn,Fd(f))=0,n=1,2,…, i.e. ( F n ) n ≥ 1 F n n ≥ 1 (F_(n))_(n >= 1)\left(F_{n}\right)_{n \geq 1}(Fn)n≥1 is D d − D d − D_(d^(-))D_{d^{-}}Dd− convergent to F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f). It follows that E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) is D d D d D_(d)D_{d}Dd-sequentially compact. By Proposition 4.6 in [5], because E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) is totally D d D d D_(d)D_{d}Dd-bounded an D d D d D_(d)D_{d}Dd-sequentially compact it follows that the set E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) is D d D d D_(d)D_{d}Dd-compact (i.e. compact with respect to the topology τ ( D d ) τ D d tau(D_(d))\tau\left(D_{d}\right)τ(Dd) ).
Because G d ( f ) ( x ) ≤ F ( x ) G d ( f ) ( x ) ≤ F ( x ) G_(d)(f)(x) <= F(x)G_{d}(f)(x) \leq F(x)Gd(f)(x)≤F(x), for all x ∈ X x ∈ X x in Xx \in Xx∈X and every F ∈ E d ( f ) F ∈ E d ( f ) F inE_(d)(f)F \in \mathcal{E}_{d}(f)F∈Ed(f), it follows that D d ( G d ( f ) , F ) = D ¯ d ( F , G d ( f ) ) = 0 D d G d ( f ) , F = D ¯ d F , G d ( f ) = 0 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)=0Dd(Gd(f),F)=D¯d(F,Gd(f))=0. Consequently, E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) is D ¯ d D ¯ d bar(D)_(d)\bar{D}_{d}D¯d-compact too. (i.e. with respect to the topology τ ( D ¯ d ) ) τ D ¯ d {: tau( bar(D)_(d)))\left.\tau\left(\bar{D}_{d}\right)\right)τ(D¯d)).
Obviously, for every F ∈ d − SLip 0 X , F | Y ∈ d − SLip 0 Y F ∈ d − SLip 0 X , F Y ∈ d − SLip 0 Y 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} YF∈d−SLip0X,F|Y∈d−SLip0Y and the set E d ( F | Y ) E d F Y E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right)Ed(F|Y) is a ( D d , D ¯ d ) D d , D ¯ d (D_(d), bar(D)_(d))\left(D_{d}, \bar{D}_{d}\right)(Dd,D¯d)-compact subset of d − SLip 0 X d − SLip 0 X d-SLip_(0)Xd-\operatorname{SLip}_{0} Xd−SLip0X, by Proposition 6 .
Now, we consider the following optimization problem:
For F ∈ d − SLip 0 X F ∈ d − SLip 0 X F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} XF∈d−SLip0X, find G 0 ∈ E d ( F | Y ) G 0 ∈ E d F Y G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G0∈Ed(F|Y) such that
(16) D d ( F , G 0 ) = inf { D d ( F , G ) : G ∈ E d ( F | Y ) } (16) D d F , G 0 = inf D d ( F , G ) : G ∈ E d F Y {:(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*}(16)Dd(F,G0)=inf{Dd(F,G):G∈Ed(F|Y)}
This problem (of best approximation) has always at least one solution, because E d ( F | Y ) E d F Y E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right)Ed(F|Y) is D d D d D_(d)D_{d}Dd-compact. Analogously, the problem of existence of an element G ¯ 0 ∈ E d ( F | Y ) G ¯ 0 ∈ E d F Y bar(G)_(0)inE_(d)(F|_(Y))\bar{G}_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G¯0∈Ed(F|Y) such that
(17) D ¯ d ( F , G ¯ 0 ) = inf { D ¯ d ( F , G ) : G ∈ E d ( F | Y ) } (17) D ¯ d F , G ¯ 0 = inf D ¯ d ( F , G ) : G ∈ E d F Y {:(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*}(17)D¯d(F,G¯0)=inf{D¯d(F,G):G∈Ed(F|Y)}
is also assured, because E d ( F | Y ) E d F Y E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right)Ed(F|Y) is D ¯ d D ¯ d bar(D)_(d)\bar{D}_{d}D¯d-compact too.
