On the existence and uniqueness of extensions of semi-Holder real-valued functions

Costica Mustata
(original), Approximation Theory, asymmetric norm, paper

Abstract

Let \((X,d)\) be a quasi-metric space, \(y_{0}\in X\) a fixed element and \(Y\) a subset of \(X\) such that \(y_{0}\in Y\). Denote by \((\Lambda_{\alpha,0}(Y,d),\Vert \cdot|_{Y,d}^{\alpha})\) the asymmetric normed cone of real-valued \(d\)-semi-H\”{o}lder functions defined on \(Y\) of exponent \(\alpha \in(0,1]\), vanishing in \(y_{0}\), and by \((\Lambda_{\alpha,0}(Y,\bar {d}),\Vert \cdot|_{Y,\bar{d}}^{\alpha})\) the similar cone if \(d\) is replaced by conjugate \(\bar{d}\) of \(d\).

Authors

Costică Mustăţa
Tiberiu Popoviciu Institute of Numerical Analysis, Romania

Keywords

Extensions, semi-Lipschitz functions, semi-Holder functions, best approximation, quasi-metric spaces.

Paper coordinates

C. Mustăţa, On the existence and uniqueness of extensions of semi-Holder real-valued functions, Rev. Anal. Numer. Theor. Approx., 39 (2010) no. 2, 134-140.

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Journal

Revue Analysis Numer Theor. Approx.

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Publishing House of the Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2010-vol39-no2-art4

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2502-059X

Online ISSN

2457-6794

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[7] A. Mennucci, On asymmetric distances, Tehnical report, Scuola Normale Superiore, Pisa, 2004.

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2010-Mustata-On the existence and uniqueness-JNAAT

ON THE EXISTENCE AND UNIQUENESS OF EXTENSIONS OF SEMI-HÖLDER REAL-VALUED FUNCTIONS

COSTICĂ MUSTĂŢA*

Abstract

Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X a fixed element and Y Y YYY a subset of X X XXX such that y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y. Denote by ( Λ α , 0 ( Y , d ) , ‖ ⋅ | Y , d α ) Λ α , 0 ( Y , d ) , ‖ ⋅ Y , d α (Lambda_(alpha,0)(Y,d),||*|_(Y,d)^(alpha))\left(\Lambda_{\alpha, 0}(Y, d), \|\left.\cdot\right|_{Y, d} ^{\alpha}\right)(Λα,0(Y,d),‖⋅|Y,dα) the asymmetric normed cone of real-valued d d ddd-semi-Hölder functions defined on Y Y YYY of exponent α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1], vanishing in y 0 y 0 y_(0)y_{0}y0, and by ( Λ α , 0 ( Y , d ¯ ) , ‖ ⋅ | Y , d ¯ α Λ α , 0 ( Y , d ¯ ) , ‖ ⋅ Y , d ¯ α Lambda_(alpha,0)(Y, bar(d)),||*|_(Y, bar(d))^(alpha)\Lambda_{\alpha, 0}(Y, \bar{d}), \|\left.\cdot\right|_{Y, \bar{d}} ^{\alpha}Λα,0(Y,d¯),‖⋅|Y,d¯α ) the similar cone if d d ddd is replaced by conjugate d ¯ d ¯ bar(d)\bar{d}d¯ of d d ddd.

One considers the following claims: (a) For every f f fff in the linear space Λ α , 0 ( Y ) = Λ α , 0 ( Y , d ) ∩ Λ α , 0 ( Y , d ¯ ) Λ α , 0 ( Y ) = Λ α , 0 ( Y , d ) ∩ Λ α , 0 ( Y , d ¯ ) Lambda_(alpha,0)(Y)=Lambda_(alpha,0)(Y,d)nnLambda_(alpha,0)(Y, bar(d))\Lambda_{\alpha, 0}(Y)=\Lambda_{\alpha, 0}(Y, d) \cap \Lambda_{\alpha, 0}(Y, \bar{d})Λα,0(Y)=Λα,0(Y,d)∩Λα,0(Y,d¯) there exist F ∈ Λ α , 0 ( X , d ) F ∈ Λ α , 0 ( X , d ) F inLambda_(alpha,0)(X,d)F \in \Lambda_{\alpha, 0}(X, d)F∈Λα,0(X,d) such that F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f and ‖ F | Y , d α = ‖ f | Y , d α F Y , d α = f Y , d α ||F|_(Y,d)^(alpha)=||f|_(Y,d)^(alpha)\left.\left\|\left.F\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha}‖F|Y,dα=‖f|Y,dα; (b) For every f ∈ Λ α , 0 ( Y ) f ∈ Λ α , 0 ( Y ) f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y)f∈Λα,0(Y) there exists F ¯ ∈ Λ α , 0 ( X , d ¯ ) F ¯ ∈ Λ α , 0 ( X , d ¯ ) bar(F)inLambda_(alpha,0)(X, bar(d))\bar{F} \in \Lambda_{\alpha, 0}(X, \bar{d})F¯∈Λα,0(X,d¯) such that F ¯ | Y = f F ¯ Y = f ( bar(F))|_(Y)=f\left.\bar{F}\right|_{Y}=fF¯|Y=f and ‖ F ¯ | Y , d ¯ α = ‖ f | Y , d ¯ α ; F ¯ Y , d ¯ α = f Y , d ¯ α ; ||( bar(F))|_(Y, bar(d))^(alpha)=||f|_(Y, bar(d))^(alpha);\left.\left\|\left.\bar{F}\right|_{Y, \bar{d}} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha} ;‖F¯|Y,d¯α=‖f|Y,d¯α; (c) The extension F F FFF in (a) is unique; (d) The extension F ¯ F ¯ bar(F)\bar{F}F¯ in (b) is unique; (e) The annihilator Y d ¯ ⊥ Y d ¯ ⊥ Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp}Yd¯⊥ of Y Y YYY in Λ α , 0 ( X , d ¯ ) Λ α , 0 ( X , d ¯ ) Lambda_(alpha,0)(X, bar(d))\Lambda_{\alpha, 0}(X, \bar{d})Λα,0(X,d¯) is proximinal for the elements of Λ α , 0 ( X ) Λ α , 0 ( X ) Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X)Λα,0(X) with respect to the distance generated by ‖ ⋅ | Y , d α ‖ ⋅ Y , d α ||*|_(Y,d)^(alpha)\|\left.\cdot\right|_{Y, d} ^{\alpha}‖⋅|Y,dα; (f) The annihilator Y d ⊥ Y d ⊥ Y_(d)^(_|_)Y_{d}^{\perp}Yd⊥ of Y Y YYY in Λ α , 0 ( X , d ) Λ α , 0 ( X , d ) Lambda_(alpha,0)(X,d)\Lambda_{\alpha, 0}(X, d)Λα,0(X,d) is proximinal for the elements of Λ α , 0 ( X ) Λ α , 0 ( X ) Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X)Λα,0(X) with respect to the distance generated by ‖ ⋅ | Y , d ¯ α ‖ ⋅ Y , d ¯ α ||*|_(Y, bar(d))^(alpha)\|\left.\cdot\right|_{Y, \bar{d}} ^{\alpha}‖⋅|Y,d¯α; (g) Y d ¯ ⊥ Y d ¯ ⊥ Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp}Yd¯⊥ in the claim (e) is Chebyshevian; (h) Y d ⊥ Y d ⊥ Y_(d)^(_|_)Y_{d}^{\perp}Yd⊥ in the claim ( f f fff ) is Chebyshevian.

Then the following equivalences hold:
( a ) ⇔ ( e ) ; ( b ) ⇔ ( f ) ; ( c ) ⇔ ( g ) ; ( d ) ⇔ ( h ) ( a ) ⇔ ( e ) ; ( b ) ⇔ ( f ) ; ( c ) ⇔ ( g ) ; ( d ) ⇔ ( h ) (a)<=>(e);(b)<=>(f);(c)<=>(g);(d)<=>(h)(a) \Leftrightarrow(e) ;(b) \Leftrightarrow(f) ;(c) \Leftrightarrow(g) ;(d) \Leftrightarrow(h)(a)⇔(e);(b)⇔(f);(c)⇔(g);(d)⇔(h).
MSC 2000. 46A22, 41A50, 41A52.
Keywords. Extensions, semi-Lipschitz functions, semi-Hölder functions, best approximation, quasi-metric spaces.

