An application of a theorem of McShane

Costica Mustata
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Costica Mustata
https://ictp.acad.ro/mustata/
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

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C. Mustăţa, An application of a theorem of McShane, Seminar on Functional Analysis and Numerical Methods, Preprint Nr. 1 (1989), 75-84 (MR # 91j: 54024).

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MR # 91j: 54024

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[1] Andrica, D., Mustata, C., An abstract Korovkin type theorem and applications, Studia Univ. „Bbeș-Bolyai”, (to appear)
[2] Aronsson, G., Extension of functions satifying Lipschitz conditions, Arkiv for Matematik 6 (1967) nr.28, 551-561
[3] Brovn, B.M., Elliot, D., Paget, D.F., Lipschitz Constants for the Bernstein Polynomials of a Lipschitz Continuous Functions, J.A.T. 49 (1982), 2, 196-199
[4] Czipser, J., Geher, L., Extension of funcitons satisfying a Lipschitz condition, Acta Math. Acad. Sci. Hungar 6 (1955), 213-220
[5] Iancu, C., Mustăța, C., Error Extimation in the Approximation of Functions by Interpolation Cubic Spline, Mathematica 29 (52) 1, (1957), 33-39.
[6] Mc Shane, E.J., Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842
[7] Mustăța, C., On the extension problem with prescribed norm, Seminar of Funcitonal Analysis and Numerical Methods, Preprint nr.4 (1981), 93-99
[8] Popoviciu, T., Sur quelques proprietes des fonctions d’une ou de deux variables reelles, Mathematica (Cluj) VIII (1933), 1-85
[9] Popoviciu, T., Sur l’approximation des fonctions converses d’ordre superieurs, Mathematica (Cluj), 10 (1934), 49-54

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1989a-Mustata-UBB-Seminar-An-application-of-a-theorem-of-McShane

AN APPLICATION OF A THEOREM OF MCSHANE

by
Costică Mustătà

Let ( X , d X , d X,dX, dX,d ) be a metric space and let Y Y YYY be a nonvoid subset of X X XXX. A function f: Y → R Y → R Y rarr RY \rightarrow RY→R is called Lipschitz on Y Y YYY if there exists a number L ⩾ 0 L ⩾ 0 L >= 0L \geqslant 0L⩾0 such that
(1)
| f ( x ) − f ( y ) | ⩽ L d ( x , y ) | f ( x ) − f ( y ) | ⩽ L d ( x , y ) |f(x)-f(y)| <= Ld(x,y)|f(x)-f(y)| \leqslant L d(x, y)|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 number L satisfying (1) is callod a Lipschitz constant for f f fff on Y Y YYY. Denote by Lip Y Y YYY the set of all real-valued Lipschitz function on Y Y YYY and by K Y ( f ) K Y ( f ) K_(Y)(f)K_{Y}(f)KY(f) the set of all Lipschitz constant for the function f f fff on Y Y YYY. It is easy to check that K Y ( f ) = [ L P , ∞ ) K Y ( f ) = L P , ∞ K_(Y)(f)=[L_(P),oo)K_{Y}(f)=\left[L_{P}, \infty\right)KY(f)=[LP,∞), where
(2) I f = sup { | f ( x ) − f ( y ) | γ d ( x , y ) ; x , y ∈ Y , x ≠ y } . (2) I f = sup { | f ( x ) − f ( y ) | γ d ( x , y ) ; x , y ∈ Y , x ≠ y } . {:(2)I_(f)=s u p{|f(x)-f(y)|gamma d(x","y);x","y in Y","x!=y}.:}\begin{equation*} I_{f}=\sup \{|f(x)-f(y)| \gamma d(x, y) ; x, y \in Y, x \neq y\} . \tag{2} \end{equation*}(2)If=sup{|f(x)−f(y)|γd(x,y);x,y∈Y,x≠y}.
If X = X X = X X=XX=XX=X then Iip X X XXX-denotes the set of all real-valued Lipschitz functions on X X XXX.
Mc Shane [6] proved the following extension theorem for Lipschitz functions 3
THEOREIA 1. Lot ( X , d X , d X,dX, dX,d ) be a metric space and lot Y Y YYY be a nonvoid subset of X X XXX. If f ∈ f ∈ f inf \inf∈ Iip X X XXX and L ⩾ 0 L ⩾ 0 L >= 0L \geqslant 0L⩾0 is a Lipschitz constant for I I III on Y Y YYY then there exists P ∈ Iip X P ∈ Iip X P in Iip XP \in \operatorname{Iip} XP∈IipX such that F | Y = P F Y = P F|_(Y)=P\left.F\right|_{Y}=PF|Y=P and | P ( x ) − P ( y ) | ⩽ L d ( x , y ) | P ( x ) − P ( y ) | ⩽ L d ( x , y ) |P(x)-P(y)| <= Ld(x,y)quad|P(x)-P(y)| \leqslant L d(x, y) \quad|P(x)−P(y)|⩽Ld(x,y), for 211 x , y ∈ X 211 x , y ∈ X 211quad x,y in X211 \quad x, y \in X211x,y∈X.