Now, because ( X , d X , d X,dX, dX,d ) is supposed ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compact, every F ∈ d F ∈ d F in dF \in dF∈d SLip 0 X SLip 0 X SLip_(0)X\operatorname{SLip}_{0} XSLip0X is bounded, and the uniform norm
(18) ‖ F ‖ ∞ = max { F ( x ) : x ∈ X } ∨ max { − F ( x ) : x ∈ X } (18) ‖ F ‖ ∞ = max { F ( x ) : x ∈ X } ∨ max { − F ( x ) : x ∈ X } {:(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*}(18)‖F‖∞=max{F(x):x∈X}∨max{−F(x):x∈X}
is well defined, by Proposition 4, (c).
Moreover, for every G ∈ E d ( F | Y ) G ∈ E d F Y G inE_(d)(F|_(Y))G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G∈Ed(F|Y), we have
(19) ‖ F − G ‖ ∞ = D d ( F , G ) ∨ D ¯ d ( F , G ) (19) ‖ F − G ‖ ∞ = D d ( F , G ) ∨ D ¯ d ( F , G ) {:(19)||F-G||_(oo)=D_(d)(F","G)vv bar(D)_(d)(F","G):}\begin{equation*} \|F-G\|_{\infty}=D_{d}(F, G) \vee \bar{D}_{d}(F, G) \tag{19} \end{equation*}(19)‖F−G‖∞=Dd(F,G)∨D¯d(F,G)
Now, we consider the following problem of uniform best approximation:
For F ∈ d − SLip 0 X F ∈ d − SLip 0 X F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} XF∈d−SLip0X, find G 0 ∈ E d ( F | Y ) G 0 ∈ E d F Y G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G0∈Ed(F|Y), such that
(20) ‖ F − G 0 ‖ ∞ = inf { ‖ F − G ‖ ∞ : G ∈ E d ( F | Y ) } (20) F − G 0 ∞ = inf ‖ F − G ‖ ∞ : G ∈ E d F Y {:(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*}(20)‖F−G0‖∞=inf{‖F−G‖∞:G∈Ed(F|Y)}
Proposition 7. Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a ( d , d ¯ ) ( d , d ¯ ) (d, bar(d))(d, \bar{d})(d,d¯)-sequentially compact quasi-metric space, θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X a fixed element, and Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y. Then for every F ∈ d F ∈ d F in dF \in dF∈d SLip 0 X SLip 0 X SLip_(0)X\operatorname{SLip}_{0} XSLip0X, there exists at least one element G 0 ∈ E d ( F | Y ) G 0 ∈ E d F Y G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G0∈Ed(F|Y), such that
‖ F − G 0 ‖ ∞ = inf { ‖ F − G ‖ ∞ : G ∈ E d ( F | Y ) } F − G 0 ∞ = inf ‖ F − G ‖ ∞ : G ∈ E d F Y ||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\}‖F−G0‖∞=inf{‖F−G‖∞:G∈Ed(F|Y)}
Proof. For every G ∈ E d ( F | Y ) G ∈ E d F Y G inE_(d)(F|_(Y))G \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G∈Ed(F|Y), using the equality (18), one obtains
inf { ‖ F − G ‖ ∞ : G ∈ E d ( F | Y ) } = = inf { D d ( F , G ) ∨ D d ( G , F ) : G ∈ E d ( F | Y ) } inf ‖ F − G ‖ ∞ : G ∈ E d F Y = = inf D d ( F , G ) ∨ D d ( G , F ) : G ∈ E d F Y {:[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}inf{‖F−G‖∞:G∈Ed(F|Y)}==inf{Dd(F,G)∨Dd(G,F):G∈Ed(F|Y)}
Because E d ( F | Y ) E d F Y E_(d)(F|_(Y))\mathcal{E}_{d}\left(\left.F\right|_{Y}\right)Ed(F|Y) is ( D d , D ¯ d D d , D ¯ d D_(d), bar(D)_(d)D_{d}, \bar{D}_{d}Dd,D¯d )-compact, the conclusion of Proposition follows.
Any solution G 0 ∈ E d ( F | Y ) G 0 ∈ E d F Y G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G0∈Ed(F|Y) of problem (20) is called an element of best uniform approximation of F F FFF by elements of E d ― ( F | Y ) E d ¯ F Y bar(E_(d))(F|_(Y))\overline{\mathcal{E}_{d}}\left(\left.F\right|_{Y}\right)Ed―(F|Y).