1. INTRODUCTION

Let X X XXX be a nonempty set and 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,∞) a function with the properties:
( QM 1 ) d ( x , y ) = d ( y , x ) = 0 QM 1 d ( x , y ) = d ( y , x ) = 0 (QM_(1))d(x,y)=d(y,x)=0\left(\mathrm{QM}_{1}\right) d(x, y)=d(y, x)=0(QM1)d(x,y)=d(y,x)=0 iff x = y x = y x=yx=yx=y,
( QM 2 ) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) QM 2 d ( x , y ) ≤ d ( x , z ) + d ( z , y ) (QM_(2))d(x,y) <= d(x,z)+d(z,y)\left(\mathrm{QM}_{2}\right) d(x, y) \leq d(x, z)+d(z, y)(QM2)d(x,y)≤d(x,z)+d(z,y),
for all x , y , z ∈ X x , y , z ∈ X x,y,z in Xx, y, z \in Xx,y,z∈X.
Then the function d d ddd is called a quasi-metric on X X XXX and the pair ( X , d X , d X,dX, dX,d ) is called quasi-metric space ([13]).
Because, in general, d ( x , y ) ≠ d ( y , x ) d ( x , y ) ≠ d ( y , x ) d(x,y)!=d(y,x)d(x, y) \neq d(y, x)d(x,y)≠d(y,x), for x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X one defines the conjugate quasi-metric d ¯ d ¯ bar(d)\bar{d}d¯ of d d ddd, by the equality 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.
Let Y Y YYY be a nonvoid subset of ( X , d X , d X,dX, dX,d ) and α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] a fixed number.
Definition 1. a) A A AAA function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is called d d ddd-semi-Hölder (of exponent α α alpha\alphaα ) if there exists a constant K Y ( f ) ≥ 0 K Y ( f ) ≥ 0 K_(Y)(f) >= 0K_{Y}(f) \geq 0KY(f)≥0 such that
(1) f ( x ) − f ( y ) ≤ K Y ( f ) d α ( x , y ) , (1) f ( x ) − f ( y ) ≤ K Y ( f ) d α ( x , y ) , {:(1)f(x)-f(y) <= K_(Y)(f)d^(alpha)(x","y)",":}\begin{equation*} f(x)-f(y) \leq K_{Y}(f) d^{\alpha}(x, y), \tag{1} \end{equation*}(1)f(x)−f(y)≤KY(f)dα(x,y),
for all x , y ∈ Y x , y ∈ Y x,y in Yx, y \in Yx,y∈Y.
b) f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is called d ¯ d ¯ bar(d)\bar{d}d¯-semi-Hölder (of exponent α α alpha\alphaα ) if there exists a constant K ¯ Y ( f ) ≥ 0 K ¯ Y ( f ) ≥ 0 bar(K)_(Y)(f) >= 0\bar{K}_{Y}(f) \geq 0K¯Y(f)≥0 such that
(2) f ( x ) − f ( y ) ≤ K ¯ Y ( f ) ⋅ d α ( y , x ) , (2) f ( x ) − f ( y ) ≤ K ¯ Y ( f ) ⋅ d α ( y , x ) , {:(2)f(x)-f(y) <= bar(K)_(Y)(f)*d^(alpha)(y","x)",":}\begin{equation*} f(x)-f(y) \leq \bar{K}_{Y}(f) \cdot d^{\alpha}(y, x), \tag{2} \end{equation*}(2)f(x)−f(y)≤K¯Y(f)⋅dα(y,x),
for all x , y ∈ Y x , y ∈ Y x,y in Yx, y \in Yx,y∈Y.
The smallest constant K Y ( f ) K Y ( f ) K_(Y)(f)K_{Y}(f)KY(f) in (1) is denoted by ‖ f | Y , d α ‖ f Y , d α ||f|_(Y,d)^(alpha)\|\left. f\right|_{Y, d} ^{\alpha}‖f|Y,dα and one shows that
(3) ‖ f | Y , d α := sup { ( f ( x ) − f ( y ) ) ∨ 0 d α ( x , y ) : d ( x , y ) > 0 ; x , y ∈ Y } . (3) ‖ f Y , d α := sup ( f ( x ) − f ( y ) ) ∨ 0 d α ( x , y ) : d ( x , y ) > 0 ; x , y ∈ Y . {:(3)||f|_(Y,d)^(alpha):=s u p{((f(x)-f(y))vv0)/(d^(alpha)(x,y)):d(x,y) > 0;x,y in Y}.:}\begin{equation*} \|\left. f\right|_{Y, d} ^{\alpha}:=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d^{\alpha}(x, y)}: d(x, y)>0 ; x, y \in Y\right\} . \tag{3} \end{equation*}(3)‖f|Y,dα:=sup{(f(x)−f(y))∨0dα(x,y):d(x,y)>0;x,y∈Y}.
Analogously one defines ‖ f | Y , d ¯ α ‖ f Y , d ¯ α ||f|_(Y, bar(d))^(alpha)\|\left. f\right|_{Y, \bar{d}} ^{\alpha}‖f|Y,d¯α.
Observe that the function f f fff is d d ddd-semi-Hölder on Y Y YYY iff − f − f -f-f−f is d ¯ d ¯ bar(d)\bar{d}d¯-semi-Hölder on Y Y YYY. Moreover
(4) ‖ f | Y , d α = ‖ − f | Y , d ¯ α (4) f Y , d α = − f Y , d ¯ α {:(4)||f|_(Y,d)^(alpha)=||-f|_(Y, bar(d))^(alpha):}\begin{equation*} \left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, \bar{d}} ^{\alpha} \tag{4} \end{equation*}(4)‖f|Y,dα=‖−f|Y,d¯α
Definition 2. ([14). Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space and Y ⊆ X Y ⊆ X Y sube XY \subseteq XY⊆X a nonempty set. The 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 on Y Y YYY 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 , x , y ∈ Y d ( x , y ) = 0 , x , y ∈ Y d(x,y)=0,x,y in Yd(x, y)=0, x, y \in Yd(x,y)=0,x,y∈Y.
The set of all ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing functions on Y Y YYY is denoted by R ≤ d Y R ≤ d Y R_( <= _(d))^(Y)\mathbb{R}_{\leq_{d}}^{Y}R≤dY and it is a cone in the linear space R Y R Y R^(Y)\mathbb{R}^{Y}RY of all real-valued functions on Y Y YYY.
The set
(5) Λ α ( Y , d ) := { f ∈ R ≤ d Y ; f is d -semi-Hölder and ‖ f | Y , d α < ∞ } (5) Λ α ( Y , d ) := f ∈ R ≤ d Y ; f  is  d -semi-Hölder and  ‖ f Y , d α < ∞ {:(5)Lambda_(alpha)(Y","d):={f inR_( <= d)^(Y);f" is "d"-semi-Hölder and "||f|_(Y,d)^(alpha) < oo}:}\begin{equation*} \Lambda_{\alpha}(Y, d):=\left\{f \in \mathbb{R}_{\leq d}^{Y} ; f \text { is } d \text {-semi-Hölder and } \|\left. f\right|_{Y, d} ^{\alpha}<\infty\right\} \tag{5} \end{equation*}(5)Λα(Y,d):={f∈R≤dY;f is d-semi-Hölder and ‖f|Y,dα<∞}
is also a cone, called the cone of d d ddd-semi-Hölder functions on Y Y YYY.
If y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y is arbitrary, but fixed, one considers the cone
