Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Paper coordinates
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).
[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
Let ( X,dX, d ) be a metric space and let YY be a nonvoid subset of XX. A function f: Y rarr RY \rightarrow R is called Lipschitz on YY if there exists a number L >= 0L \geqslant 0 such that
(1)
|f(x)-f(y)| <= Ld(x,y)|f(x)-f(y)| \leqslant L d(x, y)
for all x,y in Yx, y \in Y. A number L satisfying (1) is callod a Lipschitz constant for ff on YY. Denote by Lip YY the set of all real-valued Lipschitz function on YY and by K_(Y)(f)K_{Y}(f) the set of all Lipschitz constant for the function ff on YY. It is easy to check that K_(Y)(f)=[L_(P),oo)K_{Y}(f)=\left[L_{P}, \infty\right), where
{:(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*}
If X=XX=X then Iip XX-denotes the set of all real-valued Lipschitz functions on XX.
Mc Shane [6] proved the following extension theorem for Lipschitz functions 3
THEOREIA 1. Lot ( X,dX, d ) be a metric space and lot YY be a nonvoid subset of XX. If f inf \in Iip XX and L >= 0L \geqslant 0 is a Lipschitz constant for II on YY then there exists P in Iip XP \in \operatorname{Iip} X such that F|_(Y)=P\left.F\right|_{Y}=P and |P(x)-P(y)| <= Ld(x,y)quad|P(x)-P(y)| \leqslant L d(x, y) \quad, for 211quad x,y in X211 \quad x, y \in X.
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A function, FF, as given in Theorem 1, is called an extension of ff with preservation of Iipschitz constant, or, briefly, a Iipschitz extension of ff.
If XX is a compact metric space denote by C(X)C(X) the Benach space of all real-valued continuous functions on XX. As usually, we consider the uniform norm on C(X)C(X), i.e.
(3)
Then Itp XX is a subspace of C(X)C(X) which is dense in C(X)C(X) with respect to the uniform topology (i.e. the topology generated by the nome (2)) ( see [1] ).
Let now f in C(X)f \in C(X) and suppose that ff is Iipschitz on XX. For h in C(X)h \in C(X) consider the problem of the evaluation of the norm
knowing a Lipschitz constant L inI_(L)(f)L \in I_{L}(f) of ff and the values of ff on a subset YY of XX.
Obviously, f|_(Y)in I Delta pI\left.f\right|_{Y} \in I \Delta p I and ℏ_(f)|_(I) <= I_(I)\left.\hbar_{f}\right|_{I} \leqslant I_{I}, implying I_(I)(I)subeI_(I)(I,I)I_{I}(I) \subseteq I_{I}(I, I). By Theorem 1 if I inK_(X)(f|_(X))I \in K_{X}\left(\left.f\right|_{X}\right) then f|_(Y)\left.f\right|_{Y} has a Lipschits extension II on XX, such that II is a Iipschitz constant for i, too (i.e.) {:L inpi_(2)(F))\left.L \in \pi_{2}(F)\right).
It was shown in [4], [6] that for L inI_(I)(f|_(I))L \in I_{I}\left(\left.f\right|_{I}\right), the following functions :
{:(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*}
and
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,
and Hapachits extensions of ||_(I):}\left|\left.\right|_{I}\right. having II as Itpschitiz constant. Ior I inI_(2)(I|_(2))I \in I_{2}\left(\left.I\right|_{2}\right), denote by
(a) 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. and {:I_(U)inI_(X)(I)}\left.I_{U} \in I_{X}(I)\right\},
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the set of all Lipschitz extensions admitting LL as Lipschitz constant.
Concerning the extensions given by the formulae (5) one can prove the following result :
LEMMA 1. Let XX be a metric space, II a nonvoid subset of XX, IELip XX and L inK_(X)(f)L \in K_{X}(f). Then every F inE_(L)(f|_(Y))F \in E_{L}\left(\left.f\right|_{Y}\right) verifies the inequalities :
(7)
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 X
Proof. If LL is a Lipachitz constant for ff on XX then LL
is alsc a Lipschitz constant for f_(i)f_(i)f_{i} f_{i} on YY, so that the functions F_(i)F_{i} and F_(s)F_{s} are well defined.