Using (19), one obtains:
If F F FFF is such that
F ( x ) ≥ F d ( F | Y ) ( x ) , x ∈ X F ( x ) ≥ F d F Y ( x ) , x ∈ X F(x) >= F_(d)(F|_(Y))(x),x in XF(x) \geq F_{d}\left(\left.F\right|_{Y}\right)(x), x \in XF(x)≥Fd(F|Y)(x),x∈X
then G 0 = F d ( F | Y ) G 0 = F d F Y G_(0)=F_(d)(F|_(Y))G_{0}=F_{d}\left(\left.F\right|_{Y}\right)G0=Fd(F|Y) is the unique solution of (20), where F d ( F | Y ) F d F Y F_(d)(F|_(Y))F_{d}\left(\left.F\right|_{Y}\right)Fd(F|Y) is defined as in (12);
If F F FFF is such that
F ( x ) ≤ G d ( F | Y ) ( x ) , x ∈ X , F ( x ) ≤ G d F Y ( x ) , x ∈ X , 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,F(x)≤Gd(F|Y)(x),x∈X,
then G 0 = G d ( F | Y ) G 0 = G d F Y G_(0)=G_(d)(F|_(Y))G_{0}=G_{d}\left(\left.F\right|_{Y}\right)G0=Gd(F|Y) is the unique solution of (20), where G d ( F | Y ) G d F Y G_(d)(F|_(Y))G_{d}\left(\left.F\right|_{Y}\right)Gd(F|Y) is defined as in (13);
Finally, if F ∈ E d ( F | Y ) F ∈ E d F Y F inE_(d)(F|_(Y))F \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)F∈Ed(F|Y) i.e. ‖ F | d = ‖ F | Y | d F d = F Y d ||F|_(d)=||F|_(Y)|_(d)\left.\left.\left\|\left.F\right|_{d}=\right\| F\right|_{Y}\right|_{d}‖F|d=‖F|Y|d, then G 0 = F G 0 = F G_(0)=FG_{0}=FG0=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 , d X , d X,dX, dX,d ) is of finite diameter, i.e. such that sup { d ( x , y ) : x , y ∈ X } = diam X < ∞ sup { d ( x , y ) : x , y ∈ X } = diam X < ∞ s u p{d(x,y):x,y in X}=diam X < oo\sup \{d(x, y): x, y \in X\}=\operatorname{diam} X<\inftysup{d(x,y):x,y∈X}=diamX<∞.
For θ ∈ ( X , d ) θ ∈ ( X , d ) theta in(X,d)\theta \in(X, d)θ∈(X,d) denote c l τ ( d ) { θ } = { x ∈ X : d ( θ , x ) = 0 } c l τ ( d ) { θ } = { x ∈ X : d ( θ , x ) = 0 } cl_(tau(d)){theta}={x in X:d(theta,x)=0}c l_{\tau(d)}\{\theta\}=\{x \in X: d(\theta, x)=0\}clτ(d){θ}={x∈X:d(θ,x)=0} and c l τ ( d ¯ ) { θ } = { x ∈ X : d ( x , θ ) = 0 } c l τ ( d ¯ ) { θ } = { x ∈ X : d ( x , θ ) = 0 } 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\}clτ(d¯){θ}={x∈X:d(x,θ)=0} (see 15, p.68). Let also c l { θ } = c l τ ( d ) { θ } ∪ c l τ ( d ¯ ) { θ } c l { θ } = c l τ ( d ) { θ } ∪ c l τ ( d ¯ ) { θ } 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\}cl{θ}=clτ(d){θ}∪clτ(d¯){θ}.
The following proposition holds:
Proposition 8. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space of finite diameter, and θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X a fixed element. Then every f ∈ d − SLip 0 X f ∈ d − SLip 0 X f in d-SLip_(0)Xf \in d-\operatorname{SLip}_{0} Xf∈d−SLip0X is bounded on X ∖ cl { θ } X ∖ cl { θ } X\\cl{theta}X \backslash \operatorname{cl}\{\theta\}X∖cl{θ}.