(6) Λ α , 0 ( Y , d ) := { f ∈ Λ α ( Y , d ) : f ( y 0 ) = 0 } . (6) Λ α , 0 ( Y , d ) := f ∈ Λ α ( Y , d ) : f y 0 = 0 . {:(6)Lambda_(alpha,0)(Y","d):={f inLambda_(alpha)(Y,d):f(y_(0))=0}.:}\begin{equation*} \Lambda_{\alpha, 0}(Y, d):=\left\{f \in \Lambda_{\alpha}(Y, d): f\left(y_{0}\right)=0\right\} . \tag{6} \end{equation*}(6)Λα,0(Y,d):={f∈Λα(Y,d):f(y0)=0}.
Then the functional ‖ | Y , d α : Λ α , 0 ( Y , d ) → [ 0 , ∞ ) ‖ Y , d α : Λ α , 0 ( Y , d ) → [ 0 , ∞ ) |||_(Y,d)^(alpha):Lambda_(alpha,0)(Y,d)rarr[0,oo)\|\left.\right|_{Y, d} ^{\alpha}: \Lambda_{\alpha, 0}(Y, d) \rightarrow[0, \infty)‖|Y,dα:Λα,0(Y,d)→[0,∞) is subadditive, positively homogeneous and the equality ‖ f | Y , d α = ‖ − f | Y , d α = 0 f Y , d α = − f Y , d α = 0 ||f|_(Y,d)^(alpha)=||-f|_(Y,d)^(alpha)=0\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, d} ^{\alpha}=0‖f|Y,dα=‖−f|Y,dα=0 implies f ≡ 0 f ≡ 0 f-=0f \equiv 0f≡0. This means that ‖ ⋅ | Y , d α ‖ ⋅ Y , d α ||*|_(Y,d)^(alpha)\|\left.\cdot\right|_{Y, d} ^{\alpha}‖⋅|Y,dα is an asymmetric norm (see [13, [14]), on the cone Λ α , 0 ( Y , d ) Λ α , 0 ( Y , d ) Lambda_(alpha,0)(Y,d)\Lambda_{\alpha, 0}(Y, d)Λα,0(Y,d).
The pair ( Λ α , 0 ( Y , d ) , ‖ | Y , d α Λ α , 0 ( Y , d ) , ‖ Y , d α Lambda_(alpha,0)(Y,d),|||_(Y,d)^(alpha)\Lambda_{\alpha, 0}(Y, d), \|\left.\right|_{Y, d} ^{\alpha}Λα,0(Y,d),‖|Y,dα ) is called the asymmetric normed cone of d d ddd -semi-Hölder real-valued function on Y Y YYY (compare with [14]).
Analogously, one defines the asymmetric normed cone ( Λ α , 0 ( Y , d ¯ ) , ‖ ⋅ | Y , d ¯ α Λ α , 0 ( Y , d ¯ ) , ‖ ⋅ Y , d ¯ α Lambda_(alpha,0)(Y, bar(d)),||*|_(Y, bar(d))^(alpha)\Lambda_{\alpha, 0}(Y, \bar{d}), \|\left.\cdot\right|_{Y, \bar{d}} ^{\alpha}Λα,0(Y,d¯),‖⋅|Y,d¯α ). of all d ¯ d ¯ bar(d)\bar{d}d¯-semi-Hölder real-valued functions on Y Y YYY, vanishing at the fixed point y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y.
By the above definitions it follows that
f ∈ ( Λ α , 0 ( Y , d ) , ‖ | Y , d α ) iff − f ∈ ( Λ α , 0 ( Y , d ¯ ) , ‖ | Y , d ¯ α ) f ∈ Λ α , 0 ( Y , d ) , ‖ Y , d α  iff  − f ∈ Λ α , 0 ( Y , d ¯ ) , ‖ Y , d ¯ α f in(Lambda_(alpha,0)(Y,d),|||_(Y,d)^(alpha))" iff "-f in(Lambda_(alpha,0)(Y,( bar(d))),|||_(Y, bar(d))^(alpha))f \in\left(\Lambda_{\alpha, 0}(Y, d), \|\left.\right|_{Y, d} ^{\alpha}\right) \text { iff }-f \in\left(\Lambda_{\alpha, 0}(Y, \bar{d}), \|\left.\right|_{Y, \bar{d}} ^{\alpha}\right)f∈(Λα,0(Y,d),‖|Y,dα) iff −f∈(Λα,0(Y,d¯),‖|Y,d¯α)
and, moreover, ‖ f | Y , d α = ‖ − f | Y , d ¯ α f Y , d α = − f Y , d ¯ α ||f|_(Y,d)^(alpha)=||-f|_(Y, bar(d))^(alpha)\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, \bar{d}} ^{\alpha}‖f|Y,dα=‖−f|Y,d¯α.
Defining Λ α , 0 ( Y ) Λ α , 0 ( Y ) Lambda_(alpha,0)(Y)\Lambda_{\alpha, 0}(Y)Λα,0(Y) by
(7) Λ α , 0 ( Y ) = Λ α , 0 ( Y , d ) ∩ Λ α , 0 ( Y , d ¯ ) (7) Λ α , 0 ( Y ) = Λ α , 0 ( Y , d ) ∩ Λ α , 0 ( Y , d ¯ ) {:(7)Lambda_(alpha,0)(Y)=Lambda_(alpha,0)(Y","d)nnLambda_(alpha,0)(Y"," bar(d)):}\begin{equation*} \Lambda_{\alpha, 0}(Y)=\Lambda_{\alpha, 0}(Y, d) \cap \Lambda_{\alpha, 0}(Y, \bar{d}) \tag{7} \end{equation*}(7)Λα,0(Y)=Λα,0(Y,d)∩Λα,0(Y,d¯)
It follows that Λ α , 0 ( Y ) Λ α , 0 ( Y ) Lambda_(alpha,0)(Y)\Lambda_{\alpha, 0}(Y)Λα,0(Y) is a linear subspace. The following, theorem holds.
Theorem 3. For every f ∈ Λ α , 0 ( Y ) f ∈ Λ α , 0 ( Y ) f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y)f∈Λα,0(Y) there exist at least one function F ∈ Λ α , 0 ( Y , d ) F ∈ Λ α , 0 ( Y , d ) F inLambda_(alpha,0)(Y,d)F \in \Lambda_{\alpha, 0}(Y, d)F∈Λα,0(Y,d) and at least one function F ¯ ∈ Λ α , 0 ( Y , d ¯ ) F ¯ ∈ Λ α , 0 ( Y , d ¯ ) bar(F)inLambda_(alpha,0)(Y, bar(d))\bar{F} \in \Lambda_{\alpha, 0}(Y, \bar{d})F¯∈Λα,0(Y,d¯) such that
a) F | Y = F ¯ | Y = f F Y = F ¯ Y = f F|_(Y)=( bar(F))|_(Y)=f\left.F\right|_{Y}=\left.\bar{F}\right|_{Y}=fF|Y=F¯|Y=f.
b) ‖ F | Y , d α = ‖ f | Y , d α F Y , d α = f Y , d α ||F|_(Y,d)^(alpha)=||f|_(Y,d)^(alpha)\left.\left\|\left.F\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha}‖F|Y,dα=‖f|Y,dα and ‖ F ¯ | Y , d ¯ α = ‖ f | Y , d ¯ α F ¯ Y , d ¯ α = f Y , d ¯ α ||( bar(F))|_(Y, bar(d))^(alpha)=||f|_(Y, bar(d))^(alpha)\left.\left\|\left.\bar{F}\right|_{Y, \bar{d}} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha}‖F¯|Y,d¯α=‖f|Y,d¯α.
Proof. By Theorem 2 and Remark 3 in [11] it follows that the functions defined by the formulae:
(8) F ( f ) ( x ) = inf y ∈ Y { f ( y ) + ‖ f | Y , d α d α ( x , y ) } , x ∈ X G ( f ) ( x ) = sup y ∈ Y { f ( y ) − ‖ f | Y , d α d α ( y , x } , x ∈ X (8) F ( f ) ( x ) = inf y ∈ Y   f ( y ) + ‖ f Y , d α d α ( x , y ) , x ∈ X G ( f ) ( x ) = sup y ∈ Y   f ( y ) − ‖ f Y , d α d α ( y , x } , x ∈ X {:[(8)F(f)(x)=i n f_(y in Y){f(y)+||f|_(Y,d)^(alpha)d^(alpha)(x,y)}","x in X],[G(f)(x)=s u p_(y in Y){f(y)-||f|_(Y,d)^(alpha)d^(alpha)(y,x},x in X:}]:}\begin{align*} & F(f)(x)=\inf _{y \in Y}\left\{f(y)+\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(x, y)\right\}, x \in X \tag{8}\\ & G(f)(x)=\sup _{y \in Y}\left\{f(y)-\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(y, x\}, x \in X\right. \end{align*}(8)F(f)(x)=infy∈Y{f(y)+‖f|Y,dαdα(x,y)},x∈XG(f)(x)=supy∈Y{f(y)−‖f|Y,dαdα(y,x},x∈X