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A function, F F FFF, as given in Theorem 1, is called an extension of f f fff with preservation of Iipschitz constant, or, briefly, a Iipschitz extension of f f fff.
If X X XXX is a compact metric space denote by C ( X ) C ( X ) C(X)C(X)C(X) the Benach space of all real-valued continuous functions on X X XXX. As usually, we consider the uniform norm on C ( X ) C ( X ) C(X)C(X)C(X), i.e.
(3)
‖ h ‖ x = sup { | h ( x ) | : x ∈ x } , h ∈ o ( x ) . ‖ h ‖ x = sup { | h ( x ) | : x ∈ x } , h ∈ o ( x ) . ||h||_(x)=s u p{|h(x)|:xinx},hino(x).\|\mathrm{h}\|_{\mathrm{x}}=\sup \{|\mathrm{h}(\mathrm{x})|: \mathrm{x} \in \mathrm{x}\}, \mathrm{h} \in \mathrm{o}(\mathrm{x}) .‖h‖x=sup{|h(x)|:x∈x},h∈o(x).
Then Itp X X XXX is a subspace of C ( X ) C ( X ) C(X)C(X)C(X) which is dense in C ( X ) C ( X ) C(X)C(X)C(X) with respect to the uniform topology (i.e. the topology generated by the nome (2)) ( see [1] ).
Let now f ∈ C ( X ) f ∈ C ( X ) f in C(X)f \in C(X)f∈C(X) and suppose that f f fff is Iipschitz on X X XXX. For h ∈ C ( X ) h ∈ C ( X ) h in C(X)h \in C(X)h∈C(X) consider the problem of the evaluation of the norm
(4) ‖ f − h ‖ x . (4) ‖ f − h ‖ x . {:(4)||f-h||_(x).:}\begin{equation*} \|f-h\|_{x} . \tag{4} \end{equation*}(4)‖f−h‖x.
knowing a Lipschitz constant L ∈ I L ( f ) L ∈ I L ( f ) L inI_(L)(f)L \in I_{L}(f)L∈IL(f) of f f fff and the values of f f fff on a subset Y Y YYY of X X XXX.
Obviously, f | Y ∈ I Δ p I f Y ∈ I Δ p I f|_(Y)in I Delta pI\left.f\right|_{Y} \in I \Delta p If|Y∈IΔpI and ℏ f | I ⩽ I I ℏ f I ⩽ I I ℏ_(f)|_(I) <= I_(I)\left.\hbar_{f}\right|_{I} \leqslant I_{I}ℏf|I⩽II, implying I I ( I ) ⊆ I I ( I , I ) I I ( I ) ⊆ I I ( I , I ) I_(I)(I)subeI_(I)(I,I)I_{I}(I) \subseteq I_{I}(I, I)II(I)⊆II(I,I). By Theorem 1 if I ∈ K X ( f | X ) I ∈ K X f X I inK_(X)(f|_(X))I \in K_{X}\left(\left.f\right|_{X}\right)I∈KX(f|X) then f | Y f Y f|_(Y)\left.f\right|_{Y}f|Y has a Lipschits extension I I III on X X XXX, such that I I III is a Iipschitz constant for i, too (i.e.) L ∈ π 2 ( F ) ) L ∈ π 2 ( F ) {:L inpi_(2)(F))\left.L \in \pi_{2}(F)\right)L∈π2(F)).
It was shown in [4], [6] that for L ∈ I I ( f | I ) L ∈ I I f I L inI_(I)(f|_(I))L \in I_{I}\left(\left.f\right|_{I}\right)L∈II(f|I), the following functions :
(5) F B ( x ) = inf { P ( J ) + L a ( x , J ¯ ) ; y ∈ I } , x ∈ X , (5) F B ( x ) = inf { P ( J ) + L a ( x , J ¯ ) ;  y ∈ I } , x ∈ X , {:(5)F_(B)(x)=i n f{P(J)+La(x"," bar(J))"; "y in I}","x in X",":}\begin{equation*} F_{B}(x)=\inf \{P(J)+L a(x, \bar{J}) \text {; } y \in I\}, x \in X, \tag{5} \end{equation*}(5)FB(x)=inf{P(J)+La(x,J¯); y∈I},x∈X,
and
I 1 ( x ) = mep { x ( y ) − I A ( x , y ) , z ∈ I } , x ∈ X , I 1 ( x ) = mep { x ( y ) − I A ( x , y ) , z ∈ I } , x ∈ X , I_(1)(x)=mep{x(y)-IA(x,y),quad z in I},quad x in X,I_{1}(x)=\operatorname{mep}\{x(y)-I A(x, y), \quad z \in I\}, \quad x \in X,I1(x)=mep{x(y)−IA(x,y),z∈I},x∈X,
and Hapachits extensions of | | I I ||_(I):}\left|\left.\right|_{I}\right.||I having I I III as Itpschitiz constant. Ior I ∈ I 2 ( I | 2 ) I ∈ I 2 I 2 I inI_(2)(I|_(2))I \in I_{2}\left(\left.I\right|_{2}\right)I∈I2(I|2), denote by
(a) | u | y Y = { z ∈ I A N , | I | Y = I | Y | u | y Y = z ∈ I A N , | I | Y = I Y quad|u|y_(Y)={z in IA_(N),|I|_(Y)=I|_(Y):}\quad|u| y_{Y}=\left\{z \in I A_{N},|I|_{Y}=\left.I\right|_{Y}\right.|u|yY={z∈IAN,|I|Y=I|Y and I U ∈ I X ( I ) } I U ∈ I X ( I ) {:I_(U)inI_(X)(I)}\left.I_{U} \in I_{X}(I)\right\}IU∈IX(I)},