Proof. Let f f fff be in d d ddd-SLip 0 X 0 X _(0)X{ }_{0} X0X. By definition, we have f ( θ ) = 0 f ( θ ) = 0 f(theta)=0f(\theta)=0f(θ)=0, and for x ∈ c l τ ( d ¯ ) { θ } = { x ∈ X : d ( x , θ ) = 0 } x ∈ c l τ ( d ¯ ) { θ } = { x ∈ X : d ( x , θ ) = 0 } 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\}x∈clτ(d¯){θ}={x∈X:d(x,θ)=0}-it follows f ( x ) ≤ 0 f ( x ) ≤ 0 f(x) <= 0f(x) \leq 0f(x)≤0, because d ( x , θ ) = 0 d ( x , θ ) = 0 d(x,theta)=0d(x, \theta)=0d(x,θ)=0 implies f ( x ) ≤ f ( θ ) = 0 f ( x ) ≤ f ( θ ) = 0 f(x) <= f(theta)=0f(x) \leq f(\theta)=0f(x)≤f(θ)=0.
Analogously, for x ∈ c l τ ( d ) { θ } = { x ∈ X : d ( θ , x ) = 0 } x ∈ c l τ ( d ) { θ } = { x ∈ X : d ( θ , x ) = 0 } 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\}x∈clτ(d){θ}={x∈X:d(θ,x)=0} it follows 0 = f ( θ ) ≤ f ( x ) 0 = f ( θ ) ≤ f ( x ) 0=f(theta) <= f(x)0=f(\theta) \leq f(x)0=f(θ)≤f(x).
For every x ∈ X ∖ c l τ ( d ¯ ) { θ } x ∈ X ∖ c l τ ( d ¯ ) { θ } x in X\\cl_(tau( bar(d))){theta}x \in X \backslash c l_{\tau(\bar{d})}\{\theta\}x∈X∖clτ(d¯){θ}, we have
f ( x ) − f ( θ ) ≤ ‖ f | d d ( x , θ ) ≤ ‖ f | d diam X , f ( x ) − f ( θ ) ≤ f d d ( x , θ ) ≤ f d diam X , f(x)-f(theta) <= ||f|_(d)d(x,theta) <= ||f|_(d)diam X,f(x)-f(\theta) \leq\left.\left\|\left.f\right|_{d} d(x, \theta) \leq\right\| f\right|_{d} \operatorname{diam} X,f(x)−f(θ)≤‖f|dd(x,θ)≤‖f|ddiamX,
and consequently f ( x ) ≤ ‖ f | d diam X < ∞ f ( x ) ≤ ‖ f d diam X < ∞ f(x) <= ||f|_(d)diam X < oof(x) \leq \|\left. f\right|_{d} \operatorname{diam} X<\inftyf(x)≤‖f|ddiamX<∞.
It follows, f ( x ) ≤ ‖ f | d diam X < ∞ f ( x ) ≤ ‖ f d diam X < ∞ f(x) <= ||f|_(d)diam X < oof(x) \leq \|\left. f\right|_{d} \operatorname{diam} X<\inftyf(x)≤‖f|ddiamX<∞ for all x ∈ X ∖ c l τ ( d ¯ ) { θ } x ∈ X ∖ c l τ ( d ¯ ) { θ } x in X\\cl_(tau( bar(d))){theta}x \in X \backslash c l_{\tau(\bar{d})}\{\theta\}x∈X∖clτ(d¯){θ}.
For every x ∈ X ∖ c l τ ( d ) { θ } x ∈ X ∖ c l τ ( d ) { θ } x in X\\cl_(tau(d)){theta}x \in X \backslash c l_{\tau(d)}\{\theta\}x∈X∖clτ(d){θ} it follows
f ( θ ) − f ( x ) ≤ ‖ f | d d ( θ , x ) ≤ ‖ f | d diam X . f ( θ ) − f ( x ) ≤ f d d ( θ , x ) ≤ f d diam X . f(theta)-f(x) <= ||f|_(d)d(theta,x) <= ||f|_(d)diam X.f(\theta)-f(x) \leq\left.\left\|\left.f\right|_{d} d(\theta, x) \leq\right\| f\right|_{d} \operatorname{diam} X .f(θ)−f(x)≤‖f|dd(θ,x)≤‖f|ddiamX.