Now, let F inF_(L)(I|_( bar(Y)))F \in F_{L}\left(\left.I\right|_{\bar{Y}}\right) and suppose that there exists u in X\\Yu \in X \backslash Y such that F_(s)(u) < F(u)F_{s}(u)<F(u). Then, by the definition (5) of F_(s)F_{s}, it follows the existence of an element y in Iy \in I such that
f(y)+Ld(x,u) < P(u)f(y)+L d(x, u)<P(u)
But II and II agree on II, so that F(y)=I(y)F(y)=I(y) and
F(u)-:F(y) > Ld(x,u)F(u) \div F(y)>L d(x, u)
contradicting the fact that. II is a Lipschitz constant for FF. The inequality F_(i) <= FF_{i} \leqslant F is proved in a similar way.
Remark 1. If L inK_(I)(I)L \in K_{I}(I) then f inK_(L)(I|_(I))f \in K_{L}\left(\left.I\right|_{I}\right) so that, by Lemma 1 ,
{:(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*}
The following theorem shows that the set of Lipschitz extensions has some compactness properties:
THEOREM 2. If XX is a compact motrio space, YY a nonvoid closed subset of X,f in UpXX, f \in U p X and L inK_(X)(f)L \in K_{X}(f), then the set E_(L)(P∣I)E_{L}(P \mid I) is compact with respect to the uniform topologe of 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}\left(\left.f\right|_{Y}\right) is uniformly bounded in C(Y)C(Y). x. Proof. By Lemma 1
and ||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\},
for all F inE_(L)(f|_(Y))F \in E_{L}\left(\left.f\right|_{Y}\right).
LEMMA 3. The set E_(I)(I|_(I):}E_{I}\left(\left.I\right|_{I}\right. ) is a (uniformly) echicontzauous subset of C(X)C(X).
Proof. For epsi > 0\varepsilon>0 let delta=epsi//(L+1)\delta=\varepsilon /(L+1). Then
for all x,y in Zx, y \in Z with d(x,y) < deltad(x, y)<\delta and all F inE_(L)(I|_(I))F \in E_{L}\left(\left.I\right|_{I}\right)
LEMMA 4. The set E_(I)(I|_(Y))E_{I}\left(\left.I\right|_{Y}\right) is closed in C(X)C(X), With respect to the uniform topology of C(X)C(X).
Proof. Suppose that (F_(n))_(n >= 1)\left(F_{n}\right)_{n \geqslant 1} is a sequence in E_(1)(I|_(Y))E_{1}\left(\left.I\right|_{Y}\right) converging uniformly to a function F in C(X)F \in C(X). We have to show that FF is also in E_(I)(I|_(I))E_{I}\left(\left.I\right|_{I}\right). But
for all x,y in Xx, y \in X and all n in En \in E. Leting n rarr oon \rightarrow \infty, one obtaing
|F(x)-F(y)| <= L*d(x,y)|F(x)-F(y)| \leqslant L \cdot d(x, y)
for all x,y in Xx, y \in X. As F_(n)(y)=f(y)F_{n}(y)=f(y), for all n inXn \in \mathbb{X} and all y inIy \in \mathbb{I}, and P_(n)(y)rarr F(y)P_{n}(y) \rightarrow F(y), for n rarr oon \rightarrow \infty, it follows F(y)=f(y)F(y)=f(y), for all y in Yy \in Y, 1.e. P inZ_(L)(f|_(I))P \in Z_{L}\left(\left.f\right|_{I}\right)
Proof of Theorem 2. Taleing into account Lammas 2,3 and 4 , the theorem follows from the Arzela - Ascoli compactiness theorem .
