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“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
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C. Mustăţa, On the extension of Lipschitz functions, Seminar of Funct. Analysis and Numerical Methods, Preprint Nr. 1 (1986), 83-92.
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[1] Cobzas, S., Mustata, C., Norm Preserving Exctension of Convex Lipschitz Functions, J.A.T., 24 (1978), 236-244.
[2] Dunham, C.B., Chebyshev approximation with a null space, Proc. Amer. Math. Soc. 41 (1973) 557-558.
[3] Johnson, J.A., Banach Spaces of Lipschitz Functions and Vector-Valued Lipschitz Funcitons, Trans. Amer. Math. Soc. 148 (1970), 147-169.
[4] Shane, E.J., Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
[5] Mustata, C., Best Approximaiton and Unique Extension of Lipschitz Functions, J.A.T., 19 (1977), 222-230.
[6] Roy, A.K., Extreme Points and Linear Isometries of Banach Space of Lipschitz Functions, Canad. J. of Math. 20 (1968), 1150-1164.
[7] Singer, I., Cea mai bună aproximare în spații vectoriale normate prin elemente din subspații vectoriale, Edit. Acad. R.S. Română, București, 1967.
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ON THE EXTENSION OF LIPSCHITZ FUNCTIONS
Costică Mustăta
- Introduction. Let AA be a subset of the interval [a,b]sub R[a, b] \subset R. A function f:A rarr Rf: A \rightarrow R is called Lipschitz if there exists L >= 0L \geqslant 0 such that
{:(1)|f(x)-f(y)| <= L*|x-y|",":}\begin{equation*}
|f(x)-f(y)| \leqslant L \cdot|x-y|, \tag{1}
\end{equation*}
for all x,y in Ax, y \in A. The smallest number LL for which the inerguality
(1) holds is called the Lipschitz norm of ff and is denotod by ||f||_(L)\|\mathbf{f}\|_{L}. The Lipschitz norm of ff can be calculated also by the formula
(2) ||f||_(L)=s u p{|f(x)-f(y)|//|x-y|:x,y in A,x!=y}\|f\|_{L}=\sup \{|f(x)-f(y)| /|x-y|: x, y \in A, x \neq y\}.
(1) holds is called the Lipschitz norm of ff and is denotod by ||f||_(L)\|\mathbf{f}\|_{L}. The Lipschitz norm of ff can be calculated also by the formula
(2) ||f||_(L)=s u p{|f(x)-f(y)|//|x-y|:x,y in A,x!=y}\|f\|_{L}=\sup \{|f(x)-f(y)| /|x-y|: x, y \in A, x \neq y\}.
Denote by Lip A the set of all real valued Lipschitz func tions on A, i.e.
(3) quad Lip A={P,f:A rarr R,f\quad \operatorname{Lip} A=\{P, f: A \rightarrow R, f is Lipschitz }\}.
(3) quad Lip A={P,f:A rarr R,f\quad \operatorname{Lip} A=\{P, f: A \rightarrow R, f is Lipschitz }\}.
With the usual (i.e. pointwise ) operations of addition and multiplication by scalars, Lip A is a vector space.
A Lipschitz extension of ff to [a,b][a, b] is a Lipschitz function [a,b]rarrR[\mathrm{a}, \mathrm{b}] \rightarrow \mathrm{R} such that
F|_(A)^(')=P" and "||F||_(L)=||I||_(L)\left.F\right|_{A} ^{\prime}=P \text { and }\|F\|_{L}=\|I\|_{L}
By a result of Mo SHANE [4] every function in Iip AA has at least one Lipschitz extension in Lip[a,b]\operatorname{Lip}[\mathrm{a}, \mathrm{b}]. More exactly, the following two functions
F_(1)(x)=s u p{f(y)-||f||_(L)*|x-y|:y in A}F_{1}(x)=\sup \left\{f(y)-\|f\|_{L} \cdot|x-y|: y \in A\right\}
and.
{:(6)F_(2)(x)=i n f{f(y)+||f||_(I^('))*|x-y|quad:quad y inA}:}\begin{equation*}
F_{2}(x)=\inf \left\{f(y)+\|f\|_{I^{\prime}} \cdot|x-y| \quad: \quad y \in \mathbb{A}\right\} \tag{6}
\end{equation*}
are Lipschitz extensions of ff to [a,b][a, b]. Denoting by E(f;[a,b])E(f ;[a, b]) the set of all Lipschitz extensions of ff to [a,b][a, b], i.e.