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the set of all Lipschitz extensions admitting L L LLL as Lipschitz constant.
Concerning the extensions given by the formulae (5) one can prove the following result :
LEMMA 1. Let X X XXX be a metric space, I I III a nonvoid subset of X X XXX, IELip X X XXX and L ∈ K X ( f ) L ∈ K X ( f ) L inK_(X)(f)L \in K_{X}(f)L∈KX(f). Then every F ∈ E L ( f | Y ) F ∈ E L f Y F inE_(L)(f|_(Y))F \in E_{L}\left(\left.f\right|_{Y}\right)F∈EL(f|Y) verifies the inequalities :
(7)
F i ( x ) ⩽ F ( x ) ⩽ F s ( x ) , x ∈ X F i ( x ) ⩽ F ( x ) ⩽ F s ( x ) , x ∈ X F_(i)(x) <= F(x) <= F_(s)(x)quad,quad x in XF_{i}(x) \leqslant F(x) \leqslant F_{s}(x) \quad, \quad x \in XFi(x)⩽F(x)⩽Fs(x),x∈X
Proof. If L L LLL is a Lipachitz constant for f f fff on X X XXX then L L LLL
is alsc a Lipschitz constant for f i f i f i f i f_(i)f_(i)f_{i} f_{i}fifi on Y Y YYY, so that the functions F i F i F_(i)F_{i}Fi and F s F s F_(s)F_{s}Fs are well defined.
Now, let F ∈ F L ( I | Y ¯ ) F ∈ F L I Y ¯ F inF_(L)(I|_( bar(Y)))F \in F_{L}\left(\left.I\right|_{\bar{Y}}\right)F∈FL(I|Y¯) and suppose that there exists u ∈ X ∖ Y u ∈ X ∖ Y u in X\\Yu \in X \backslash Yu∈X∖Y such that F s ( u ) < F ( u ) F s ( u ) < F ( u ) F_(s)(u) < F(u)F_{s}(u)<F(u)Fs(u)<F(u). Then, by the definition (5) of F s F s F_(s)F_{s}Fs, it follows the existence of an element y ∈ I y ∈ I y in Iy \in Iy∈I such that
f ( y ) + L d ( x , u ) < P ( u ) f ( y ) + L d ( x , u ) < P ( u ) f(y)+Ld(x,u) < P(u)f(y)+L d(x, u)<P(u)f(y)+Ld(x,u)<P(u)
But I I III and I I III agree on I I III, so that F ( y ) = I ( y ) F ( y ) = I ( y ) F(y)=I(y)F(y)=I(y)F(y)=I(y) and
F ( u ) ÷ F ( y ) > L d ( x , u ) F ( u ) ÷ F ( y ) > L d ( x , u ) F(u)-:F(y) > Ld(x,u)F(u) \div F(y)>L d(x, u)F(u)÷F(y)>Ld(x,u)
contradicting the fact that. I I III is a Lipschitz constant for F F FFF. The inequality F i ⩽ F F i ⩽ F F_(i) <= FF_{i} \leqslant FFi⩽F is proved in a similar way.
Remark 1. If L ∈ K I ( I ) L ∈ K I ( I ) L inK_(I)(I)L \in K_{I}(I)L∈KI(I) then f ∈ K L ( I | I ) f ∈ K L I I f inK_(L)(I|_(I))f \in K_{L}\left(\left.I\right|_{I}\right)f∈KL(I|I) so that, by Lemma 1 ,
(8) P i ( x ) ⩽ f ( x ) ⩽ F s ( x ) , x ∈ X (8) P i ( x ) ⩽ f ( x ) ⩽ F s ( x ) , x ∈ X {:(8)P_(i)(x) <= f(x) <= F_(s)(x)quad","quad x in X:}\begin{equation*} P_{i}(x) \leqslant f(x) \leqslant F_{s}(x) \quad, \quad x \in X \tag{8} \end{equation*}(8)Pi(x)⩽f(x)⩽Fs(x),x∈X
The following theorem shows that the set of Lipschitz extensions has some compactness properties:
THEOREM 2. If X X XXX is a compact motrio space, Y Y YYY a nonvoid closed subset of X , f ∈ U p X X , f ∈ U p X X,f in UpXX, f \in U p XX,f∈UpX and L ∈ K X ( f ) L ∈ K X ( f ) L inK_(X)(f)L \in K_{X}(f)L∈KX(f), then the set
E L ( P ∣ I ) E L ( P ∣ I ) E_(L)(P∣I)E_{L}(P \mid I)EL(P∣I) is compact with respect to the uniform topologe of C ( X ) C ( X ) C(X)C(X)C(X).
We intend to apply Arrela - Ascoli compactness theorem and we shall divide the proof into three lemmas.
LEWLL 2. The set E L ( f | Y ) E L f Y E_(L)(f|_(Y))E_{L}\left(\left.f\right|_{Y}\right)EL(f|Y) is uniformly bounded in C ( Y ) C ( Y ) C(Y)C(Y)C(Y). x. Proof. By Lemma 1
and ‖ F i ‖ 2 ⩽ max { ‖ F i ‖ x , ‖ F s ‖ x } F i 2 ⩽ max F i x , F s x ||F_(i)||_(2) <= max{||F_(i)||_(x),||F_(s)||_(x)}\left\|F_{i}\right\|_{2} \leqslant \max \left\{\left\|F_{i}\right\|_{x},\left\|F_{s}\right\|_{x}\right\}‖Fi‖2⩽max{‖Fi‖x,‖Fs‖x},
for all F ∈ E L ( f | Y ) F ∈ E L f Y F inE_(L)(f|_(Y))F \in E_{L}\left(\left.f\right|_{Y}\right)F∈EL(f|Y).
LEMMA 3. The set E I ( I | I E I I I E_(I)(I|_(I):}E_{I}\left(\left.I\right|_{I}\right.EI(I|I ) is a (uniformly) echicontzauous subset of C ( X ) C ( X ) C(X)C(X)C(X).
Proof. For ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 let δ = ε / ( L + 1 ) δ = ε / ( L + 1 ) delta=epsi//(L+1)\delta=\varepsilon /(L+1)δ=ε/(L+1). Then
| F ( x ) − F ( y ) | ⩽ L d ( x , y ) < L ⋅ ε L + 1 < ε | F ( x ) − F ( y ) | ⩽ L d ( x , y ) < L ⋅ ε L + 1 < ε |F(x)-F(y)| <= Ld(x,y) < (L*epsi)/(L+1) < epsi|F(x)-F(y)| \leqslant L d(x, y)<\frac{L \cdot \varepsilon}{L+1}<\varepsilon|F(x)−F(y)|⩽Ld(x,y)<L⋅εL+1<ε
for all x , y ∈ Z x , y ∈ Z x,y in Zx, y \in Zx,y∈Z with d ( x , y ) < δ d ( x , y ) < δ d(x,y) < deltad(x, y)<\deltad(x,y)<δ and all F ∈ E L ( I | I ) F ∈ E L I I F inE_(L)(I|_(I))F \in E_{L}\left(\left.I\right|_{I}\right)F∈EL(I|I)
LEMMA 4. The set E I ( I | Y ) E I I Y E_(I)(I|_(Y))E_{I}\left(\left.I\right|_{Y}\right)EI(I|Y) is closed in C ( X ) C ( X ) C(X)C(X)C(X), With respect to the uniform topology of C ( X ) C ( X ) C(X)C(X)C(X).
Proof. Suppose that ( F n ) n ⩾ 1 F n n ⩾ 1 (F_(n))_(n >= 1)\left(F_{n}\right)_{n \geqslant 1}(Fn)n⩾1 is a sequence in E 1 ( I | Y ) E 1 I Y E_(1)(I|_(Y))E_{1}\left(\left.I\right|_{Y}\right)E1(I|Y) converging uniformly to a function F ∈ C ( X ) F ∈ C ( X ) F in C(X)F \in C(X)F∈C(X). We have to show that F F FFF is also in E I ( I | I ) E I I I E_(I)(I|_(I))E_{I}\left(\left.I\right|_{I}\right)EI(I|I). But
