Rev. Anal. Num´er. Th´eor. Approx., vol. 39 (2010) no. 2, pp. 134–140 ictp.acad.ro/jnaat
ON THE EXISTENCE AND UNIQUENESS OF EXTENSIONS OF SEMI-H ¨OLDER REAL-VALUED FUNCTIONS
COSTIC ˘A MUST ˘AT¸ A∗
Abstract. Let (X, d) be a quasi-metric space,y0 ∈ X a fixed element andY a subset ofX such thaty0∈Y. Denote by (Λα,0(Y, d),k · |αY,d) the asymmetric normed cone of real-valued d-semi-H¨older functions defined on Y of exponent α ∈ (0,1], vanishing in y0, and by (Λα,0(Y,d),¯ k · |αY,d¯) the similar cone if d is replaced by conjugate ¯dofd.
One considers the following claims:
(a) For everyfin the linear space Λα,0(Y) = Λα,0(Y, d)∩Λα,0(Y,d) there exist¯ F∈Λα,0(X, d) such thatF|Y =f andkF|αY,d=kf|αY,d;
(b) For everyf∈Λα,0(Y) there exists ¯F ∈Λα,0(X,d) such that ¯¯ F|Y =f and kF|¯αY,d¯=kf|αY,d¯;
(c) The extensionF in (a) is unique;
(d) The extension ¯F in (b) is unique;
(e) The annihilator Yd¯⊥ of Y in Λα,0(X,d) is proximinal for the elements of¯ Λα,0(X) with respect to the distance generated byk · |αY,d;
(f) The annihilator Yd⊥ of Y in Λα,0(X, d) is proximinal for the elements of Λα,0(X) with respect to the distance generated byk · |αY,d¯;
(g) Yd¯⊥in the claim (e) is Chebyshevian;
(h) Yd⊥in the claim (f) is Chebyshevian.
Then the following equivalences hold:
(a)⇔(e); (b)⇔(f); (c)⇔(g); (d)⇔(h).
MSC 2000. 46A22, 41A50, 41A52.
Keywords. Extensions, semi-Lipschitz functions, semi-H¨older functions, best approximation, quasi-metric spaces.
1. INTRODUCTION
Let X be a nonempty set and d : X ×X → [0,∞) a function with the properties:
(QM1) d(x, y) =d(y, x) = 0 iffx=y, (QM2) d(x, y)≤d(x, z) +d(z, y), for all x, y, z∈X.
Then the function d is called a quasi-metric on X and the pair (X, d) is calledquasi-metric space ([13]).
∗“T. Popoviciu” Institute of Numerical Analysis, Cluj-Napoca, Romania, e-mail:
Because, in general,d(x, y)6=d(y, x),forx, y∈Xone defines the conjugate quasi-metricdof d, by the equality ¯d(x, y) =d(y, x),for all x, y∈X.
LetY be a nonvoid subset of (X, d) and α∈(0,1] a fixed number.
Definition 1. a) A function f : Y → R is called d-semi-H¨older (of exponent α) if there exists a constant KY(f)≥0 such that
(1) f(x)−f(y)≤KY(f)dα(x, y), for allx, y∈Y.
b) f : Y → R is called d-semi-H¨older (of exponent α) if there exists a constant KY(f)≥0 such that
(2) f(x)−f(y)≤KY(f)·dα(y, x), for allx, y∈Y.
The smallest constantKY(f) in (1) is denoted bykf|αY,dand one shows that (3) kf|αY,d:= supn(f(x)−f(y))∨0
dα(x,y) :d(x, y)>0;x, y∈Yo . Analogously one defines kf|αY,d.
Observe that the function f is d-semi-H¨older onY iff −f is d-semi-H¨older on Y.Moreover
(4) kf|αY,d=k−f|αY,d.
Definition 2. ([14]). Let (X, d) be a quasi-metric space and Y ⊆ X a nonempty set. The function f :Y →Ris called≤d-increasing on Y if f(x)≤ f(y) whenever d(x, y) = 0, x, y∈Y.
The set of all ≤d-increasing functions on Y is denoted by RY≤d and it is a cone in the linear space RY of all real-valued functions on Y.
The set
(5) Λα(Y, d) :={f ∈RY≤d;f is d-semi-H¨older and kf|αY,d <∞}
is also a cone, called the cone ofd-semi-H¨older functions onY.
Ify0 ∈Y is arbitrary, but fixed, one considers the cone (6) Λα,0(Y, d) :={f ∈Λα(Y, d) :f(y0) = 0}.