Then f ( x ) ≥ − ‖ f | d diam X > − ∞ f ( x ) ≥ − ‖ f d diam X > − ∞ f(x) >= -||f|_(d)diam X > -oof(x) \geq-\|\left. f\right|_{d} \operatorname{diam} X>-\inftyf(x)≥−‖f|ddiamX>−∞, for all x ∈ X ∖ c l τ ( d ) { θ } x ∈ X ∖ c l τ ( d ) { θ } x in X\\cl_(tau(d)){theta}x \in X \backslash c l_{\tau(d)}\{\theta\}x∈X∖clτ(d){θ}. Consequently − ‖ f | d diam X ≤ f ( x ) ≤ ‖ f | d diam X , x ∈ X ∖ c l { θ } − f d diam X ≤ f ( x ) ≤ f d diam X , x ∈ X ∖ c l { θ } -||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\}−‖f|ddiamX≤f(x)≤‖f|ddiamX,x∈X∖cl{θ}.
Now, let ( X , d X , d X,dX, dX,d ) be a quasi-metric space of finite diameter, θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X a fixed element, and Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y. Then, for every F ∈ d − SLip 0 X F ∈ d − SLip 0 X F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} XF∈d−SLip0X, it follows F | Y ∈ d − SLip 0 Y F Y ∈ d − SLip 0 Y F|_(Y)in d-SLip_(0)Y\left.F\right|_{Y} \in d-\operatorname{SLip}_{0} YF|Y∈d−SLip0Y, and the set
E d ( F | Y ) = { G ∈ d − SLip 0 X : G | Y = F | Y , ‖ G | d = ‖ F | Y | d } E d F Y = G ∈ d − SLip 0 X : G Y = F Y , G d = F Y d 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\}Ed(F|Y)={G∈d−SLip0X:G|Y=F|Y,‖G|d=‖F|Y|d}
is non empty.
This set is also ( D d , D ¯ d D d , D ¯ d D_(d), bar(D)_(d)D_{d}, \bar{D}_{d}Dd,D¯d )-compact and the following proposition holds:
Proposition 9. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space of finite diameter, θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X a fixed element, and Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y. Then for every F ∈ d − SLip 0 X F ∈ d − SLip 0 X F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} XF∈d−SLip0X, there exists at least one element G 0 ∈ E d ( F | Y ) G 0 ∈ E d F Y G_(0)inE_(d)(F|_(Y))G_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G0∈Ed(F|Y) such that
‖ ( F − G 0 ) | X ∖ c l { θ } ‖ ∞ = inf { ‖ ( F − G ) | X ∖ c l { θ } ‖ ∞ : G ∈ E d ( F | Y ) } . F − G 0 X ∖ c l { θ } ∞ = inf ( F − G ) X ∖ c l { θ } ∞ : G ∈ E d F Y . ||(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\} .‖(F−G0)|X∖cl{θ}‖∞=inf{‖(F−G)|X∖cl{θ}‖∞:G∈Ed(F|Y)}.
The proof is immediate.
Example 10. Let X = [ − 10 , 10 ] X = [ − 10 , 10 ] X=[-10,10]X=[-10,10]X=[−10,10] and the quasi-metric d : X × X → [ 0 , ∞ ) d : X × X → [ 0 , ∞ ) d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty)d:X×X→[0,∞) defined by
d ( x , y ) = { y − x if x ≤ y 2 ( x − y ) if x > y d ( x , y ) = y − x  if  x ≤ y 2 ( x − y )  if  x > y 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.d(x,y)={y−x if x≤y2(x−y) if x>y
Consider θ = 0 θ = 0 theta=0\theta=0θ=0 and Y = { − 1 , 0 , 1 } Y = { − 1 , 0 , 1 } Y={-1,0,1}Y=\{-1,0,1\}Y={−1,0,1}. Then the function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R
f ( y ) = { − 1 , y = − 1 , 0 , y = 0 , 3 , y = 1 , f ( y ) = − 1 , y = − 1 , 0 , y = 0 , 3 , y = 1 , f(y)={[-1","y=-1","],[0","y=0","],[3","y=1","]:}f(y)=\left\{\begin{aligned} -1, & y=-1, \\ 0, & y=0, \\ 3, & y=1, \end{aligned}\right.f(y)={−1,y=−1,0,y=0,3,y=1,
is in d − SLip 0 Y d − SLip 0 Y d-SLip_(0)Yd-\operatorname{SLip}_{0} Yd−SLip0Y and ‖ f | d = 3 ‖ f d = 3 ||f|_(d)=3\|\left. f\right|_{d}=3‖f|d=3.