| F ( x ) − F ( y ) | ⩽ | P ( x ) − F n ( x ) | + | F n ( x ) − F n ( y ) | 2 + + | F n ( y ) − F ( y ) | ⩽ 2 | P − F n | + L ˙ ⋅ d ( x , y ) | F ( x ) − F ( y ) | ⩽ P ( x ) − F n ( x ) + F n ( x ) − F n ( y ) 2 + + F n ( y ) − F ( y ) ⩽ 2 P − F n + L ˙ ⋅ d ( x , y ) {:[|F(x)-F(y)| <= |P(x)-F_(n)(x)|+|F_(n)(x)-F_(n)(y)|^(2)+],[+|F_(n)(y)-F(y)| <= 2|P-F_(n)|+L^(˙)*d(x","y)]:}\begin{aligned} |F(x)-F(y)| & \leqslant\left|P(x)-F_{n}(x)\right|+\left|F_{n}(x)-F_{n}(y)\right|^{2}+ \\ & +\left|F_{n}(y)-F(y)\right| \leqslant 2\left|P-F_{n}\right|+\dot{L} \cdot d(x, y) \end{aligned}|F(x)−F(y)|⩽|P(x)−Fn(x)|+|Fn(x)−Fn(y)|2++|Fn(y)−F(y)|⩽2|P−Fn|+L˙⋅d(x,y)
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X and all n ∈ E n ∈ E n in En \in En∈E. Leting n → ∞ n → ∞ n rarr oon \rightarrow \inftyn→∞, one obtaing
| F ( x ) − F ( y ) | ⩽ L ⋅ d ( x , y ) | F ( x ) − F ( y ) | ⩽ L ⋅ d ( x , y ) |F(x)-F(y)| <= L*d(x,y)|F(x)-F(y)| \leqslant L \cdot d(x, y)|F(x)−F(y)|⩽L⋅d(x,y)
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X. As F n ( y ) = f ( y ) F n ( y ) = f ( y ) F_(n)(y)=f(y)F_{n}(y)=f(y)Fn(y)=f(y), for all n ∈ X n ∈ X n inXn \in \mathbb{X}n∈X and all y ∈ I y ∈ I y inIy \in \mathbb{I}y∈I, and P n ( y ) → F ( y ) P n ( y ) → F ( y ) P_(n)(y)rarr F(y)P_{n}(y) \rightarrow F(y)Pn(y)→F(y), for n → ∞ n → ∞ n rarr oon \rightarrow \inftyn→∞, it follows F ( y ) = f ( y ) F ( y ) = f ( y ) F(y)=f(y)F(y)=f(y)F(y)=f(y), for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y, 1.e. P ∈ Z L ( f | I ) P ∈ Z L f I P inZ_(L)(f|_(I))P \in Z_{L}\left(\left.f\right|_{I}\right)P∈ZL(f|I)
Proof of Theorem 2. Taleing into account Lammas 2,3 and 4 , the theorem follows from the Arzela - Ascoli compactiness theorem .
corotitary 1. Let X X XXX be a compact metric space, Y Y YYY a nonvoid closed subset of X , f ∈ I L p X , L ∈ K X ( f ) X , f ∈ I L p X , L ∈ K X ( f ) X,f in ILpX,L inK_(X)(f)X, f \in I L p X, L \in K_{X}(f)X,f∈ILpX,L∈KX(f) and h ∈ G ( X ) h ∈ G ( X ) h in G(X)h \in G(X)h∈G(X). Then there exist at least two eunctions F 1 , F 2 ∈ F L ( e | X ) F 1 , F 2 ∈ F L e X F_(1),F_(2)inF_(L)(e|_(X))F_{1}, F_{2} \in F_{L}\left(\left.e\right|_{X}\right)F1,F2∈FL(e|X) such that :
(9) ‖ F 1 − h ‖ X ⩽ ‖ F − h ‖ Z ⩽ ‖ F 2 − h ‖ X , (9) F 1 − h X ⩽ ‖ F − h ‖ Z ⩽ F 2 − h X , {:(9)||F_(1)-h||_(X) <= ||F-h||_(Z) <= ||F_(2)-h||_(X)",":}\begin{equation*} \left\|F_{1}-h\right\|_{X} \leqslant\|F-h\|_{Z} \leqslant\left\|F_{2}-h\right\|_{X}, \tag{9} \end{equation*}(9)‖F1−h‖X⩽‖F−h‖Z⩽‖F2−h‖X,
for all F ∈ E L ( f | I ) 4 F ∈ E L f I 4 F inE_(L)(f|_(I))4F \in E_{L}\left(\left.f\right|_{I}\right) 4F∈EL(f|I)4. In particular
(10) ‖ F 1 − h ‖ X ≤ ‖ e − h ‖ X ≤ ‖ D 2 − h ‖ X . (10) F 1 − h X ≤ ‖ e − h ‖ X ≤ D 2 − h X . {:(10)||F_(1)-h||_(X) <= ||e-h||_(X) <= ||D_(2)-h||_(X).:}\begin{equation*} \left\|F_{1}-h\right\|_{X} \leq\|e-h\|_{X} \leq\left\|D_{2}-h\right\|_{X} . \tag{10} \end{equation*}(10)‖F1−h‖X≤‖e−h‖X≤‖D2−h‖X.
pactity of the set
Proof. The Corollary follows by the compactity of the I L ( f | I ) I L f I I_(L)(f|_(I))\mathrm{I}_{\mathrm{L}}\left(\left.\mathrm{f}\right|_{\mathrm{I}}\right)IL(f|I) and the coatinuity of the function d ( h , I ) = ‖ h − I ‖ X d ( h , I ) = ‖ h − I ‖ X d(h,I)=||h-I||_(X)\mathrm{d}(\mathrm{h}, \mathrm{I})=\|\mathrm{h}-\mathrm{I}\|_{\mathrm{X}}d(h,I)=‖h−I‖X, P ∈ E L ( P | Y ) P ∈ E L P Y P inE_(L)(P|_(Y))P \in E_{L}\left(\left.P\right|_{Y}\right)P∈EL(P|Y).
COROILARY 2. , Let X X XXX be a compact metric space, Y Y YYY a nonvoid closed subset of X , f ∈ L i p ― X , L ∈ K X ( f ) X , f ∈ L i p ¯ X , L ∈ K X ( f ) X,f in bar(Lip)X,L inK_(X)(f)X, f \in \overline{L i p} X, L \in K_{X}(f)X,f∈Lip―X,L∈KX(f) and h ∈ G ( X ) h ∈ G ( X ) h in G(X)h \in G(X)h∈G(X). Then
(11) min { ‖ h − F i ‖ X , ‖ h − F s ‖ X } ⩽ ‖ h − f ‖ X ⩽ min h − F i X , h − F s X ⩽ ‖ h − f ‖ X ⩽ quad min{||h-F_(i)||_(X),||h-F_(s)||_(X)} <= ||h-f||_(X) <=\quad \min \left\{\left\|\mathrm{h}-\mathrm{F}_{\mathrm{i}}\right\|_{\mathrm{X}},\left\|\mathrm{h}-\mathrm{F}_{\mathrm{s}}\right\|_{\mathrm{X}}\right\} \leqslant\|\mathrm{h}-\mathrm{f}\|_{\mathrm{X}} \leqslantmin{‖h−Fi‖X,‖h−Fs‖X}⩽‖h−f‖X⩽
≤ max { ‖ h − F i ‖ X , ‖ h − F B ‖ I } ≤ max h − F i X , h − F B I <= max{||h-F_(i)||_(X),||(h)-F_(B)||_(I)}\leq \max \left\{\left\|\mathrm{h}-\mathrm{F}_{i}\right\|_{\mathrm{X}},\left\|\mathrm{~h}-\mathrm{F}_{\mathrm{B}}\right\|_{\mathrm{I}}\right\}≤max{‖h−Fi‖X,‖ h−FB‖I}
where the functions F i F i F_(i)F_{i}Fi and F B F B F_(B)F_{B}FB are defined by (5).
Proof. Corollary 2 follows from the inequalities (8) and Corollary 1.
COROLLARY 3. Iet X X XXX be a compact metric space, I a nonvoid closed subset of X , P ∈ Lip Z , L ∈ K ू ( f ) X , P ∈ Lip Z , L ∈ K ू ( f ) X,P in Lip Z,L inK_(ू)(f)X, P \in \operatorname{Lip} Z, L \in K_{ू}(f)X,P∈LipZ,L∈Kू(f) and h ∈ C ( X ) h ∈ C ( X ) h in C(X)h \in C(X)h∈C(X). Then there oxist at least two function F 1 , F 2 ∈ F L ( f | Y ) F 1 , F 2 ∈ F L f Y F_(1),F_(2)inF_(L)(f|_(Y))F_{1}, F_{2} \in F_{L}\left(\left.f\right|_{Y}\right)F1,F2∈FL(f|Y) such that