are elements of Λ α , 0 ( X , d ) Λ α , 0 ( X , d ) Lambda_(alpha,0)(X,d)\Lambda_{\alpha, 0}(X, d)Λα,0(X,d) and, respectively, the functions given by
(9) F ¯ ( f ) ( x ) = inf y ∈ Y { f ( y ) + ‖ f | Y , d ¯ α d α ( y , x ) } , x ∈ X G ¯ ( f ) ( x ) = sup y ∈ Y { f ( y ) − ‖ f | Y , d ¯ α d α ( x , y ) } , x ∈ X (9) F ¯ ( f ) ( x ) = inf y ∈ Y   f ( y ) + ‖ f Y , d ¯ α d α ( y , x ) , x ∈ X G ¯ ( f ) ( x ) = sup y ∈ Y   f ( y ) − ‖ f Y , d ¯ α d α ( x , y ) , x ∈ X {:[(9) bar(F)(f)(x)=i n f_(y in Y){f(y)+||f|_(Y, bar(d))^(alpha)d^(alpha)(y,x)}","x in X],[ bar(G)(f)(x)=s u p_(y in Y){f(y)-||f|_(Y, bar(d))^(alpha)d^(alpha)(x,y)}","x in X]:}\begin{align*} & \bar{F}(f)(x)=\inf _{y \in Y}\left\{f(y)+\|\left. f\right|_{Y, \bar{d}} ^{\alpha} d^{\alpha}(y, x)\right\}, x \in X \tag{9}\\ & \bar{G}(f)(x)=\sup _{y \in Y}\left\{f(y)-\|\left. f\right|_{Y, \bar{d}} ^{\alpha} d^{\alpha}(x, y)\right\}, x \in X \end{align*}(9)F¯(f)(x)=infy∈Y{f(y)+‖f|Y,d¯αdα(y,x)},x∈XG¯(f)(x)=supy∈Y{f(y)−‖f|Y,d¯αdα(x,y)},x∈X
are elements of Λ α , 0 ( X , d ¯ ) Λ α , 0 ( X , d ¯ ) Lambda_(alpha,0)(X, bar(d))\Lambda_{\alpha, 0}(X, \bar{d})Λα,0(X,d¯) such that
(10) F ( f ) | Y = G ( f ) | Y = f and ‖ F ( f ) | Y , d α = ‖ G ( f ) | Y , d α = ‖ f | Y , d α (10) F ( f ) Y = G ( f ) Y = f  and  F ( f ) Y , d α = G ( f ) Y , d α = ‖ f Y , d α {:(10)F(f)|_(Y)=G(f)|_(Y)=f" and "||F(f)|_(Y,d)^(alpha)=||G(f)|_(Y,d)^(alpha)=||f|_(Y,d)^(alpha):}\begin{equation*} \left.F(f)\right|_{Y}=\left.G(f)\right|_{Y}=f \text { and }\left.\left\|\left.F(f)\right|_{Y, d} ^{\alpha}=\right\| G(f)\right|_{Y, d} ^{\alpha}=\|\left. f\right|_{Y, d} ^{\alpha} \tag{10} \end{equation*}(10)F(f)|Y=G(f)|Y=f and ‖F(f)|Y,dα=‖G(f)|Y,dα=‖f|Y,dα
respectively
(11) F ¯ ( f ) | Y = G ¯ ( f ) | Y = f and ‖ F ¯ ( f ) | Y , d ¯ α = ‖ G ¯ ( f ) | Y , d ¯ α = ‖ f | Y , d ¯ α (11) F ¯ ( f ) Y = G ¯ ( f ) Y = f  and  F ¯ ( f ) Y , d ¯ α = G ¯ ( f ) Y , d ¯ α = ‖ f Y , d ¯ α {:(11)( bar(F))(f)|_(Y)=( bar(G))(f)|_(Y)=f" and "||( bar(F))(f)|_(Y, bar(d))^(alpha)=||( bar(G))(f)|_(Y, bar(d))^(alpha)=||f|_(Y, bar(d))^(alpha):}\begin{equation*} \left.\bar{F}(f)\right|_{Y}=\left.\bar{G}(f)\right|_{Y}=f \text { and }\left.\left\|\left.\bar{F}(f)\right|_{Y, \bar{d}} ^{\alpha}=\right\| \bar{G}(f)\right|_{Y, \bar{d}} ^{\alpha}=\|\left. f\right|_{Y, \bar{d}} ^{\alpha} \tag{11} \end{equation*}(11)F¯(f)|Y=G¯(f)|Y=f and ‖F¯(f)|Y,d¯α=‖G¯(f)|Y,d¯α=‖f|Y,d¯α
For f ∈ Λ α , 0 ( Y ) f ∈ Λ α , 0 ( Y ) f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y)f∈Λα,0(Y) let us consider the following (nonempty) sets of extensions:
(12) E d ( f ) := { H ∈ Λ α , 0 ( X , d ) : H | Y = f and ‖ H | Y , d α = ‖ f | Y , d α } (12) E d ( f ) := H ∈ Λ α , 0 ( X , d ) : H Y = f  and  H Y , d α = f Y , d α {:(12)E_(d)(f):={H inLambda_(alpha,0)(X,d):H|_(Y)=f" and "||H|_(Y,d)^(alpha)=||f|_(Y,d)^(alpha)}:}\begin{equation*} \mathcal{E}_{d}(f):=\left\{H \in \Lambda_{\alpha, 0}(X, d):\left.H\right|_{Y}=f \text { and }\left.\left\|\left.H\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha}\right\} \tag{12} \end{equation*}(12)Ed(f):={H∈Λα,0(X,d):H|Y=f and ‖H|Y,dα=‖f|Y,dα}
and
(13) E d ¯ ( f ) := { H ¯ ∈ ∧ α , 0 ( X , d ¯ ) : H ¯ | Y = f and ‖ H ¯ | Y , d ¯ α = ‖ f | Y , d ¯ α } (13) E d ¯ ( f ) := H ¯ ∈ ∧ α , 0 ( X , d ¯ ) : H ¯ Y = f  and  H ¯ Y , d ¯ α = f Y , d ¯ α {:(13)E_( bar(d))(f):={( bar(H))in^^_(alpha,0)(X,( bar(d))):( bar(H))|_(Y)=f" and "||( bar(H))|_(Y, bar(d))^(alpha)=||f|_(Y, bar(d))^(alpha)}:}\begin{equation*} \mathcal{E}_{\bar{d}}(f):=\left\{\bar{H} \in \wedge_{\alpha, 0}(X, \bar{d}):\left.\bar{H}\right|_{Y}=f \text { and }\left.\left\|\left.\bar{H}\right|_{Y, \bar{d}} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha}\right\} \tag{13} \end{equation*}(13)Ed¯(f):={H¯∈∧α,0(X,d¯):H¯|Y=f and ‖H¯|Y,d¯α=‖f|Y,d¯α}
The sets E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) and E d ¯ ( f ) E d ¯ ( f ) E_( bar(d))(f)\mathcal{E}_{\bar{d}}(f)Ed¯(f) are convex and
(14) F ( f ) ( x ) ≥ H ( x ) ≥ G ( f ) ( x ) , x ∈ X (14) F ( f ) ( x ) ≥ H ( x ) ≥ G ( f ) ( x ) , x ∈ X {:(14)F(f)(x) >= H(x) >= G(f)(x)","x in X:}\begin{equation*} F(f)(x) \geq H(x) \geq G(f)(x), x \in X \tag{14} \end{equation*}(14)F(f)(x)≥H(x)≥G(f)(x),x∈X
for all H ∈ E d ( f ) H ∈ E d ( f ) H inE_(d)(f)H \in \mathcal{E}_{d}(f)H∈Ed(f);
(15) F ¯ ( f ) ( x ) ≥ H ¯ ( x ) ≥ G ¯ ( f ) ( x ) , x ∈ H , (15) F ¯ ( f ) ( x ) ≥ H ¯ ( x ) ≥ G ¯ ( f ) ( x ) , x ∈ H , {:(15) bar(F)(f)(x) >= bar(H)(x) >= bar(G)(f)(x)","x in H",":}\begin{equation*} \bar{F}(f)(x) \geq \bar{H}(x) \geq \bar{G}(f)(x), x \in H, \tag{15} \end{equation*}(15)F¯(f)(x)≥H¯(x)≥G¯(f)(x),x∈H,
for all H ¯ ∈ E d ¯ ( f ) H ¯ ∈ E d ¯ ( f ) bar(H)inE_( bar(d))(f)\bar{H} \in \mathcal{E}_{\bar{d}}(f)H¯∈Ed¯(f).
Also, for F ∈ Λ α , 0 ( X ) , F | Y ∈ Λ α , 0 ( Y ) F ∈ Λ α , 0 ( X ) , F Y ∈ Λ α , 0 ( Y ) F inLambda_(alpha,0)(X),F|_(Y)inLambda_(alpha,0)(Y)F \in \Lambda_{\alpha, 0}(X),\left.F\right|_{Y} \in \Lambda_{\alpha, 0}(Y)F∈Λα,0(X),F|Y∈Λα,0(Y) and