(7) E(f;[a,b])={F in Lip[a,b]:F|_(A)=f:}\mathbb{E}(f ;[a, b])=\left\{F \in \operatorname{Lip}[a, b]:\left.F\right|_{A}=f\right. and {:||F||_(L)=||f||_(L)}\left.\|F\|_{L}=\|f\|_{L}\right\}
the following assertions hold true
(a) quadF_(1)(x) <= F(x) <= F_(2)(x),x in[a,b]\quad F_{1}(x) \leq F(x) \leq F_{2}(x), x \in[a, b], for all F inR(f;[a,b])F \in \mathbb{R}(f ;[a, b]);
(b) quad E(f;[a,b])\quad E(f ;[a, b]) is a convex subset of Lip[a,b]\operatorname{Lip}[a, b];
(c) The functions F_(1)\mathrm{F}_{1} and F_(2)\mathrm{F}_{2} are extreme points of E(f;[a,b])E(f ;[a, b]).
(7) E(f;[a,b])={F in Lip[a,b]:F|_(A)=f:}\mathbb{E}(f ;[a, b])=\left\{F \in \operatorname{Lip}[a, b]:\left.F\right|_{A}=f\right. and {:||F||_(L)=||f||_(L)}\left.\|F\|_{L}=\|f\|_{L}\right\}
the following assertions hold true
(a) quadF_(1)(x) <= F(x) <= F_(2)(x),x in[a,b]\quad F_{1}(x) \leq F(x) \leq F_{2}(x), x \in[a, b], for all F inR(f;[a,b])F \in \mathbb{R}(f ;[a, b]);
(b) quad E(f;[a,b])\quad E(f ;[a, b]) is a convex subset of Lip[a,b]\operatorname{Lip}[a, b];
(c) The functions F_(1)\mathrm{F}_{1} and F_(2)\mathrm{F}_{2} are extreme points of E(f;[a,b])E(f ;[a, b]).
By the definition of the Lipschitz noxm ( or by (2) ), ||f||_(L)=0\|f\|_{L}=0 if and only if f=f= constant and therefore ||\|." II is not actually a norm on Lip A\operatorname{Lip} A but it is a norm on the space Lip _(0){ }_{0} of all functions in Lip AA vanishing at a fixed point x_(0)in Ax_{0} \in A. The space Lip _(A)^(A){ }_{\mathrm{A}}{ }^{\mathrm{A}} with the Lipschitz norm is a dual Banach space (see [3]).
2. Lipschitz extensions from finite subsets of [a,b][a, b].
2. Lipschitz extensions from finite subsets of [a,b][a, b].
Let C[a,b]C[a, b] be the space of all real valued continuous functions on [a,b][a, b] and let
{:(8)u={x_(0),x_(1),dots,x_(n)}quad","quad a <= x_(0) < x_(1) < dots < x_(n) <= b quad",":}\begin{equation*}
u=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\} \quad, \quad a \leqslant x_{0}<x_{1}<\ldots<x_{n} \leqslant b \quad, \tag{8}
\end{equation*}
be a linite subset of [a,b][\mathrm{a}, \mathrm{b}]. Then obviously, the restriction f|_(11)\left.f\right|_{11} of aa function f in C[a,b]f \in C[a, b] to MM is in Lip M\operatorname{Lip} M and
(9) quad|f|_(M)||_(L)=max{|f(x_(i))-f(x_(j))|//|x_(i)-x_(j)|:i,j=0,1,dots n,i!=j}\quad|f|_{M} \|_{L}=\max \left\{\left|f\left(x_{i}\right)-f\left(x_{j}\right)\right| /\left|x_{i}-x_{j}\right|: i, j=0,1, \ldots n, i \neq j\right\} Let U:C[a,b]rarr Lip MU: C[a, b] \rightarrow \operatorname{Lip} M be the restriction operator, i.e.
(10)
(9) quad|f|_(M)||_(L)=max{|f(x_(i))-f(x_(j))|//|x_(i)-x_(j)|:i,j=0,1,dots n,i!=j}\quad|f|_{M} \|_{L}=\max \left\{\left|f\left(x_{i}\right)-f\left(x_{j}\right)\right| /\left|x_{i}-x_{j}\right|: i, j=0,1, \ldots n, i \neq j\right\} Let U:C[a,b]rarr Lip MU: C[a, b] \rightarrow \operatorname{Lip} M be the restriction operator, i.e.