(12) ‖ h − F 1 ‖ x = ain { ‖ h − F 1 ‖ x , ‖ h − F s ‖ x } and (12) h − F 1 x = ain h − F 1 x , h − F s x  and  {:[(12)||h-F_(1)||_(x)=ain{||h-F_(1)||_(x),||h-F_(s)||_(x)}],[" and "]:}\begin{align*} & \left\|h-F_{1}\right\|_{x}=\operatorname{ain}\left\{\left\|h-F_{1}\right\|_{x},\left\|h-F_{s}\right\|_{x}\right\} \tag{12}\\ & \text { and } \end{align*}(12)‖h−F1‖x=ain{‖h−F1‖x,‖h−Fs‖x} and 
‖ h − H 2 ‖ I = max { ‖ h − F i ‖ I , ‖ h − F S ‖ I } h − H 2 I = max h − F i I , h − F S I ||h-H_(2)||_(I)=max{||h-F_(i)||_(I),||(h)-F_(S)||_(I)}\left\|\mathrm{h}-\mathrm{H}_{2}\right\|_{\mathrm{I}}=\max \left\{\left\|\mathrm{h}-\mathrm{F}_{i}\right\|_{\mathrm{I}},\left\|\mathrm{~h}-\mathrm{F}_{\mathrm{S}}\right\|_{\mathrm{I}}\right\}‖h−H2‖I=max{‖h−Fi‖I,‖ h−FS‖I}
Proof. By Corollary 1, there exist two functions F 1 , F 2 ∈ F L ( f | Y ) F 1 , F 2 ∈ F L f Y F_(1),F_(2)inF_(L)(f|_(Y))F_{1}, F_{2} \in F_{L}\left(\left.f\right|_{Y}\right)F1,F2∈FL(f|Y) verifying the inequalities (9), for all I ∈ E L ( f i I ) I ∈ E L f i I I inE_(L)(fi_(I))I \in E_{L}\left(f i_{I}\right)I∈EL(fiI). But then ∣ F 1 − h ‖ x ≤ ‖ F 2 − h ‖ x ∣ F 1 − h x ≤ F 2 − h ‖ x ∣F_(1)-h||_(x) <= ||F_(2)-h||_(x)\mid F_{1}-h\left\|_{x} \leq\right\| F_{2}-h \|_{x}∣F1−h‖x≤‖F2−h‖x and ‖ F x − h ‖ x ≤ ‖ F 2 − h ‖ z F x − h x ≤ F 2 − h z ||F_(x)-h||_(x) <= ||F_(2)-h||_(z)\left\|F_{x}-h\right\|_{x} \leq\left\|F_{2}-h\right\|_{z}‖Fx−h‖x≤‖F2−h‖z, so that
max { ‖ h − F i ‖ x , ‖ h − F a ‖ x } ≤ ‖ F 2 − h ‖ x . max h − F i x , h − F a x ≤ F 2 − h x . max{||h-F_(i)||_(x),||(h)-F_(a)||_(x)} <= ||F_(2)-h||_(x).\max \left\{\left\|\mathrm{h}-\mathrm{F}_{\mathrm{i}}\right\|_{\mathrm{x}},\left\|\mathrm{~h}-\mathrm{F}_{\mathrm{a}}\right\|_{\mathrm{x}}\right\} \leq\left\|\mathrm{F}_{2}-\mathrm{h}\right\|_{\mathrm{x}} .max{‖h−Fi‖x,‖ h−Fa‖x}≤‖F2−h‖x.
On the other hand, by (8) , it follows
‖ F 2 − h ‖ X ≤ max { ‖ F i − h ‖ X , ‖ F B − h ‖ X } , F 2 − h X ≤ max F i − h X , F B − h X , ||F_(2)-h||_(X) <= max{||F_(i)-h||_(X),||F_(B)-h||_(X)},\left\|F_{2}-h\right\|_{X} \leq \max \left\{\left\|F_{i}-h\right\|_{X},\left\|F_{B}-h\right\|_{X}\right\},‖F2−h‖X≤max{‖Fi−h‖X,‖FB−h‖X},
which shows that the second equality in (12) is true. The first one is proved similarly.
Zemark 2. Corollary 3 ahows that the evaluations (11) are exact in the class E L ( f | Y ) E L f Y E_(L)(f|_(Y))E_{L}\left(\left.f\right|_{Y}\right)EL(f|Y).
A δ δ delta\deltaδ - net in a metric space ( X , d ) ( X , d ) (X,d)(X, d)(X,d) is a subset Z Z ZZZ of X X XXX such that for every x ∈ X x ∈ X x in Xx \in Xx∈X there exists z ∈ Z z ∈ Z z in Zz \in Zz∈Z such that d ( x , z ) < δ d ( x , z ) < δ d(x,z) < deltad(x, z)<\deltad(x,z)<δ.
THEOREM 3. If X X XXX is a compact metric space, Y Y YYY is a δ δ delta\deltaδ-net in X , f ∈ Lip X X , f ∈ Lip X X,f in Lip XX, f \in \operatorname{Lip} XX,f∈LipX and L ∈ K X ( f ) L ∈ K X ( f ) L inK_(X)(f)L \in K_{X}(f)L∈KX(f), then
(13) | ‖ P s − h ‖ X − ‖ P i − h ‖ Z | ⩽ 2 L δ P s − h X − P i − h Z ⩽ 2 L δ quad|||P_(s)-h||_(X)-||P_(i)-h||_(Z)| <= 2L delta\quad\left|\left\|P_{s}-h\right\|_{X}-\left\|P_{i}-h\right\|_{Z}\right| \leqslant 2 L \delta|‖Ps−h‖X−‖Pi−h‖Z|⩽2Lδ.
Proof. Let x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X. As Y Y YYY is a δ δ delta\deltaδ-net in X X XXX there exists J 0 ∈ I J 0 ∈ I J_(0)in IJ_{0} \in IJ0∈I such that d ( x 0 , J 0 ) < δ d x 0 , J 0 < δ d(x_(0),J_(0)) < deltad\left(x_{0}, J_{0}\right)<\deltad(x0,J0)<δ, so that
P i ( x 0 ) − F s ( x 0 ) = inf { f ( y ) + L d ( x 0 , y ) : y ∈ I } − P i x 0 − F s x 0 = inf f ( y ) + L d x 0 , y : y ∈ I − P_(i)(x_(0))-F_(s)(x_(0))=i n f{f(y)+Ld(x_(0),y):y in I}-P_{i}\left(x_{0}\right)-F_{s}\left(x_{0}\right)=\inf \left\{f(y)+L d\left(x_{0}, y\right): y \in I\right\}-Pi(x0)−Fs(x0)=inf{f(y)+Ld(x0,y):y∈I}−
− sup { f ( y ) − L d ( x 0 , y ) : y ∈ I } = − sup f ( y ) − L d x 0 , y : y ∈ I = -s u p{f(y)-Ld(x_(0),y):y in I}=-\sup \left\{f(y)-L d\left(x_{0}, y\right): y \in I\right\}=−sup{f(y)−Ld(x0,y):y∈I}=
= inf { f ( y ) + L d ( x 0 , y ) : y ∈ Y } + inf { − f ( y ) + L d ( x 0 , y ) : y ∈ I } = inf f ( y ) + L d x 0 , y : y ∈ Y + inf − f ( y ) + L d x 0 , y : y ∈ I =i n f{f(y)+Ld(x_(0),y):y in Y}+i n f{-f(y)+Ld(x_(0),y):y in I}=\inf \left\{f(y)+L d\left(x_{0}, y\right): y \in Y\right\}+\inf \left\{-f(y)+L d\left(x_{0}, y\right): y \in I\right\}=inf{f(y)+Ld(x0,y):y∈Y}+inf{−f(y)+Ld(x0,y):y∈I}
⩽ f ( y 0 ) + L d ( x 0 , y 0 ) = f ( y 0 ) + L d ( x 0 , y 0 ) = 2 L d ( x 0 , y 0 ) ⩽ 2 L δ ⩽ f y 0 + L d x 0 , y 0 = f y 0 + L d x 0 , y 0 = 2 L d x 0 , y 0 ⩽ 2 L δ <= f(y_(0))+Ld(x_(0),y_(0))=f(y_(0))+Ld(x_(0),y_(0))=2Ld(x_(0),y_(0)) <= 2L delta\leqslant f\left(y_{0}\right)+L d\left(x_{0}, y_{0}\right)=f\left(y_{0}\right)+L d\left(x_{0}, y_{0}\right)=2 L d\left(x_{0}, y_{0}\right) \leqslant 2 L \delta⩽f(y0)+Ld(x0,y0)=f(y0)+Ld(x0,y0)=2Ld(x0,y0)⩽2Lδ.
As x 0 x 0 x_(0)x_{0}x0 was arbitrarily choosen in X X XXX, it follows
2 L δ ⩾ ‖ F s − F i ‖ X ⩾ | ‖ F s − h ‖ X − ‖ F i − h ‖ X | . 2 L δ ⩾ F s − F i X ⩾ F s − h X − F i − h X . 2L delta >= ||F_(s)-F_(i)||_(X) >= |||F_(s)-h||_(X)-||F_(i)-h||_(X)|.2 L \delta \geqslant\left\|F_{s}-F_{i}\right\|_{X} \geqslant\left|\left\|F_{s}-h\right\|_{X}-\left\|F_{i}-h\right\|_{X}\right| .2Lδ⩾‖Fs−Fi‖X⩾|‖Fs−h‖X−‖Fi−h‖X|.