F − H ∈ Λ α , 0 ( X , d ¯ ) , for all H ∈ E d ( F | Y ) , F − H ¯ ∈ Λ α , 0 ( X , d ) for all H ¯ ∈ E d ¯ ( F | Y ) . F − H ∈ Λ α , 0 ( X , d ¯ ) ,  for all  H ∈ E d F Y , F − H ¯ ∈ Λ α , 0 ( X , d )  for all  H ¯ ∈ E d ¯ F Y . {:[F-H inLambda_(alpha,0)(X"," bar(d))","" for all "H inE_(d)(F|_(Y))","],[F- bar(H)inLambda_(alpha,0)(X","d)" for all " bar(H)inE_( bar(d))(F|_(Y)).]:}\begin{aligned} & F-H \in \Lambda_{\alpha, 0}(X, \bar{d}), \text { for all } H \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right), \\ & F-\bar{H} \in \Lambda_{\alpha, 0}(X, d) \text { for all } \bar{H} \in \mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right) . \end{aligned}F−H∈Λα,0(X,d¯), for all H∈Ed(F|Y),F−H¯∈Λα,0(X,d) for all H¯∈Ed¯(F|Y).
Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X fixed and Y ⊆ X Y ⊆ X Y sube XY \subseteq XY⊆X such that y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y.
Let
(16) Y d ⊥ := { G ∈ Λ α , 0 ( X , d ) : G | Y = 0 } (16) Y d ⊥ := G ∈ Λ α , 0 ( X , d ) : G Y = 0 {:(16)Y_(d)^(_|_):={G inLambda_(alpha,0)(X,d):G|_(Y)=0}:}\begin{equation*} Y_{d}^{\perp}:=\left\{G \in \Lambda_{\alpha, 0}(X, d):\left.G\right|_{Y}=0\right\} \tag{16} \end{equation*}(16)Yd⊥:={G∈Λα,0(X,d):G|Y=0}
and
(17) Y d ¯ ⊥ := { G ¯ ∈ ∧ α , d ( X , d ¯ ) : G ¯ | Y = 0 } . (17) Y d ¯ ⊥ := G ¯ ∈ ∧ α , d ( X , d ¯ ) : G ¯ Y = 0 . {:(17)Y_( bar(d))^(_|_):={( bar(G))in^^_(alpha,d)(X,( bar(d))):( bar(G))|_(Y)=0}.:}\begin{equation*} Y_{\bar{d}}^{\perp}:=\left\{\bar{G} \in \wedge_{\alpha, d}(X, \bar{d}):\left.\bar{G}\right|_{Y}=0\right\} . \tag{17} \end{equation*}(17)Yd¯⊥:={G¯∈∧α,d(X,d¯):G¯|Y=0}.
Obviously, for F ∈ Λ α , 0 ( X ) F ∈ Λ α , 0 ( X ) F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X)F∈Λα,0(X)
(18) F − E d ( F | Y ) ⊂ Λ α , 0 ( X , d ¯ ) (18) F − E d F Y ⊂ Λ α , 0 ( X , d ¯ ) {:(18)F-E_(d)(F|_(Y))subLambda_(alpha,0)(X"," bar(d)):}\begin{equation*} F-\mathcal{E}_{d}\left(\left.F\right|_{Y}\right) \subset \Lambda_{\alpha, 0}(X, \bar{d}) \tag{18} \end{equation*}(18)F−Ed(F|Y)⊂Λα,0(X,d¯)
and
(19) F − E d ¯ ( F | Y ) ⊂ Λ α , 0 ( X , d ) . (19) F − E d ¯ F Y ⊂ Λ α , 0 ( X , d ) . {:(19)F-E_( bar(d))(F|_(Y))subLambda_(alpha,0)(X","d).:}\begin{equation*} F-\mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right) \subset \Lambda_{\alpha, 0}(X, d) . \tag{19} \end{equation*}(19)F−Ed¯(F|Y)⊂Λα,0(X,d).
In the sequel we prove a result of Phelps type ([1], [10, 12]) concerning the existence and uniqueness of the extensions preserving the smallest semi-Hölder constants and a problem of best approximation by elements of Y d ⊥ Y d ⊥ Y_(d)^(_|_)Y_{d}^{\perp}Yd⊥ and Y d ¯ ⊥ Y d ¯ ⊥ Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp}Yd¯⊥, respectively.
Let ( X , ‖ ‖ ) ( X , ‖ ‖ ) (X,||||)(X,\| \|)(X,‖‖) be an asymmetric norm (see [13, [14]) and let M M MMM be a nonempty set of X X XXX. The set M M MMM is called proximinal for x ∈ X x ∈ X x in Xx \in Xx∈X iff there exists at least one element m 0 ∈ M m 0 ∈ M m_(0)in Mm_{0} \in Mm0∈M such that
| | x − m 0 | = inf { | | x − m ∣: m ∈ M } = ρ ( x , M ) . x − m 0 = inf { | | x − m ∣: m ∈ M } = ρ ( x , M ) . ||x-m_(0)|=i n f{||x-m∣:m in M}=rho(x,M).:}\left|\left|x-m_{0}\right|=\inf \{| | x-m \mid: m \in M\}=\rho(x, M) .\right.||x−m0|=inf{||x−m∣:m∈M}=ρ(x,M).
If M M MMM is proximinal for x x xxx, then the set P M ( x ) = { m 0 ∈ M : ‖ x − m 0 ∣= ρ ( x , M ) } P M ( x ) = m 0 ∈ M : ‖ x − m 0 ∣= ρ ( x , M ) } P_(M)(x)={m_(0)in M:||x-m_(0)∣=:}rho(x,M)}P_{M}(x)=\left\{m_{0} \in M: \| x-m_{0} \mid=\right. \rho(x, M)\}PM(x)={m0∈M:‖x−m0∣=ρ(x,M)} is called the set of elements of best approximations for x x xxx in M M MMM. If card P M ( x ) = 1 P M ( x ) = 1 P_(M)(x)=1P_{M}(x)=1PM(x)=1 then the set M M MMM is called Chebyshevian for x x xxx.
The set M M MMM is called proximinal if M M MMM is proximinal for every x ∈ X x ∈ X x in Xx \in Xx∈X, and Chebyshevian if M M MMM is Chebyshevian for every x ∈ X x ∈ X x in Xx \in Xx∈X.
Now, consider the following two problems of best approximation:
P d ― ( F ) P d ¯ ( F ) P_( bar(d))(F)\mathbf{P}_{\overline{\mathrm{d}}}(\mathbf{F})Pd―(F). For F ∈ Λ α , 0 ( X ) F ∈ Λ α , 0 ( X ) F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X)F∈Λα,0(X) find G 0 ∈ Y d ⊥ G 0 ∈ Y d ⊥ G_(0)inY_(d)^(_|_)G_{0} \in Y_{d}^{\perp}G0∈Yd⊥ such that
(20) ‖ F − G 0 | Y , d ¯ α = inf { ‖ F − G | Y , d ¯ α : G ∈ Y d ⊥ } = ρ d ¯ ( F , Y d ⊥ ) (20) ‖ F − G 0 Y , d ¯ α = inf ‖ F − G Y , d ¯ α : G ∈ Y d ⊥ = ρ d ¯ F , Y d ⊥ {:(20)||F-G_(0)|_(Y, bar(d))^(alpha)=i n f{||F-G|_(Y, bar(d))^(alpha):G inY_(d)^(_|_)}=rho_( bar(d))(F,Y_(d)^(_|_)):}\begin{equation*} \| F-\left.G_{0}\right|_{Y, \bar{d}} ^{\alpha}=\inf \left\{\| F-\left.G\right|_{Y, \bar{d}} ^{\alpha}: G \in Y_{d}^{\perp}\right\}=\rho_{\bar{d}}\left(F, Y_{d}^{\perp}\right) \tag{20} \end{equation*}(20)‖F−G0|Y,d¯α=inf{‖F−G|Y,d¯α:G∈Yd⊥}=ρd¯(F,Yd⊥)
and
P d ( F ) P d ( F ) P_(d)(F)\mathbf{P}_{\mathrm{d}}(\mathbf{F})Pd(F). For F ∈ Λ α , 0 ( X ) F ∈ Λ α , 0 ( X ) F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X)F∈Λα,0(X) find G ¯ 0 ∈ Y d ¯ ⊥ G ¯ 0 ∈ Y d ¯ ⊥ bar(G)_(0)inY_( bar(d))^(_|_)\bar{G}_{0} \in Y_{\bar{d}}^{\perp}G¯0∈Yd¯⊥ such that