corotitary 1. Let XX be a compact metric space, YY a nonvoid closed subset of X,f in ILpX,L inK_(X)(f)X, f \in I L p X, L \in K_{X}(f) and h in G(X)h \in G(X). Then there exist at least two eunctions F_(1),F_(2)inF_(L)(e|_(X))F_{1}, F_{2} \in F_{L}\left(\left.e\right|_{X}\right) such that :
pactity of the set
Proof. The Corollary follows by the compactity of the I_(L)(f|_(I))\mathrm{I}_{\mathrm{L}}\left(\left.\mathrm{f}\right|_{\mathrm{I}}\right) and the coatinuity of the function d(h,I)=||h-I||_(X)\mathrm{d}(\mathrm{h}, \mathrm{I})=\|\mathrm{h}-\mathrm{I}\|_{\mathrm{X}}, P inE_(L)(P|_(Y))P \in E_{L}\left(\left.P\right|_{Y}\right).
COROILARY 2. , Let XX be a compact metric space, YY a nonvoid closed subset of X,f in bar(Lip)X,L inK_(X)(f)X, f \in \overline{L i p} X, L \in K_{X}(f) and h in G(X)h \in G(X). Then
(11) 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}} \leqslant
where the functions F_(i)F_{i} and F_(B)F_{B} are defined by (5).
Proof. Corollary 2 follows from the inequalities (8) and Corollary 1.
COROLLARY 3. Iet XX be a compact metric space, I a nonvoid closed subset of X,P in Lip Z,L inK_(ू)(f)X, P \in \operatorname{Lip} Z, L \in K_{ू}(f) and h in C(X)h \in C(X). Then there oxist at least two function F_(1),F_(2)inF_(L)(f|_(Y))F_{1}, F_{2} \in F_{L}\left(\left.f\right|_{Y}\right) such that
{:[(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*}
Proof. By Corollary 1, there exist two functions F_(1),F_(2)inF_(L)(f|_(Y))F_{1}, F_{2} \in F_{L}\left(\left.f\right|_{Y}\right) verifying the inequalities (9), for all I inE_(L)(fi_(I))I \in E_{L}\left(f i_{I}\right). But then ∣F_(1)-h||_(x) <= ||F_(2)-h||_(x)\mid F_{1}-h\left\|_{x} \leq\right\| F_{2}-h \|_{x} and ||F_(x)-h||_(x) <= ||F_(2)-h||_(z)\left\|F_{x}-h\right\|_{x} \leq\left\|F_{2}-h\right\|_{z}, so that
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}\left(\left.f\right|_{Y}\right).
A delta\delta - net in a metric space (X,d)(X, d) is a subset ZZ of XX such that for every x in Xx \in X there exists z in Zz \in Z such that d(x,z) < deltad(x, z)<\delta.
THEOREM 3. If XX is a compact metric space, YY is a delta\delta-net in X,f in Lip XX, f \in \operatorname{Lip} X and L inK_(X)(f)L \in K_{X}(f), then
(13) 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.
Proof. Let x_(0)in Xx_{0} \in X. As YY is a delta\delta-net in XX there exists J_(0)in IJ_{0} \in I such that d(x_(0),J_(0)) < deltad\left(x_{0}, J_{0}\right)<\delta, so that 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\}- -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\}= =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\} <= 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.
As x_(0)x_{0} was arbitrarily choosen in XX, it follows
1^(0)1^{0} Let x=[a,b],d(x,y)=|x-y|x=[a, b], d(x, y)=|x-y| and let zz be a division Delta_(X)\Delta_{X} of the interval [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. Let ff be a Iipachitz function on [a,b][a, b] with Iigschitz constant LL and lat f_(1)=f(x_(1)),1=0,1,dots,nf_{1}=f\left(x_{1}\right), 1=0,1, \ldots, n. It was shown in [5] that there exists a unique finterpolation cubic spline ss having the form :
for x in[x_(i-1),x_(i)],1=1,2,dots,nx \in\left[x_{i-1}, x_{i}\right], 1=1,2, \ldots, n and such that s(x_(i))=f_(i)s\left(x_{i}\right)=f_{i}, 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}=q.