EXAMPIRS .

1 0 1 0 1^(0)1^{0}10 Let x = [ a , b ] , d ( x , y ) = | x − y | x = [ a , b ] , d ( x , y ) = | x − y | x=[a,b],d(x,y)=|x-y|x=[a, b], d(x, y)=|x-y|x=[a,b],d(x,y)=|x−y| and let z z zzz be a division Δ X Δ X Delta_(X)\Delta_{X}ΔX of the interval [ a , b ] , Δ Z = a = x 0 < x 1 < … < x n = b [ a , b ] , Δ Z = a = x 0 < x 1 < … < x n = b [a,b],Delta_(Z)=a=x_(0) < x_(1) < dots < x_(n)=b[a, b], \Delta_{Z}=a=x_{0}<x_{1}<\ldots<x_{n}=b[a,b],ΔZ=a=x0<x1<…<xn=b. Let f f fff be a Iipachitz function on [ a , b ] [ a , b ] [a,b][a, b][a,b] with Iigschitz constant L L LLL and lat f 1 = f ( x 1 ) , 1 = 0 , 1 , … , n f 1 = f x 1 , 1 = 0 , 1 , … , n f_(1)=f(x_(1)),1=0,1,dots,nf_{1}=f\left(x_{1}\right), 1=0,1, \ldots, nf1=f(x1),1=0,1,…,n. It was shown in [5] that there exists a unique finterpolation cubic spline s s sss having the form :
s ( x ) = M 1 − M 1 − 1 6 h 1 ( x − x i − 1 ) 3 + M i − 1 2 ( x − x i − 1 ) 2 + m 1 ( x − x i − 1 ) + f i − 1 s ( x ) = M 1 − M 1 − 1 6 h 1 x − x i − 1 3 + M i − 1 2 x − x i − 1 2 + m 1 x − x i − 1 + f i − 1 {:[s(x)=(M_(1)-M_(1-1))/(6h_(1))(x-x_(i-1))^(3)+(M_(i-1))/(2)(x-x_(i-1))^(2)+m_(1)(x-x_(i-1))],[+f_(i-1)]:}\begin{aligned} s(x)=\frac{M_{1}-M_{1-1}}{6 h_{1}}\left(x-x_{i-1}\right)^{3}+\frac{M_{i-1}}{2}\left(x-x_{i-1}\right)^{2} & +m_{1}\left(x-x_{i-1}\right) \\ & +f_{i-1} \end{aligned}s(x)=M1−M1−16h1(x−xi−1)3+Mi−12(x−xi−1)2+m1(x−xi−1)+fi−1
for x ∈ [ x i − 1 , x i ] , 1 = 1 , 2 , … , n x ∈ x i − 1 , x i , 1 = 1 , 2 , … , n x in[x_(i-1),x_(i)],1=1,2,dots,nx \in\left[x_{i-1}, x_{i}\right], 1=1,2, \ldots, nx∈[xi−1,xi],1=1,2,…,n and such that s ( x i ) = f i s x i = f i s(x_(i))=f_(i)s\left(x_{i}\right)=f_{i}s(xi)=fi, s ′ ( x 1 ) = m 1 , s ′ ′ ( x 1 ) = M 1 , i = 1 , 2 , … , n ; m 0 = p , M 0 = q s ′ x 1 = m 1 , s ′ ′ x 1 = M 1 , i = 1 , 2 , … , n ; m 0 = p , M 0 = q s^(')(x_(1))=m_(1),s^('')(x_(1))=M_(1),i=1,2,dots,n;m_(0)=p,M_(0)=qs^{\prime}\left(x_{1}\right)=m_{1}, s^{\prime \prime}\left(x_{1}\right)=M_{1}, i=1,2, \ldots, n ; m_{0}=p, M_{0}=qs′(x1)=m1,s′′(x1)=M1,i=1,2,…,n;m0=p,M0=q.
The evaluation (11) give: :
(14) min { ‖ s − F i ‖ X , ‖ s − F s ‖ X } ≤ ‖ s − f ‖ 1 2 ≤ max { ‖ s − F 1 ‖ X , ‖ s − F s ‖ X } . (14) min s − F i X , s − F s X ≤ ‖ s − f ‖ 1 2 ≤ max s − F 1 X , s − F s X . {:[(14)min{||s-F_(i)||_(X),||s-F_(s)||_(X)} <= ||s-f||],[(1)/(2) <= max{||s-F_(1)||_(X),||s-F_(s)||_(X)}.]:}\begin{align*} \min \left\{\left\|s-F_{i}\right\|_{X},\left\|s-F_{s}\right\|_{X}\right\} & \leq\|s-f\| \tag{14}\\ \frac{1}{2} & \leq \max \left\{\left\|s-F_{1}\right\|_{X},\left\|s-F_{s}\right\|_{X}\right\} . \end{align*}(14)min{‖s−Fi‖X,‖s−Fs‖X}≤‖s−f‖12≤max{‖s−F1‖X,‖s−Fs‖X}.
By Corollary 3 the evaluations (14) are exact in the class of Ilpschitz functions on [ a , b ] [ a , b ] [a,b][\mathrm{a}, \mathrm{b}][a,b] admitting a Lipschitz constant L L LLL and taking the values f i f i f_(i)f_{i}fi on the knets x i , i = 0 , 1 , … , n x i , i = 0 , 1 , … , n x_(i),i=0,1,dots,nx_{i}, i=0,1, \ldots, nxi,i=0,1,…,n.
All of the terms invelved in the inequalities (14) can be efectively calculated, as F i F i F_(i)F_{i}Fi and F s F s F_(s)F_{s}Fs are polygenal lines on every subinterval [ x i − 1 , x i ] x i − 1 , x i [x_(i-1),x_(i)]\left[x_{i-1}, x_{i}\right][xi−1,xi] and s − P i , s − P s s − P i , s − P s s-P_(i),s-P_(s)s-P_{i}, s-P_{s}s−Pi,s−Ps are polynoiiials of degree at most 3 on every such subinterval (see [5]).
2 0 2 0 2^(0)2^{0}20 Let X = [ 0 , 1 ] X = [ 0 , 1 ] X=[0,1]X=[0,1]X=[0,1] and fer given a , b ∈ R a , b ∈ R a,b in Ra, b \in Ra,b∈R and K ⩾ 0 K ⩾ 0 K >= 0K \geqslant 0K⩾0 consider the following set of Iipschity functions:
F = { f ∈ Lip [ 0 , 1 ] , f ( 0 ) = a , f ( 1 ) = b , L f ≤ K } F = f ∈ Lip [ 0 , 1 ] , f ( 0 ) = a , f ( 1 ) = b , L f ≤ K F={f in Lip[0,1],f(0)=a,f(1)=b,L_(f) <= K}\mathcal{F}=\left\{f \in \operatorname{Lip}[0,1], f(0)=a, f(1)=b, L_{f} \leq K\right\}F={f∈Lip[0,1],f(0)=a,f(1)=b,Lf≤K}