Then the functional k |αY,d : Λα,0(Y, d) → [0,∞) is subadditive, positively homogeneous and the equalitykf|αY,d =k−f|αY,d= 0 impliesf ≡0.This means that k·|αY,d is an asymmetric norm (see [13], [14]), on the cone Λα,0(Y, d).
The pair
Λα,0(Y, d),k |αY,d
is called the asymmetric normed cone of d- semi-H¨older real-valued function onY (compare with [14]).
Analogously, one defines the asymmetric normed cone (Λα,0(Y, d),k · |α
Y,d).
of all d-semi-H¨older real-valued functions on Y, vanishing at the fixed point y0 ∈Y.
By the above definitions it follows that
f ∈(Λα,0(Y, d),k |αY,d) iff −f ∈(Λα,0(Y, d),k |αY,d) and, moreover, kf|αY,d=k−f|αY,d.
Defining Λα,0(Y) by
(7) Λα,0(Y) = Λα,0(Y, d)∩Λα,0(Y, d),
It follows that Λα,0(Y) is a linear subspace. The following, theorem holds.
Theorem 3. For every f ∈ Λα,0(Y) there exist at least one function F ∈ Λα,0(Y, d) and at least one function F ∈Λα,0(Y, d) such that
a) F|Y = F Y =f.
b) kF|αY,d =kf|αY,d and F
α
Y,d =kf|αY,d.
Proof. By Theorem 2 and Remark 3 in [11] it follows that the functions defined by the formulae:
F(f)(x) = inf
y∈Y{f(y) +kf|αY,ddα(x, y)}, x∈X, (8)
G(f)(x) = sup
y∈Y
{f(y)− kf|αY,ddα(y, x}, x∈X, are elements of Λα,0(X, d) and, respectively, the functions given by
F(f)(x) = inf
y∈Y{f(y) +kf|αY,ddα(y, x)}, x∈X, (9)
G(f)(x) = sup
y∈Y
{f(y)− kf|αY,ddα(x, y)}, x∈X are elements of Λα,0(X, d) such that
(10) F(f)|Y = G(f)|Y =f and kF(f)|αY,d=kG(f)|αY,d=kf|αY,d, respectively
(11) F(f)
Y = G(f)
Y =f and F(f)
α Y,d=
G(f)
α
Y,d=kf|αY,d.
Forf ∈Λα,0(Y) let us consider the following (nonempty) sets of extensions:
(12) Ed(f) :={H∈Λα,0(X, d) : H|Y =f and kH|αY,d =kf|αY,d} and
(13) Ed(f) :={H ∈ ∧α,0(X, d) : H
Y =f and H
α
Y,d=kf|αY,d}.
The sets Ed(f) andEd(f) are convex and
(14) F(f)(x)≥H(x)≥G(f)(x), x∈X for all H∈ Ed(f);
(15) F(f)(x)≥H(x)≥G(f)(x), x∈H, for all H∈ Ed(f).
Also, for F ∈Λα,0(X), F|Y ∈Λα,0(Y) and
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) be a quasi-metric space,y0∈X fixed andY ⊆Xsuch that y0∈Y.
Let
(16) Yd⊥:={G∈Λα,0(X, d) : G|Y = 0}
and
(17) Yd⊥:={G∈ ∧α,d(X, d) : G
Y = 0}.
Obviously, forF ∈Λα,0(X)
(18) F − Ed(F|Y)⊂Λα,0(X, d) and
(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-H¨older constants and a problem of best approximation by elements of Yd⊥ and Yd⊥, respectively.
Let (X,k |) be an asymmetric norm (see [13], [14]) and letM be a nonempty set ofX. The setM is calledproximinal for x∈X iff there exists at least one element m0 ∈M such that
kx−m0|= inf{kx−m|:m∈M}=ρ(x, M).
If M is proximinal for x, then the set PM(x) = {m0 ∈ M : kx−m0| = ρ(x, M)} is called the set of elements of best approximations for x in M. If cardPM(x) = 1 then the set M is called Chebyshevian forx.
The set M is called proximinal if M is proximinal for every x ∈ X, and Chebyshevian ifM is Chebyshevian for every x∈X.
Now, consider the following two problems of best approximation:
Pd(F).ForF ∈Λα,0(X) find G0∈Yd⊥ such that
(20) kF−G0|αY,d = inf{kF −G|αY,d :G∈Yd⊥}=ρd(F, Yd⊥) and
Pd(F).ForF ∈Λα,0(X) find G0∈Y⊥
d such that
(21)
F −G0
α
Y,d= inf{
F −G
α
X,d:G∈Yd⊥}=ρd(F, Yd⊥).