By Corollary 3 the evaluations (14) are exact in the class of Ilpschitz functions on [a,b][\mathrm{a}, \mathrm{b}] admitting a Lipschitz constant LL and taking the values f_(i)f_{i} on the knets x_(i),i=0,1,dots,nx_{i}, i=0,1, \ldots, n.
All of the terms invelved in the inequalities (14) can be efectively calculated, as F_(i)F_{i} and F_(s)F_{s} are polygenal lines on every subinterval [x_(i-1),x_(i)]\left[x_{i-1}, x_{i}\right] and s-P_(i),s-P_(s)s-P_{i}, s-P_{s} are polynoiiials of degree at most 3 on every such subinterval (see [5]). 2^(0)2^{0} Let X=[0,1]X=[0,1] and fer given a,b in Ra, b \in R and K >= 0K \geqslant 0 consider the following set of Iipschity functions:
where I_(f)I_{f} is the smallest Lipschitz censitant of ff given by (2). Let B_(n): widetilde(F)longrightarrow c[0,1]B_{n}: \widetilde{F} \longrightarrow c[0,1] be the Bernstein polynomial (restricted to F\mathscr{F} ) defined by
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} .
B.M. Brown , D. Elliet and D.F. Paget [3] proved that B_(n)( widetilde(F))subeFB_{n}(\widetilde{\mathcal{F}}) \subseteq \mathcal{F}, for all n=1,2,dotsn=1,2, \ldots.
Taking inte account the formulae *\cdot (5) and applying Lamma 1 it follows that
for all f inFf \in \mathcal{F}.
But, the set F\mathcal{F} contains the functions F_(1)F_{1} and F_(s)F_{s} and conseguently it contains wil the Bernstein polynomials, B_(n)(P_(i))B_{n}\left(P_{i}\right) and B_(n)(B_(s))quad,n=1,2,dotsB_{n}\left(B_{s}\right) \quad, n=1,2, \ldots
The function F_(i)F_{i} being cenvex, the pelynomials B_(n)(F_(i))B_{n}\left(F_{i}\right) are zenver too, and
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] .
Analogously, F_(s)F_{s} is concal, the polynomials B_(n)(F_(s))B_{n}\left(F_{s}\right) are concave too , and
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]
1 ANDRICA, D., UUSTÁTA, C., An abstract Korovkin type theorem and applications, "Studia Univ. Babes-Bolyai", (6o appear).
2 ARONSSON, G., Extension of functions satisfying Lipschitz conditions, Arkiv för Matematik 6 (1967) Mr.28, 551 - 561 .
3 BROWN, B.U., ELLIOT, D., PAGET, D.F., Lipschitz Constants for the Bernstein Polynomials of a Lipechitz Continuous Functions, J. A. T. 49 (1982) 2, 196-199 .
4 CZIPSER, J., GEHKR, L., Extension of fungtions satisfying a Tipschitz condition, Acta Math. Acad. Sci. Hungar 6 (1955), 213-220.
5 IANCU, C., MUSTATA, C., Error Estimation in the Approximation of Functions by Interpolation Cubic Spline , Mathematica 29 (52) 1, (1987), 33-39:.
6 Mc SHANE, E.J., Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 237 - 842 .
7 MUSTĂTA, C., On the extension problem with prescribed norz , Seminar of Functional Analysis and Numerical Methods, Praprint Nr. 4 (1981), 93 - 99 .
8 POPOVICIU, T., Sur quelques propriótés des fonctions d'une ou de deux variables Ifelles, Mathematica (Claj) VIII (1933), 1-85.
9 POPOVICIU, T., Sur I'approximation des fonctions convexes d'ordre superiours, Mathematioa (Cluj) 10 (1934), 49-54.
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