(10)
U(f)=f|_(M)," for "f in C[a,b]U(f)=\left.f\right|_{M}, \text { for } f \in C[a, b]
By the above quated result of HC SHANE, f|_(M)\left.f\right|_{M} has at least a Lipschitz extension F in Lip[a,b]F \in \operatorname{Lip}[a, b]. Let V:Lip M rarrP(Lip[a,b])V: \operatorname{Lip} M \rightarrow \mathscr{P}(\operatorname{Lip}[a, b]) the extension operator defined by
{:(11)V(g)=E(g;[a","b])","quad g in Lip M",":}\begin{equation*}
V(g)=E(g ;[a, b]), \quad g \in \operatorname{Lip} M, \tag{11}
\end{equation*}
and let W:C[a,b]rarrG^(rho)(Lip[a,b])W: C[a, b] \rightarrow \mathcal{G}^{\rho}(\operatorname{Lip}[a, b]) be the composition of UU and VV
{:(12)W=V_(0)U:}\begin{equation*}
W=V_{0} U \tag{12}
\end{equation*}
In general WW is a multivalued operator (point to set) . A function f in C[a,b]f \in C[a, b] such that f in W(f)f \in W(f) is called a fix point of WW.
Obviously, the set of fix points of the operator WW is non-void.
Indeed, if g in Lip Mg \in \operatorname{Lip} M and ff is a Lipschitz extension of GG to [a,b][a, b], then f in W(f)f \in W(f). The fix points of the operator WW are characterized in Theorem 2.1 below.
Obviously, the set of fix points of the operator WW is non-void.
Indeed, if g in Lip Mg \in \operatorname{Lip} M and ff is a Lipschitz extension of GG to [a,b][a, b], then f in W(f)f \in W(f). The fix points of the operator WW are characterized in Theorem 2.1 below.
" For "x,y in[a,b],x!=y" and "f in C[a,b]", put "\text { For } x, y \in[a, b], x \neq y \text { and } f \in C[a, b] \text {, put }
{:(13)[x","y;f]=(f(x)-f(y))//(x-y):}\begin{equation*}
[x, y ; f]=(f(x)-f(y)) /(x-y) \tag{13}
\end{equation*}
and
{:(14)f(x","y;f)(t)=[x","y;f]*(t-x)+f(x)","quad t in[a","b]:}\begin{equation*}
f(x, y ; f)(t)=[x, y ; f] \cdot(t-x)+f(x), \quad t \in[a, b] \tag{14}
\end{equation*}
2.1 THEORTM, Let finC[a,b]\mathrm{f} \in \mathrm{C}[\mathrm{a}, \mathrm{b}] and let M be the set ( e ).
Then f in W(f)f \in W(f) if and only if there exists an index k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\} such that
(15) s u p{|[x,y;f]|:quad x,y in[a,b],quad x!in y}=|[x_(k),x_(k+1);f]|\sup \{|[x, y ; f]|: \quad x, y \in[a, b], \quad x \notin y\}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|.
(15) s u p{|[x,y;f]|:quad x,y in[a,b],quad x!in y}=|[x_(k),x_(k+1);f]|\sup \{|[x, y ; f]|: \quad x, y \in[a, b], \quad x \notin y\}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|.
Proof. If f in W(f)f \in W(f) then f in E(f|_(M);[a,b])f \in E\left(\left.f\right|_{M} ;[a, b]\right) and
||I||_(L)=||f||_(M)||_(L)=max{|[x_(j),x_(j+1);f]|:j=0,1,dots,n-1}\|I\|_{L}=\|f\|_{M} \|_{L}=\max \left\{\left|\left[x_{j}, x_{j+1} ; f\right]\right|: j=0,1, \ldots, n-1\right\}
so that, there exists k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\} such that ||f||_(M)||_(L)=\|f\|_{M} \|_{L}=
|[x_(k),x_(k+1);f]|\left|\left[x_{k}, x_{k+1} ; f\right]\right|, and the relation (15) holds.