where I f I f I_(f)I_{f}If is the smallest Lipschitz censitant of f f fff given by (2). Let B n : F ~ ⟶ c [ 0 , 1 ] B n : F ~ ⟶ c [ 0 , 1 ] B_(n): widetilde(F)longrightarrow c[0,1]B_{n}: \widetilde{F} \longrightarrow c[0,1]Bn:F~⟶c[0,1] be the Bernstein polynomial (restricted to F F F\mathscr{F}F ) defined by
B n ( f ) ( x ) = ∑ k = 0 n C n k f ( k n ) ( 1 − x ) n − k ⋅ x k , x ∈ [ 0 , 1 ] , f ∈ F . B n ( f ) ( x ) = ∑ k = 0 n   C n k f k n ( 1 − x ) n − k ⋅ x k , x ∈ [ 0 , 1 ] , f ∈ F . B_(n)(f)(x)=sum_(k=0)^(n)C_(n)^(k)f((k)/(n))(1-x)^(n-k)*x^(k),x in[0,1],f inF.B_{n}(f)(x)=\sum_{k=0}^{n} C_{n}^{k} f\left(\frac{k}{n}\right)(1-x)^{n-k} \cdot x^{k}, x \in[0,1], f \in \mathcal{F} .Bn(f)(x)=∑k=0nCnkf(kn)(1−x)n−k⋅xk,x∈[0,1],f∈F.
B.M. Brown , D. Elliet and D.F. Paget [3] proved that B n ( F ~ ) ⊆ F B n ( F ~ ) ⊆ F B_(n)( widetilde(F))subeFB_{n}(\widetilde{\mathcal{F}}) \subseteq \mathcal{F}Bn(F~)⊆F, for all n = 1 , 2 , … n = 1 , 2 , … n=1,2,dotsn=1,2, \ldotsn=1,2,….
Taking inte account the formulae ⋅ ⋅ *\cdot⋅ (5) and applying Lamma 1 it follows that
F i ( x ) = max { a − L f x , b − L f ( 1 − x ) } ≤ ⩽ B a ( f ) ( x ) ⩽ min { a + L f x , b + L f ( 1 − x ) } = F s ( x ) F i ( x ) = max a − L f x , b − L f ( 1 − x ) ≤ ⩽ B a ( f ) ( x ) ⩽ min a + L f x , b + L f ( 1 − x ) = F s ( x ) {:[F_(i)(x)=max{a-L_(f)x,b-L_(f)(1-x)} <= ],[ <= B_(a)(f)(x) <= min{a+L_(f)x,b+L_(f)(1-x)}=F_(s)(x)]:}\begin{aligned} F_{i}(x) & =\max \left\{a-L_{f} x, b-L_{f}(1-x)\right\} \leq \\ & \leqslant B_{a}(f)(x) \leqslant \min \left\{a+L_{f} x, b+L_{f}(1-x)\right\}=F_{s}(x) \end{aligned}Fi(x)=max{a−Lfx,b−Lf(1−x)}≤⩽Ba(f)(x)⩽min{a+Lfx,b+Lf(1−x)}=Fs(x)
for x ∈ [ 0 , 1 ] x ∈ [ 0 , 1 ] x in[0,1]x \in[0,1]x∈[0,1], implying
a + b 2 − L f 2 ⩽ B n ( f ) ( x ) ⩽ a + b 2 + L f 2 a + b 2 − L f 2 ⩽ B n ( f ) ( x ) ⩽ a + b 2 + L f 2 (a+b)/(2)-(L_(f))/(2) <= B_(n)(f)(x) <= (a+b)/(2)+(L_(f))/(2)\frac{a+b}{2}-\frac{L_{f}}{2} \leqslant B_{n}(f)(x) \leqslant \frac{a+b}{2}+\frac{L_{f}}{2}a+b2−Lf2⩽Bn(f)(x)⩽a+b2+Lf2
for all x ∈ [ 0 , 1 ] x ∈ [ 0 , 1 ] x in[0,1]x \in[0,1]x∈[0,1], and consequently
a + b 2 − E 2 ⩽ B n ( f ) ( x ) − ⩽ a + b 2 + E 2 , x ∈ [ 0 , 1 ] a + b 2 − E 2 ⩽ B n ( f ) ( x ) − ⩽ a + b 2 + E 2 , x ∈ [ 0 , 1 ] (a+b)/(2)-(E)/(2) <= B_(n)(f)(x)^(-) <= (a+b)/(2)+(E)/(2),x in[0,1]\frac{a+b}{2}-\frac{E}{2} \leqslant B_{n}(f)(x)^{-} \leqslant \frac{a+b}{2}+\frac{E}{2}, x \in[0,1]a+b2−E2⩽Bn(f)(x)−⩽a+b2+E2,x∈[0,1]
(because L p ≤ K L p ≤ K L_(p) <= KL_{p} \leq KLp≤K, for all f ∈ F f ∈ F f inFf \in \mathcal{F}f∈F ).
Therefore
‖ B n ( P ) ‖ ⩽ max { | a + b + K | 2 , | a + b − K | 2 } , B n ( P ) ⩽ max | a + b + K | 2 , | a + b − K | 2 , ||B_(n)(P)|| <= max{(|a+b+K|)/(2),(|a+b-K|)/(2)},\left\|B_{n}(P)\right\| \leqslant \max \left\{\frac{|a+b+K|}{2}, \frac{|a+b-K|}{2}\right\},‖Bn(P)‖⩽max{|a+b+K|2,|a+b−K|2},
for all f ∈ F f ∈ F f inFf \in \mathcal{F}f∈F.
But, the set F F F\mathcal{F}F contains the functions F 1 F 1 F_(1)F_{1}F1 and F s F s F_(s)F_{s}Fs and conseguently it contains wil the Bernstein polynomials, B n ( P i ) B n P i B_(n)(P_(i))B_{n}\left(P_{i}\right)Bn(Pi) and B n ( B s ) , n = 1 , 2 , … B n B s , n = 1 , 2 , … B_(n)(B_(s))quad,n=1,2,dotsB_{n}\left(B_{s}\right) \quad, n=1,2, \ldotsBn(Bs),n=1,2,…
  • The function F i F i F_(i)F_{i}Fi being cenvex, the pelynomials B n ( F i ) B n F i B_(n)(F_(i))B_{n}\left(F_{i}\right)Bn(Fi) are zenver too, and
B n ( F i ) ( x ) ⩾ F i ( x ) , x ∈ [ 0 , 1 ] . B n F i ( x ) ⩾ F i ( x ) , x ∈ [ 0 , 1 ] . B_(n)(F_(i))(x) >= F_(i)(x)quad,quad x in[0,1].B_{n}\left(F_{i}\right)(x) \geqslant F_{i}(x) \quad, \quad x \in[0,1] .Bn(Fi)(x)⩾Fi(x),x∈[0,1].
Analogously, F s F s F_(s)F_{s}Fs is concal, the polynomials B n ( F s ) B n F s B_(n)(F_(s))B_{n}\left(F_{s}\right)Bn(Fs) are concave too , and
B n ( F s ) ( x ) ≤ F s ( x ) , x ∈ [ 0 , 2 ] B n F s ( x ) ≤ F s ( x ) , x ∈ [ 0 , 2 ] B_(n)(F_(s))(x) <= F_(s)(x),quad x in[0,2]B_{n}\left(F_{s}\right)(x) \leq F_{s}(x), \quad x \in[0,2]Bn(Fs)(x)≤Fs(x),x∈[0,2]
(see c.g. [8], [9] ).