Let
(22) PY⊥
d
(F) :={G0 ∈Yd⊥:
F−G0
α
X,d=ρd(F, Yd⊥)}
and
(23) PY⊥
d (F) :={G0 ∈Yd⊥:kF−G0|α
X,d=ρd(F, Yd⊥)}.
The following theorem holds.
Theorem 4. If F ∈Λα,0(X) then
(24) PY⊥
d
(F) =F− Ed(F|Y),
(25) PY⊥
d (F) =F− Ed(F|Y) and
(26) ρd(F, Yd⊥) =kF|Y αY,d,
(27) ρd(F, Yd⊥) =kF|Y
α Y,d . Proof. LetF ∈Λα,0(X)(= Λα,0(X, d)∩Λα,0(X, d)) Then, for every G∈Yd⊥,
kF|Y
αY,d =kF|Y −G Y
αY,d ≤ F−G
α X,d. Taking the infimum with respect toG∈Y⊥
d ,one obtainskF|Y |αY,d≤ρd(F, Y⊥
d ).
On the other hand, for everyH ∈ Ed(F|Y), kF|Y
αY,d =kH|αX,d=kF −(F −H)|αX,d. Because F−H ∈Y⊥
d ,it followskF|Y |αY,d≥ρd(F, Yd⊥).
Consequently,Y⊥
d is proximinal with respect to the distanceρd(ρd-proximinal in short) and
ρd(F, Yd⊥) =kF|Y αY,d. Now, letG0 ∈PY⊥
d
(F).Then (F−G0)|Y = F|Y and
F−G0
α X,d= kF|Y |αY,d.This means thatF−G0∈ Ed(F|Y),i.e., G0∈F− Ed(F|Y).Conse- quently, G0∈PY⊥
d
(F) impliesG0 ∈F − Ed(F|Y).
Taking into account the first part of the proof it follows PY⊥
d
(F) = F − Ed(F|Y).
Analogously, one obtainsρd(F, Yd⊥) =kF|Y |α
Y,dandPY⊥
d (F) =F−Ed(F|Y).
By the equalities (22) and (23) it follows.
Corollary 5. Let F ∈Λα,0(X) andY ⊂X such that y0∈Y. Then a) cardEd(F|Y) = 1 iff Y⊥
d is ρd-Chebyshevian;
b) cardEd(F|Y) = 1 iff Yd⊥ is ρd-Chebyshevian.
Remark6. Observe that the linear space Λα,0(X) = Λα,0(X, d)∩Λα,0(X, d) is a Banach space with respect to the norm
(28) kF|αX = max{kF|αX,d,kF|αX,d}.
In fact this space in the space of all real-valued Lipschitz functions defined on the quasi-metric space (X, dα),vanishing at a fixed pointy0 ∈X. Obviously, (29) kFkαX = sup
n|F(x)−F(y)|
dα(x,y) :d(x, y)>0; x, y∈X o
is a norm on Λα,0(X).
Corollary 7. For every element f in the space Λα,0(Y) = Λα,0(Y, d)∩ Λα,0(Y, d) there exists F ∈Λα,0(X) such that
F|Y =f and kFkαX =kfkαY .
The set of all extensions of f ∈Λα,0(Y) preserving the norm kfkαY (of the form (29), is denoted by E(f),i.e.,
(30) E(f) : ={F∈Λα,0(X) : F|Y =f and kFkαX =kfkαY}.
Denote by
(31) Y⊥:={G∈Λα,0(X) : G|Y = 0}.
the annihilator of the set Y in Banach space Λα,0(X), and one considers the following problem of best approximation:
P.ForF ∈Λα,0(X) findG0 ∈Y⊥ such that
kF−G0kαX = inf{kF −GkαX :G∈Y⊥}=ρ(F, Y⊥).
Corollary 8. The subspace Y⊥is proximinal in Λα,0(X) and the set of elements of best approximation for F ∈Λα,0(X) is
PY⊥(F) =F− E(F|Y).
The distance of F toY⊥ is given by
ρ(F, Y⊥) =kF|Y kαY .
The subspace Y⊥ is Chebyshevian for F iff cardE(F|Y) = 1.
Forf in the linear space Λα,0(Y),the equalitiesF(f)(x) =F(f)(x), x∈X and G(f)(x) = G(f)(x), x ∈ X are verified iff kf|αY,d = kf|αY,d. This means that kf|αY,d=k−f|αY,d and, consequently,
kfkαY = max{kf|αY,d;kf|αY,d}=kf|αY,d.
By Theorem 3 in [14], it follows that Λa,0(Y) is a Banach space and (Y, dα) is a metric space.
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Received by the editors: April 13, 2010.