Conversely, if the relation (15) holds for a index k in{0,1,dotsk \in\{0,1, \ldots, n-1}n-1\}, then
|[x_(k),x_(k+1);f]|=s u p{|[x,y;f]|:x,y in[a,b],x!=y} >=\left|\left[x_{k}, x_{k+1} ; f\right]\right|=\sup \{|[x, y ; f]|: x, y \in[a, b], x \neq y\} \geqslant
>= max{|[x_(j),x_(j+1);f]|:j=0,1,dots,n-1}=||f|_(M)||_(L)\geqslant \max \left\{\left|\left[x_{j}, x_{j+1} ; f\right]\right|: j=0,1, \ldots, n-1\right\}=\left\|\left.f\right|_{M}\right\|_{L}.
Therefore, |f|||_(I)=|[x_(k),x_(k+1);f]|=||f||_(L)|f|\left\|_{I}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|=\right\| f \|_{L}, which shows that f in W(f)f \in W(f).
||I||_(L)=||f||_(M)||_(L)=max{|[x_(j),x_(j+1);f]|:j=0,1,dots,n-1}\|I\|_{L}=\|f\|_{M} \|_{L}=\max \left\{\left|\left[x_{j}, x_{j+1} ; f\right]\right|: j=0,1, \ldots, n-1\right\}
so that, there exists k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\} such that ||f||_(M)||_(L)=\|f\|_{M} \|_{L}=
|[x_(k),x_(k+1);f]|\left|\left[x_{k}, x_{k+1} ; f\right]\right|, and the relation (15) holds.
Conversely, if the relation (15) holds for a index k in{0,1,dotsk \in\{0,1, \ldots, n-1}n-1\}, then
|[x_(k),x_(k+1);f]|=s u p{|[x,y;f]|:x,y in[a,b],x!=y} >=\left|\left[x_{k}, x_{k+1} ; f\right]\right|=\sup \{|[x, y ; f]|: x, y \in[a, b], x \neq y\} \geqslant
>= max{|[x_(j),x_(j+1);f]|:j=0,1,dots,n-1}=||f|_(M)||_(L)\geqslant \max \left\{\left|\left[x_{j}, x_{j+1} ; f\right]\right|: j=0,1, \ldots, n-1\right\}=\left\|\left.f\right|_{M}\right\|_{L}.
Therefore, |f|||_(I)=|[x_(k),x_(k+1);f]|=||f||_(L)|f|\left\|_{I}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|=\right\| f \|_{L}, which shows that f in W(f)f \in W(f).
Theoret 2.1 has some corollaries.
2.2 COROITARY . If the relation (15) holds for an index k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\} then
2.2 COROITARY . If the relation (15) holds for an index k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\} then
f(x)=L(x_(k),x_(k+1);f)(x)" for all "x in[x_(k),x_(k+1)]f(x)=\mathcal{L}\left(x_{k}, x_{k+1} ; f\right)(x) \text { for all } x \in\left[x_{k}, x_{k+1}\right]
Proof. If f(x^('))!=ℓ(x_(k),x_(k+1);f)(x^('))f\left(x^{\prime}\right) \neq \ell\left(x_{k}, x_{k+1} ; f\right)\left(x^{\prime}\right), for an x^(')in(x_(k),x_(k+1))x^{\prime} \in\left(x_{k}, x_{k+1}\right)
then
then
{:[max{|[x_(k),x^(');f]|,:}{:|[x^('),x_(k+1);f]|} > |[x_(k),x_(k+1);f]|=],[*=s u p{|[x","y;f]|:x","y in[a","b]","x!=y}]:}\begin{aligned}
\max \left\{\left|\left[x_{k}, x^{\prime} ; f\right]\right|,\right. & \left.\left|\left[x^{\prime}, x_{k+1} ; f\right]\right|\right\}>\left|\left[x_{k}, x_{k+1} ; f\right]\right|= \\
\cdot & =\sup \{|[x, y ; f]|: x, y \in[a, b], x \neq y\}
\end{aligned}
which is a contradiction.
2.3 COROILARY. If f in W(f)f \in W(f) then f(x)=F(x),x in[x_(k),x_(k+1)]f(x)=F(x), x \in\left[x_{k}, x_{k+1}\right], for 311. F in W(f)F \in W(f), where k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\} is the index for which relation (25) is true.
2.3 COROILARY. If f in W(f)f \in W(f) then f(x)=F(x),x in[x_(k),x_(k+1)]f(x)=F(x), x \in\left[x_{k}, x_{k+1}\right], for 311. F in W(f)F \in W(f), where k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\} is the index for which relation (25) is true.