The functions
F d ( f ) ( x ) = inf y ∈ Y { f ( y ) + 3 d ( x , y ) } = { − 4 − 3 x , x ∈ [ − 10 , − 1 ] , 6 x + 5 , x ∈ ( − 1 , − 5 9 ] , − 3 x , x ∈ ( − 5 9 , 0 ] , 6 x , x ∈ ( 0 , 2 3 ] , 6 − 3 x , x ∈ ( 2 3 , 1 ] , 6 x − 3 , x ∈ ( 1 , 10 ] . F d ( f ) ( x ) = inf y ∈ Y   { f ( y ) + 3 d ( x , y ) } = − 4 − 3 x , x ∈ [ − 10 , − 1 ] , 6 x + 5 , x ∈ − 1 , − 5 9 , − 3 x , x ∈ − 5 9 , 0 , 6 x , x ∈ 0 , 2 3 , 6 − 3 x , x ∈ 2 3 , 1 , 6 x − 3 , x ∈ ( 1 , 10 ] . {:[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}Fd(f)(x)=infy∈Y{f(y)+3d(x,y)}={−4−3x,x∈[−10,−1],6x+5,x∈(−1,−59],−3x,x∈(−59,0],6x,x∈(0,23],6−3x,x∈(23,1],6x−3,x∈(1,10].
and, respectively
G d ( f ) ( x ) = sup y ∈ Y { f ( y ) − 3 d ( y , x ) } = = { 6 x + 5 , x ∈ [ − 10 , − 1 ] − 3 x + 4 , x ∈ ( − 1 , − 4 9 ] 6 x , x ∈ ( − 4 9 , 0 ] − 3 x , x ∈ ( 0 , 1 3 ] 6 x − 3 , x ∈ ( 1 3 , 1 ] − 3 x − 6 , x ∈ ( 1 , 10 ] G d ( f ) ( x ) = sup y ∈ Y   { f ( y ) − 3 d ( y , x ) } = = 6 x + 5 , x ∈ [ − 10 , − 1 ] − 3 x + 4 , x ∈ − 1 , − 4 9 6 x , x ∈ − 4 9 , 0 − 3 x , x ∈ 0 , 1 3 6 x − 3 , x ∈ 1 3 , 1 − 3 x − 6 , x ∈ ( 1 , 10 ] {:[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}Gd(f)(x)=supy∈Y{f(y)−3d(y,x)}=={6x+5,x∈[−10,−1]−3x+4,x∈(−1,−49]6x,x∈(−49,0]−3x,x∈(0,13]6x−3,x∈(13,1]−3x−6,x∈(1,10]
verifies the conditions:
F d ( f ) | Y = G d ( f ) | Y = f ‖ F d ( f ) | d = ‖ G d ( f ) | d = ‖ f | d = 3 F d ( f ) Y = G d ( f ) Y = f ‖ F d ( f ) d = G d ( f ) d = f d = 3 {:[F_(d)(f)|_(Y)=G_(d)(f)|_(Y)=f],[||F_(d)(f)|_(d)=||G_(d)(f)|_(d)=||f|_(d)=3]:}\begin{aligned} \left.F_{d}(f)\right|_{Y} & =\left.G_{d}(f)\right|_{Y}=f \\ \|\left. F_{d}(f)\right|_{d} & =\left.\left\|\left.G_{d}(f)\right|_{d}=\right\| f\right|_{d}=3 \end{aligned}Fd(f)|Y=Gd(f)|Y=f‖Fd(f)|d=‖Gd(f)|d=‖f|d=3
and
F d ( f ) ( x ) ≥ H ( x ) ≥ G d ( f ) ( x ) , x ∈ [ − 10 , 10 ] F d ( f ) ( x ) ≥ H ( x ) ≥ G d ( f ) ( x ) , x ∈ [ − 10 , 10 ] F_(d)(f)(x) >= H(x) >= G_(d)(f)(x),x in[-10,10]F_{d}(f)(x) \geq H(x) \geq G_{d}(f)(x), x \in[-10,10]Fd(f)(x)≥H(x)≥Gd(f)(x),x∈[−10,10]
where H ∈ E d ( f ) H ∈ E d ( f ) H inE_(d)(f)H \in \mathcal{E}_{d}(f)H∈Ed(f) is an arbitrary extension of f f fff.