(21) ‖ F − G ¯ 0 | Y , d α = inf { ‖ F − G ¯ | X , d α : G ¯ ∈ Y d ¯ ⊥ } = ρ d ( F , Y d ¯ ⊥ ) . (21) ‖ F − G ¯ 0 Y , d α = inf ‖ F − G ¯ X , d α : G ¯ ∈ Y d ¯ ⊥ = ρ d F , Y d ¯ ⊥ . {:(21)||F- bar(G)_(0)|_(Y,d)^(alpha)=i n f{||F-( bar(G))|_(X,d)^(alpha):( bar(G))inY_( bar(d))^(_|_)}=rho_(d)(F,Y_( bar(d))^(_|_)).:}\begin{equation*} \| F-\left.\bar{G}_{0}\right|_{Y, d} ^{\alpha}=\inf \left\{\| F-\left.\bar{G}\right|_{X, d} ^{\alpha}: \bar{G} \in Y_{\bar{d}}^{\perp}\right\}=\rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right) . \tag{21} \end{equation*}(21)‖F−G¯0|Y,dα=inf{‖F−G¯|X,dα:G¯∈Yd¯⊥}=ρd(F,Yd¯⊥).
Let
(22) P Y d ¯ ⊥ ( F ) := { G ¯ 0 ∈ Y d ¯ ⊥ : ‖ F − G ¯ 0 | X , d α = ρ d ( F , Y d ¯ ⊥ ) } (22) P Y d ¯ ⊥ ( F ) := G ¯ 0 ∈ Y d ¯ ⊥ : ‖ F − G ¯ 0 X , d α = ρ d F , Y d ¯ ⊥ {:(22)P_(Y_( bar(d))^(_|_))(F):={ bar(G)_(0)inY_( bar(d))^(_|_):||F- bar(G)_(0)|_(X,d)^(alpha)=rho_(d)(F,Y_( bar(d))^(_|_))}:}\begin{equation*} P_{Y_{\bar{d}}^{\perp}}(F):=\left\{\bar{G}_{0} \in Y_{\bar{d}}^{\perp}: \| F-\left.\bar{G}_{0}\right|_{X, d} ^{\alpha}=\rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right)\right\} \tag{22} \end{equation*}(22)PYd¯⊥(F):={G¯0∈Yd¯⊥:‖F−G¯0|X,dα=ρd(F,Yd¯⊥)}
and
(23) P Y d ⊥ ( F ) := { G 0 ∈ Y d ⊥ : ‖ F − G 0 | X , d ¯ α = ρ d ¯ ( F , Y d ⊥ ) } . (23) P Y d ⊥ ( F ) := G 0 ∈ Y d ⊥ : ‖ F − G 0 X , d ¯ α = ρ d ¯ F , Y d ⊥ . {:(23)P_(Y_(d)^(_|_))(F):={G_(0)inY_(d)^(_|_):||F-G_(0)|_(X, bar(d))^(alpha)=rho_( bar(d))(F,Y_(d)^(_|_))}.:}\begin{equation*} P_{Y_{d}^{\perp}}(F):=\left\{G_{0} \in Y_{d}^{\perp}: \| F-\left.G_{0}\right|_{X, \bar{d}} ^{\alpha}=\rho_{\bar{d}}\left(F, Y_{d}^{\perp}\right)\right\} . \tag{23} \end{equation*}(23)PYd⊥(F):={G0∈Yd⊥:‖F−G0|X,d¯α=ρd¯(F,Yd⊥)}.
The following theorem holds.
Theorem 4. If F ∈ Λ α , 0 ( X ) F ∈ Λ α , 0 ( X ) F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X)F∈Λα,0(X) then
and
(24) P Y d ¯ ⊥ ( F ) = F − E d ( F | Y ) (25) P Y d ¯ ⊥ ( F ) = F − E d ¯ ( F | Y ) (24) P Y d ¯ ⊥ ( F ) = F − E d F Y (25) P Y d ¯ ⊥ ( F ) = F − E d ¯ F Y {:[(24)P_(Y_( bar(d))^(_|_))(F)=F-E_(d)(F|_(Y))],[(25)P_(Y_( bar(d))^(_|_))(F)=F-E_( bar(d))(F|_(Y))]:}\begin{align*} & P_{Y_{\bar{d}}^{\perp}}(F)=F-\mathcal{E}_{d}\left(\left.F\right|_{Y}\right) \tag{24}\\ & P_{Y_{\bar{d}}^{\perp}}(F)=F-\mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right) \tag{25} \end{align*}(24)PYd¯⊥(F)=F−Ed(F|Y)(25)PYd¯⊥(F)=F−Ed¯(F|Y)
and
(26) ρ d ( F , Y d ¯ ⊥ ) = ‖ F | Y | Y , d α (27) ρ d ¯ ( F , Y d ⊥ ) = ‖ F | Y | Y , d ¯ α (26) ρ d F , Y d ¯ ⊥ = ‖ F Y Y , d α (27) ρ d ¯ F , Y d ⊥ = ‖ F Y Y , d ¯ α {:[(26)rho_(d)(F,Y_( bar(d))^(_|_))=||F|_(Y)|_(Y,d)^(alpha)],[(27)rho_( bar(d))(F,Y_(d)^(_|_))=||F|_(Y)|_(Y, bar(d))^(alpha)]:}\begin{align*} & \rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha} \tag{26}\\ & \rho_{\bar{d}}\left(F, Y_{d}^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|_{Y, \bar{d}} ^{\alpha} \tag{27} \end{align*}(26)ρd(F,Yd¯⊥)=‖F|Y|Y,dα(27)ρd¯(F,Yd⊥)=‖F|Y|Y,d¯α
Proof. Let F ∈ Λ α , 0 ( X ) ( = Λ α , 0 ( X , d ) ∩ Λ α , 0 ( X , d ¯ ) ) F ∈ Λ α , 0 ( X ) = Λ α , 0 ( X , d ) ∩ Λ α , 0 ( X , d ¯ ) F inLambda_(alpha,0)(X)(=Lambda_(alpha,0)(X,d)nnLambda_(alpha,0)(X,( bar(d))))F \in \Lambda_{\alpha, 0}(X)\left(=\Lambda_{\alpha, 0}(X, d) \cap \Lambda_{\alpha, 0}(X, \bar{d})\right)F∈Λα,0(X)(=Λα,0(X,d)∩Λα,0(X,d¯))
Then, for every G ¯ ∈ Y d ¯ ⊥ G ¯ ∈ Y d ¯ ⊥ bar(G)inY_( bar(d))^(_|_)\bar{G} \in Y_{\bar{d}}^{\perp}G¯∈Yd¯⊥,
‖ F | Y | Y , d α = ‖ F | Y − G ¯ | Y | Y , d α ≤ ‖ F − G ¯ | X , d α F Y Y , d α = F Y − G ¯ Y Y , d α ≤ ‖ F − G ¯ X , d α ||F|_(Y)|_(Y,d)^(alpha)=||F|_(Y)-( bar(G))|_(Y)|_(Y,d)^(alpha) <= ||F-( bar(G))|_(X,d)^(alpha)\left.\left\|\left.\left.F\right|_{Y}\right|_{Y, d} ^{\alpha}=\right\| F\right|_{Y}-\left.\left.\bar{G}\right|_{Y}\right|_{Y, d} ^{\alpha} \leq \| F-\left.\bar{G}\right|_{X, d} ^{\alpha}‖F|Y|Y,dα=‖F|Y−G¯|Y|Y,dα≤‖F−G¯|X,dα
Taking the infimum with respect to G ¯ ∈ Y d ¯ ⊥ G ¯ ∈ Y d ¯ ⊥ bar(G)inY_( bar(d))^(_|_)\bar{G} \in Y_{\bar{d}}^{\perp}G¯∈Yd¯⊥, one obtains ‖ F | Y | Y , d α ≤ ρ d ( F , Y d ¯ ⊥ ) ‖ F Y Y , d α ≤ ρ d F , Y d ¯ ⊥ ||F|_(Y)|_(Y,d)^(alpha) <= rho_(d)(F,Y_( bar(d))^(_|_))\|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha} \leq \rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right)‖F|Y|Y,dα≤ρd(F,Yd¯⊥). On the other hand, for every H ∈ E d ( F | Y ) H ∈ E d F Y H inE_(d)(F|_(Y))H \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)H∈Ed(F|Y),
‖ F | Y | Y , d α = ‖ H | X , d α = ‖ F − ( F − H ) | X , d α F Y Y , d α = H X , d α = ‖ F − ( F − H ) X , d α ||F|_(Y)|_(Y,d)^(alpha)=||H|_(X,d)^(alpha)=||F-(F-H)|_(X,d)^(alpha)\left.\left\|\left.\left.F\right|_{Y}\right|_{Y, d} ^{\alpha}=\right\| H\right|_{X, d} ^{\alpha}=\| F-\left.(F-H)\right|_{X, d} ^{\alpha}‖F|Y|Y,dα=‖H|X,dα=‖F−(F−H)|X,dα
Because F − H ∈ Y d ¯ ⊥ F − H ∈ Y d ¯ ⊥ F-H inY_( bar(d))^(_|_)F-H \in Y_{\bar{d}}^{\perp}F−H∈Yd¯⊥, it follows ‖ F | Y | Y , d α ≥ ρ d ( F , Y d ⊥ ) ‖ F Y Y , d α ≥ ρ d F , Y d ⊥ ||F|_(Y)|_(Y,d)^(alpha) >= rho_(d)(F,Y_(d)^(_|_))\|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha} \geq \rho_{d}\left(F, Y_{d}^{\perp}\right)‖F|Y|Y,dα≥ρd(F,Yd⊥).