Proof. If F in W(+-)F \in W( \pm) then
s u p{|[x,y;F]|:quadx,yin[a,b],x!=y}=||F||_(I)=||f||_(M)||_(L)=∣[x_(k),x_(k+1);F||^(F)∣:}\sup \{|[\mathrm{x}, \mathrm{y} ; \mathrm{F}]|: \quad \mathrm{x}, \mathrm{y} \in[\mathrm{a}, \mathrm{b}], \mathrm{x} \neq \mathrm{y}\}=\|\mathrm{F}\|_{\mathrm{I}}=\|\mathrm{f}\|_{\mathrm{M}} \|_{\mathrm{L}}=\mid\left[\mathrm{x}_{\mathrm{k}}, \mathrm{x}_{\mathrm{k}+1} ; \mathrm{F} \|^{\mathrm{F}} \mid\right.
and Corollary - 2.3 follows from Corollary 2.2 .
2.4 COROTLARY. If a=x_(0)a=x_{0} and x_(n)=bx_{n}=b, then
(a) s u p{|[x,y;f]|:x,y in[a,b],x!=y}=|[x_(0),x_(n);f]|\sup \{|[x, y ; f]|: x, y \in[a, b], x \neq y\}=\left|\left[x_{0}, x_{n} ; f\right]\right|, implies f(x)=f(x_(0),x_(n);f)(x)f(x)=f\left(x_{0}, x_{n} ; f\right)(x), for all x in[a,b]x \in[a, b], and
(b) s u p{|[x,y;e]|:x,y in{a,b],x!=y}=|[x_(k),x_(k+1);f]|\sup \{|[x, y ; e]|: x, y \in\{a, b], x \neq y\}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|, k=0,1,dots,n-1k=0,1, \ldots, n-1, implies f(x)=f(x_(k),x_(k+1);f)(x)f(x)=f\left(x_{k}, x_{k+1} ; f\right)(x), x in[x_(k),x_(k+1)]quad,k=0,1,dots,n-1x \in\left[x_{k}, x_{k+1}\right] \quad, k=0,1, \ldots, n-1.
s u p{|[x,y;F]|:quadx,yin[a,b],x!=y}=||F||_(I)=||f||_(M)||_(L)=∣[x_(k),x_(k+1);F||^(F)∣:}\sup \{|[\mathrm{x}, \mathrm{y} ; \mathrm{F}]|: \quad \mathrm{x}, \mathrm{y} \in[\mathrm{a}, \mathrm{b}], \mathrm{x} \neq \mathrm{y}\}=\|\mathrm{F}\|_{\mathrm{I}}=\|\mathrm{f}\|_{\mathrm{M}} \|_{\mathrm{L}}=\mid\left[\mathrm{x}_{\mathrm{k}}, \mathrm{x}_{\mathrm{k}+1} ; \mathrm{F} \|^{\mathrm{F}} \mid\right.
and Corollary - 2.3 follows from Corollary 2.2 .
2.4 COROTLARY. If a=x_(0)a=x_{0} and x_(n)=bx_{n}=b, then
(a) s u p{|[x,y;f]|:x,y in[a,b],x!=y}=|[x_(0),x_(n);f]|\sup \{|[x, y ; f]|: x, y \in[a, b], x \neq y\}=\left|\left[x_{0}, x_{n} ; f\right]\right|, implies f(x)=f(x_(0),x_(n);f)(x)f(x)=f\left(x_{0}, x_{n} ; f\right)(x), for all x in[a,b]x \in[a, b], and
(b) s u p{|[x,y;e]|:x,y in{a,b],x!=y}=|[x_(k),x_(k+1);f]|\sup \{|[x, y ; e]|: x, y \in\{a, b], x \neq y\}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|, k=0,1,dots,n-1k=0,1, \ldots, n-1, implies f(x)=f(x_(k),x_(k+1);f)(x)f(x)=f\left(x_{k}, x_{k+1} ; f\right)(x), x in[x_(k),x_(k+1)]quad,k=0,1,dots,n-1x \in\left[x_{k}, x_{k+1}\right] \quad, k=0,1, \ldots, n-1.
Combining Corollaries 2.3 and 2.4 it follows
2.5 COROLLARY. If f in Iip[a,b],x_(o)=a,x_(n)=bf \in \operatorname{Iip}[a, b], x_{o}=a, x_{n}=b and ||f||_(L)==∣[x_(k),x_(k+1);f],k=0,1,dots,n-1\|f\|_{L}= =\mid\left[x_{k}, x_{k+1} ; f\right], k=0,1, \ldots, n-1, then W(f)={f}W(f)=\{f\}.