Obviously, ( X , d X , d X,dX, dX,d ) is ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compact and E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) is compact in the uniform topology.
Let F ∈ d − SLip 0 X F ∈ d − SLip 0 X F in d-SLip_(0)XF \in d-\operatorname{SLip}_{0} XF∈d−SLip0X such that F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f.
Then
E d ( F | Y ) = E d ( f ) E d F Y = E d ( f ) E_(d)(F|_(Y))=E_(d)(f)\mathcal{E}_{d}\left(\left.F\right|_{Y}\right)=\mathcal{E}_{d}(f)Ed(F|Y)=Ed(f)
If
F ( x ) ≥ F d ( f ) ( x ) , ∀ x ∈ [ − 10 , 10 ] F ( x ) ≥ F d ( f ) ( x ) , ∀ x ∈ [ − 10 , 10 ] F(x) >= F_(d)(f)(x),AA x in[-10,10]F(x) \geq F_{d}(f)(x), \forall x \in[-10,10]F(x)≥Fd(f)(x),∀x∈[−10,10]
then
‖ F − F d ( f ) ‖ ∞ = inf { ‖ F − H ‖ ∞ : H ∈ E d ( F | Y ) } F − F d ( f ) ∞ = inf ‖ F − H ‖ ∞ : H ∈ E d F Y ||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\}‖F−Fd(f)‖∞=inf{‖F−H‖∞:H∈Ed(F|Y)}
For example, let F F FFF be the function
F ( x ) = { F d ( f ) ( x ) , x ∈ [ − 1 , 1 ] , − 4 x − 5 , x ∈ [ − 10 , − 1 ) 7 x − 4 , x ∈ ( 1 , 10 ) F ( x ) = F d ( f ) ( x ) , x ∈ [ − 1 , 1 ] , − 4 x − 5 , x ∈ [ − 10 , − 1 ) 7 x − 4 , x ∈ ( 1 , 10 ) 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.F(x)={Fd(f)(x),x∈[−1,1],−4x−5,x∈[−10,−1)7x−4,x∈(1,10)
Then
‖ F − F d ( f ) ‖ ∞ = max x ∈ [ − 10 , 1 ] { − x − 1 } ∨ max x ∈ [ 1 , 10 ] { x − 1 } = = 9 F − F d ( f ) ∞ = max x ∈ [ − 10 , 1 ]   { − x − 1 } ∨ max x ∈ [ 1 , 10 ]   { x − 1 } = = 9 {:[||F-F_(d)(f)||_(oo)=max_(x in[-10,1]){-x-1}vvmax_(x in[1,10]){x-1}=],[=9]:}\begin{aligned} \left\|F-F_{d}(f)\right\|_{\infty} & =\max _{x \in[-10,1]}\{-x-1\} \vee \max _{x \in[1,10]}\{x-1\}= \\ & =9 \end{aligned}‖F−Fd(f)‖∞=maxx∈[−10,1]{−x−1}∨maxx∈[1,10]{x−1}==9
Similarly, if F ( x ) ≤ G d ( f ) ( x ) , ∀ x ∈ [ − 10 , 10 ] F ( x ) ≤ G d ( f ) ( x ) , ∀ x ∈ [ − 10 , 10 ] F(x) <= G_(d)(f)(x),AA x in[-10,10]F(x) \leq G_{d}(f)(x), \forall x \in[-10,10]F(x)≤Gd(f)(x),∀x∈[−10,10]
then
‖ F − G d ( f ) ‖ ∞ = inf { ‖ F − H ‖ ∞ : H ∈ E d ( F | Y ) } F − G d ( f ) ∞ = inf ‖ F − H ‖ ∞ : H ∈ E d F Y ||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\}‖F−Gd(f)‖∞=inf{‖F−H‖∞:H∈Ed(F|Y)}

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Received by the editors: October 20, 2006.
  1. ∗ ∗ ^(**){ }^{*}∗ 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.
2007

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