Because

and lim n → ∞ ‖ B n ( F i ) − F i ‖ = 0 lim n → ∞ ‖ B n ( F s ) − F s ‖ = 0 max x ∈ [ 0 , 1 ] F s ( x ) = a + b 2 + K 2 x ∈ [ 0 , 1 ]  and  lim n → ∞   B n F i − F i = 0 lim n → ∞   B n F s − F s = 0 max x ∈ [ 0 , 1 ]   F s ( x ) = a + b 2 + K 2 x ∈ [ 0 , 1 ] " and "{:[lim_(n rarr oo)||B_(n)(F_(i))-F_(i)||=0],[lim_(n rarr oo)||B_(n)(F_(s))-F_(s)||=0],[max_(x in[0,1])F_(s)(x)=(a+b)/(2)+(K)/(2)],[x in[0","1]]:}\text { and } \begin{aligned} & \lim _{n \rightarrow \infty}\left\|B_{n}\left(F_{i}\right)-F_{i}\right\|=0 \\ & \lim _{n \rightarrow \infty}\left\|B_{n}\left(F_{s}\right)-F_{s}\right\|=0 \\ & \max _{x \in[0,1]} F_{s}(x)=\frac{a+b}{2}+\frac{K}{2} \\ & x \in[0,1] \end{aligned} and limn→∞‖Bn(Fi)−Fi‖=0limn→∞‖Bn(Fs)−Fs‖=0maxx∈[0,1]Fs(x)=a+b2+K2x∈[0,1]
it follows the formulae
lim n → ∞ sup I ∈ F ‖ B n ( e ) ‖ = max { | a + b + K | 2 , | a + b − K | 2 } lim n → ∞   sup I ∈ F   B n ( e ) = max | a + b + K | 2 , | a + b − K | 2 lim_(n rarr oo)s u p_(I inF)||B_(n)(e)||=max{(|a+b+K|)/(2),(|a+b-K|)/(2)}\lim _{n \rightarrow \infty} \sup _{I \in \mathcal{F}}\left\|B_{n}(e)\right\|=\max \left\{\frac{|a+b+K|}{2}, \frac{|a+b-K|}{2}\right\}limn→∞supI∈F‖Bn(e)‖=max{|a+b+K|2,|a+b−K|2}

REFERENCES

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