3. Faces and extreme points.
2.5 COROLLARY. If f in Iip[a,b],x_(o)=a,x_(n)=bf \in \operatorname{Iip}[a, b], x_{o}=a, x_{n}=b and ||f||_(L)==∣[x_(k),x_(k+1);f],k=0,1,dots,n-1\|f\|_{L}= =\mid\left[x_{k}, x_{k+1} ; f\right], k=0,1, \ldots, n-1, then W(f)={f}W(f)=\{f\}.
3. Faces and extreme points.
Let XX be a normed space and B(X)={XGX:||x|| <= 1}B(X)=\{X G X:\|x\| \leq 1\} its closed unit ball. A subset A sube B(X)A \subseteq B(X) is called an extremal subset (a face) of B(X)B(X) if alphaf_(1)+(1-alpha)f_(2)in A\alpha f_{1}+(1-\alpha) f_{2} \in A for f_(1),f_(2)in B(X)f_{1}, f_{2} \in B(X) and a number alpha,0 < alpha < 1\alpha, 0<\alpha<1, implies f_(1),f_(2)in Af_{1}, f_{2} \in A. If AA contains exactly one point ff, then ff is called an extreme point of B(X)B(X).
Let MM be a finite set of real numbers
and
M={x_(0),x_(1),dots,x_(n)}quad,quadx_(0) < x_(1) < dots < x_(n)M=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\} \quad, \quad x_{0}<x_{1}<\ldots<x_{n}
Lip_(0)M^(')={f,quad:}\operatorname{Lip}_{0} M^{\prime}=\left\{f, \quad\right. f :M rarr R,quad f: M \rightarrow R, \quad f is Lipschitz and {:f(x_(0))=0}\left.f\left(x_{0}\right)=0\right\}.
3.1 THEOREM. The function f in B(:}f \in B\left(\right. Lip {:_(0)M)\left._{0} M\right) is an extreme point of B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right) if and only if
and
M={x_(0),x_(1),dots,x_(n)}quad,quadx_(0) < x_(1) < dots < x_(n)M=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\} \quad, \quad x_{0}<x_{1}<\ldots<x_{n}
Lip_(0)M^(')={f,quad:}\operatorname{Lip}_{0} M^{\prime}=\left\{f, \quad\right. f :M rarr R,quad f: M \rightarrow R, \quad f is Lipschitz and {:f(x_(0))=0}\left.f\left(x_{0}\right)=0\right\}.
3.1 THEOREM. The function f in B(:}f \in B\left(\right. Lip {:_(0)M)\left._{0} M\right) is an extreme point of B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right) if and only if
{:(16){:||f||_(L)=1=∣[x_(k),x_(k+1);f]]:}\begin{equation*}
\left.\|f\|_{L}=1=\mid\left[x_{k}, x_{k+1} ; f\right]\right] \tag{16}
\end{equation*}
for k=0,1,dots,n-1k=0,1, \ldots, n-1.
Proof. Suppose that relation (16) do not hold and let kin{0,1,dots,n-1}\mathbb{k} \in\{0,1, \ldots, n-1\}𝕜 and epsi > 0\varepsilon>0 be such that
Proof. Suppose that relation (16) do not hold and let kin{0,1,dots,n-1}\mathbb{k} \in\{0,1, \ldots, n-1\}𝕜 and epsi > 0\varepsilon>0 be such that
|[x_(k),x_(k+1);f]|=1-epsi.\left|\left[x_{k}, x_{k+1} ; f\right]\right|=1-\varepsilon .
Let
f_(1)(x)={[f(x)",",x in{x_(0),x_(1),dots,x_(k)}],[f(x)+delta",",x in{x_(k+1),dots,x_(n)}]:}f_{1}(x)= \begin{cases}f(x), & x \in\left\{x_{0}, x_{1}, \ldots, x_{k}\right\} \\ f(x)+\delta, & x \in\left\{x_{k+1}, \ldots, x_{n}\right\}\end{cases}
and
f_(2)(x)={[f(x)",",x in{x_(0),x_(1),dots,x_(k)}],[f(x)-delta",",x in{x_(k+1),dots,x_(n)}]:}f_{2}(x)= \begin{cases}f(x), & x \in\left\{x_{0}, x_{1}, \ldots, x_{k}\right\} \\ f(x)-\delta, & x \in\left\{x_{k+1}, \ldots, x_{n}\right\}\end{cases}
where delta=(epsi//2)(x_(k+1)-x_(k))\delta=(\varepsilon / 2)\left(x_{k+1}-x_{k}\right).