Consequently, Y d ¯ ⊥ Y d ¯ ⊥ Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp}Yd¯⊥ is proximinal with respect to the distance ρ d ρ d rho_(d)\rho_{d}ρd ( ρ d ρ d rho_(d)\rho_{d}ρd-proximinal in short) and
ρ d ( F , Y d ¯ ⊥ ) = ‖ F | Y | Y , d α ρ d F , Y d ¯ ⊥ = ‖ F Y Y , d α rho_(d)(F,Y_( bar(d))^(_|_))=||F|_(Y)|_(Y,d)^(alpha)\rho_{d}\left(F, Y_{\bar{d}}^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha}ρd(F,Yd¯⊥)=‖F|Y|Y,dα
Now, let G ¯ 0 ∈ P Y d ¯ ⊥ ( F ) G ¯ 0 ∈ P Y d ¯ ⊥ ( F ) bar(G)_(0)inP_(Y_( bar(d))^(_|_))(F)\bar{G}_{0} \in P_{Y_{\bar{d}}^{\perp}}(F)G¯0∈PYd¯⊥(F). Then ( F − G ¯ 0 ) | Y = F | Y F − G ¯ 0 Y = F Y (F- bar(G)_(0))|_(Y)=F|_(Y)\left.\left(F-\bar{G}_{0}\right)\right|_{Y}=\left.F\right|_{Y}(F−G¯0)|Y=F|Y and ‖ F − G ¯ 0 | X , d α = ‖ F | Y | Y , d α ‖ F − G ¯ 0 X , d α = ‖ F Y Y , d α ||F- bar(G)_(0)|_(X,d)^(alpha)=||F|_(Y)|_(Y,d)^(alpha)\| F-\left.\bar{G}_{0}\right|_{X, d} ^{\alpha}= \|\left.\left. F\right|_{Y}\right|_{Y, d} ^{\alpha}‖F−G¯0|X,dα=‖F|Y|Y,dα. This means that F − G ¯ 0 ∈ E d ( F | Y ) F − G ¯ 0 ∈ E d F Y F- bar(G)_(0)inE_(d)(F|_(Y))F-\bar{G}_{0} \in \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)F−G¯0∈Ed(F|Y), i.e., G ¯ 0 ∈ F − E d ( F | Y ) G ¯ 0 ∈ F − E d F Y bar(G)_(0)in F-E_(d)(F|_(Y))\bar{G}_{0} \in F-\mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G¯0∈F−Ed(F|Y). Consequently, G ¯ 0 ∈ P Y d ¯ ⊥ ( F ) G ¯ 0 ∈ P Y d ¯ ⊥ ( F ) bar(G)_(0)inP_(Y_( bar(d))^(_|_))(F)\bar{G}_{0} \in P_{Y_{\bar{d}}^{\perp}}(F)G¯0∈PYd¯⊥(F) implies G ¯ 0 ∈ F − E d ( F | Y ) G ¯ 0 ∈ F − E d F Y bar(G)_(0)in F-E_(d)(F|_(Y))\bar{G}_{0} \in F-\mathcal{E}_{d}\left(\left.F\right|_{Y}\right)G¯0∈F−Ed(F|Y).
Taking into account the first part of the proof it follows P Y d ¯ ⊥ ( F ) = F − E d ( F | Y ) P Y d ¯ ⊥ ( F ) = F − E d F Y P_(Y_( bar(d))^(_|_))(F)=F-E_(d)(F|_(Y))P_{Y_{\bar{d}}^{\perp}}(F)=F- \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)PYd¯⊥(F)=F−Ed(F|Y).
Analogously, one obtains ρ d ¯ ( F , Y d ⊥ ) = ‖ F | Y | Y , d ¯ α ρ d ¯ F , Y d ⊥ = ‖ F Y Y , d ¯ α rho_( bar(d))(F,Y_(d)^(_|_))=||F|_(Y)|_(Y, bar(d))^(alpha)\rho_{\bar{d}}\left(F, Y_{d}^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|_{Y, \bar{d}} ^{\alpha}ρd¯(F,Yd⊥)=‖F|Y|Y,d¯α and P Y d ⊥ ( F ) = F − E d ¯ ( F | Y ) P Y d ⊥ ( F ) = F − E d ¯ F Y P_(Y_(d)^(_|_))(F)=F-E_( bar(d))(F|_(Y))P_{Y_{d}^{\perp}}(F)=F-\mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right)PYd⊥(F)=F−Ed¯(F|Y).
By the equalities (22) and (23) it follows.
Corollary 5. Let F ∈ Λ α , 0 ( X ) F ∈ Λ α , 0 ( X ) F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X)F∈Λα,0(X) and Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X such that y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y. Then
a) card E d ( F | Y ) = 1 card E d F Y = 1 cardE_(d)(F|_(Y))=1\operatorname{card} \mathcal{E}_{d}\left(\left.F\right|_{Y}\right)=1cardEd(F|Y)=1 iff Y d ¯ ⊥ Y d ¯ ⊥ Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp}Yd¯⊥ is ρ d ρ d rho_(d)\rho_{d}ρd-Chebyshevian;
b) card E d ¯ ( F | Y ) = 1 card E d ¯ F Y = 1 cardE_( bar(d))(F|_(Y))=1\operatorname{card} \mathcal{E}_{\bar{d}}\left(\left.F\right|_{Y}\right)=1cardEd¯(F|Y)=1 iff Y d ¯ ⊥ Y d ¯ ⊥ Y_( bar(d))^(_|_)Y_{\bar{d}}^{\perp}Yd¯⊥ is ρ d ¯ ρ d ¯ rho_( bar(d))\rho_{\bar{d}}ρd¯-Chebyshevian.
Remark 6. Observe that the linear space Λ α , 0 ( X ) = Λ α , 0 ( X , d ) ∩ Λ α , 0 ( X , d ¯ ) Λ α , 0 ( X ) = Λ α , 0 ( X , d ) ∩ Λ α , 0 ( X , d ¯ ) Lambda_(alpha,0)(X)=Lambda_(alpha,0)(X,d)nnLambda_(alpha,0)(X, bar(d))\Lambda_{\alpha, 0}(X)=\Lambda_{\alpha, 0}(X, d) \cap \Lambda_{\alpha, 0}(X, \bar{d})Λα,0(X)=Λα,0(X,d)∩Λα,0(X,d¯) is a Banach space with respect to the norm
(28) ‖ F | X α = max { ‖ F | X , d α , ‖ F | X , d ¯ α } (28) ‖ F X α = max F X , d α , F X , d ¯ α {:(28)||F|_(X)^(alpha)=max{||F|_(X,d)^(alpha),||F|_(X, bar(d))^(alpha)}:}\begin{equation*} \|\left. F\right|_{X} ^{\alpha}=\max \left\{\left.\left\|\left.F\right|_{X, d} ^{\alpha},\right\| F\right|_{X, \bar{d}} ^{\alpha}\right\} \tag{28} \end{equation*}(28)‖F|Xα=max{‖F|X,dα,‖F|X,d¯α}
In fact this space in the space of all real-valued Lipschitz functions defined on the quasi-metric space ( X , d α X , d α X,d^(alpha)X, d^{\alpha}X,dα ), vanishing at a fixed point y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X. Obviously,
(29) ‖ F ‖ X α = sup { | F ( x ) − F ( y ) | d α ( x , y ) : d ( x , y ) > 0 ; x , y ∈ X } (29) ‖ F ‖ X α = sup | F ( x ) − F ( y ) | d α ( x , y ) : d ( x , y ) > 0 ; x , y ∈ X {:(29)||F||_(X)^(alpha)=s u p{(|F(x)-F(y)|)/(d^(alpha)(x,y)):d(x,y) > 0;x,y in X}:}\begin{equation*} \|F\|_{X}^{\alpha}=\sup \left\{\frac{|F(x)-F(y)|}{d^{\alpha}(x, y)}: d(x, y)>0 ; x, y \in X\right\} \tag{29} \end{equation*}(29)‖F‖Xα=sup{|F(x)−F(y)|dα(x,y):d(x,y)>0;x,y∈X}
is a norm on Λ α , 0 ( X ) Λ α , 0 ( X ) Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X)Λα,0(X).
Corollary 7. For every element f f fff in the space Λ α , 0 ( Y ) = Λ α , 0 ( Y , d ) ∩ Λ α , 0 ( Y , d ¯ ) Λ α , 0 ( Y ) = Λ α , 0 ( Y , d ) ∩ Λ α , 0 ( Y , d ¯ ) Lambda_(alpha,0)(Y)=Lambda_(alpha,0)(Y,d)nnLambda_(alpha,0)(Y, bar(d))\Lambda_{\alpha, 0}(Y)=\Lambda_{\alpha, 0}(Y, d) \cap \Lambda_{\alpha, 0}(Y, \bar{d})Λα,0(Y)=Λα,0(Y,d)∩Λα,0(Y,d¯) there exists F ∈ Λ α , 0 ( X ) F ∈ Λ α , 0 ( X ) F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X)F∈Λα,0(X) such that
F | Y = f and ‖ F ‖ X α = ‖ f ‖ Y α F Y = f  and  ‖ F ‖ X α = ‖ f ‖ Y α F|_(Y)=f" and "||F||_(X)^(alpha)=||f||_(Y)^(alpha)\left.F\right|_{Y}=f \text { and }\|F\|_{X}^{\alpha}=\|f\|_{Y}^{\alpha}F|Y=f and ‖F‖Xα=‖f‖Yα
The set of all extensions of f ∈ Λ α , 0 ( Y ) f ∈ Λ α , 0 ( Y ) f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y)f∈Λα,0(Y) preserving the norm ‖ f ‖ Y α ‖ f ‖ Y α ||f||_(Y)^(alpha)\|f\|_{Y}^{\alpha}‖f‖Yα (of the form (29), is denoted by E ( f ) E ( f ) E(f)\mathcal{E}(f)E(f), i.e.,
(30) E ( f ) := { F ∈ Λ α , 0 ( X ) : F | Y = f and ‖ F ‖ X α = ‖ f ‖ Y α } . (30) E ( f ) := F ∈ Λ α , 0 ( X ) : F Y = f  and  ‖ F ‖ X α = ‖ f ‖ Y α . {:(30)E(f):={F inLambda_(alpha,0)(X):F|_(Y)=f" and "||F||_(X)^(alpha)=||f||_(Y)^(alpha)}.:}\begin{equation*} \mathcal{E}(f):=\left\{F \in \Lambda_{\alpha, 0}(X):\left.F\right|_{Y}=f \text { and }\|F\|_{X}^{\alpha}=\|f\|_{Y}^{\alpha}\right\} . \tag{30} \end{equation*}(30)E(f):={F∈Λα,0(X):F|Y=f and ‖F‖Xα=‖f‖Yα}.
Denote by
(31) Y ⊥ := { G ∈ Λ α , 0 ( X ) : G | Y = 0 } . (31) Y ⊥ := G ∈ Λ α , 0 ( X ) : G Y = 0 . {:(31)Y^(_|_):={G inLambda_(alpha,0)(X):G|_(Y)=0}.:}\begin{equation*} Y^{\perp}:=\left\{G \in \Lambda_{\alpha, 0}(X):\left.G\right|_{Y}=0\right\} . \tag{31} \end{equation*}(31)Y⊥:={G∈Λα,0(X):G|Y=0}.
the annihilator of the set Y Y YYY in Banach space Λ α , 0 ( X ) Λ α , 0 ( X ) Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X)Λα,0(X), and one considers the following problem of best approximation:
P. For F ∈ Λ α , 0 ( X ) F ∈ Λ α , 0 ( X ) F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X)F∈Λα,0(X) find G 0 ∈ Y ⊥ G 0 ∈ Y ⊥ G_(0)inY^(_|_)G_{0} \in Y^{\perp}G0∈Y⊥ such that
‖ F − G 0 ‖ X α = inf { ‖ F − G ‖ X α : G ∈ Y ⊥ } = ρ ( F , Y ⊥ ) . F − G 0 X α = inf ‖ F − G ‖ X α : G ∈ Y ⊥ = ρ F , Y ⊥ . ||F-G_(0)||_(X)^(alpha)=i n f{||F-G||_(X)^(alpha):G inY^(_|_)}=rho(F,Y^(_|_)).\left\|F-G_{0}\right\|_{X}^{\alpha}=\inf \left\{\|F-G\|_{X}^{\alpha}: G \in Y^{\perp}\right\}=\rho\left(F, Y^{\perp}\right) .‖F−G0‖Xα=inf{‖F−G‖Xα:G∈Y⊥}=ρ(F,Y⊥).
Corollary 8. The subspace Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is proximinal in Λ α , 0 ( X ) Λ α , 0 ( X ) Lambda_(alpha,0)(X)\Lambda_{\alpha, 0}(X)Λα,0(X) and the set of elements of best approximation for F ∈ Λ α , 0 ( X ) F ∈ Λ α , 0 ( X ) F inLambda_(alpha,0)(X)F \in \Lambda_{\alpha, 0}(X)F∈Λα,0(X) is
P Y ⊥ ( F ) = F − E ( F | Y ) . P Y ⊥ ( F ) = F − E F Y . P_(Y^(_|_))(F)=F-E(F|_(Y)).P_{Y^{\perp}}(F)=F-\mathcal{E}\left(\left.F\right|_{Y}\right) .PY⊥(F)=F−E(F|Y).
The distance of F F FFF to Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is given by
ρ ( F , Y ⊥ ) = ‖ F | Y ‖ Y α ρ F , Y ⊥ = F Y Y α rho(F,Y^(_|_))=||F|_(Y)||_(Y)^(alpha)\rho\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|_{Y}^{\alpha}ρ(F,Y⊥)=‖F|Y‖Yα
The subspace Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is Chebyshevian for F F FFF iff card E ( F | Y ) = 1 card E F Y = 1 cardE(F|_(Y))=1\operatorname{card} \mathcal{E}\left(\left.F\right|_{Y}\right)=1cardE(F|Y)=1.
For f f fff in the linear space Λ α , 0 ( Y ) Λ α , 0 ( Y ) Lambda_(alpha,0)(Y)\Lambda_{\alpha, 0}(Y)Λα,0(Y), the equalities F ( f ) ( x ) = F ¯ ( f ) ( x ) , x ∈ X F ( f ) ( x ) = F ¯ ( f ) ( x ) , x ∈ X F(f)(x)= bar(F)(f)(x),x in XF(f)(x)=\bar{F}(f)(x), x \in XF(f)(x)=F¯(f)(x),x∈X and G ( f ) ( x ) = G ¯ ( f ) ( x ) , x ∈ X G ( f ) ( x ) = G ¯ ( f ) ( x ) , x ∈ X G(f)(x)= bar(G)(f)(x),x in XG(f)(x)=\bar{G}(f)(x), x \in XG(f)(x)=G¯(f)(x),x∈X are verified iff ‖ f | Y , d α = ‖ f | Y , d ¯ α f Y , d α = f Y , d ¯ α ||f|_(Y,d)^(alpha)=||f|_(Y, bar(d))^(alpha)\left.\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\| f\right|_{Y, \bar{d}} ^{\alpha}‖f|Y,dα=‖f|Y,d¯α. This means that ‖ f | Y , d α = ‖ − f | Y , d α f Y , d α = − f Y , d α ||f|_(Y,d)^(alpha)=||-f|_(Y,d)^(alpha)\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, d} ^{\alpha}‖f|Y,dα=‖−f|Y,dα and, consequently,
‖ f ‖ Y α = max { ‖ f | Y , d α ; ‖ f | Y , d ¯ α } = ‖ f | Y , d α ‖ f ‖ Y α = max f Y , d α ; f Y , d ¯ α = ‖ f Y , d α ||f||_(Y)^(alpha)=max{||f|_(Y,d)^(alpha);||f|_(Y, bar(d))^(alpha)}=||f|_(Y,d)^(alpha)\|f\|_{Y}^{\alpha}=\max \left\{\left.\left\|\left.f\right|_{Y, d} ^{\alpha} ;\right\| f\right|_{Y, \bar{d}} ^{\alpha}\right\}=\|\left. f\right|_{Y, d} ^{\alpha}‖f‖Yα=max{‖f|Y,dα;‖f|Y,d¯α}=‖f|Y,dα
By Theorem 3 in [14], it follows that Λ a , 0 ( Y ) Λ a , 0 ( Y ) Lambda_(a,0)(Y)\Lambda_{a, 0}(Y)Λa,0(Y) is a Banach space and ( Y , d α Y , d α Y,d^(alpha)Y, d^{\alpha}Y,dα ) is a metric space.

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Received by the editors: April 13, 2010.
  1. *"T. Popoviciu" Institute of Numerical Analysis, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro, cmustata2001@yahoo.com.
2010

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