Because
Because
{:[(|f(x_(k+1))+delta-f(x_(k))|)/(x_(k+1)-x_(k)) <= 1-epsi//2 < 1quad" and "],[(|f(x_(k+1))-delta-f(x_(k))|)/(x_(k+1)-x_(k)) <= 1-epsi//2 < 1", it follows that "]:}\begin{aligned}
& \frac{\left|f\left(x_{k+1}\right)+\delta-f\left(x_{k}\right)\right|}{x_{k+1}-x_{k}} \leqslant 1-\varepsilon / 2<1 \quad \text { and } \\
& \frac{\left|f\left(x_{k+1}\right)-\delta-f\left(x_{k}\right)\right|}{x_{k+1}-x_{k}} \leqslant 1-\varepsilon / 2<1 \text {, it follows that }
\end{aligned}
||f_(1)||_(L)=1=||f_(2)||_(J)\left\|f_{1}\right\|_{L}=1=\left\|f_{2}\right\|_{J}. But f=(1//2)(f_(1)+f_(2))f=(1 / 2)\left(f_{1}+f_{2}\right), so that ff is not an extreme point of B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right).
Suppose now that condition (16) is fulfilled and there exist two functions g_(1),g_(2)in B(Lip_(0)M)g_{1}, g_{2} \in B\left(\operatorname{Lip}_{0} M\right) such that g_(1)!=f!=g_(2)g_{1} \neq f \neq g_{2} and f=(1//2)(g_(1)+g_(2))f=(1 / 2)\left(g_{1}+g_{2}\right). Let x_(i)x_{i} be the smallest element of MM for which g_(1)(x_(i))!=f(x_(i))g_{1}\left(x_{i}\right) \neq f\left(x_{i}\right). As g_(1)(x_(0))=f(x_(0))=0g_{1}\left(x_{0}\right)=f\left(x_{0}\right)=0 and g_(2)(x_(0))=0==f(x_(0))g_{2}\left(x_{0}\right)=0= =f\left(x_{0}\right), it follows i >= 1i \geqslant 1.
Case I. f(x_(i)) > f(x_(i-1))f\left(x_{i}\right)>f\left(x_{i-1}\right) and g_(1)(x_(i)) > f(x_(i))g_{1}\left(x_{i}\right)>f\left(x_{i}\right). In this case
(g(x_(i))-g_(1)(x_(i-1)))/(x_(i)-x_(i-1)) > (f(x_(i))-f(x_(i-1)))/(x_(i)-x_(i-1))=1\frac{g\left(x_{i}\right)-g_{1}\left(x_{i-1}\right)}{x_{i}-x_{i-1}}>\frac{f\left(x_{i}\right)-f\left(x_{i-1}\right)}{x_{i}-x_{i-1}}=1
which implies ||g_(1)||_(L) > 1\left\|g_{1}\right\|_{L}>1, i.e. quadg_(1)!in B(Lip_(0)M)\quad g_{1} \notin B\left(\operatorname{Lip}_{0} M\right).
||E_(1)||_(L) > g_(1)!in B(Lip M)\left\|\mathrm{E}_{1}\right\|_{L}>\mathrm{g}_{1} \notin B(\operatorname{Lip} M). . . . . . .
||E_(1)||_(L) > g_(1)!in B(Lip M)\left\|\mathrm{E}_{1}\right\|_{L}>\mathrm{g}_{1} \notin B(\operatorname{Lip} M). . . . . . .
Case II. f(x_(i)) > f(x_(i-1))f\left(x_{i}\right)>f\left(x_{i-1}\right) and g_(1)(x_(i)) < f(x_(i))g_{1}\left(x_{i}\right)<f\left(x_{i}\right). In this case g_(2)(x_(1))=2f(x_(1))-g_(1)(x_(i)) > f(x_(i))g_{2}\left(x_{1}\right)=2 f\left(x_{1}\right)-g_{1}\left(x_{i}\right)>f\left(x_{i}\right) and
{:[(g_(2)(x_(i))-g_(2)(x_(i-1)))/(x_(i)-x_(i-1)) > (f(x_(i))-[2f(x_(i-1))-g_(1)(x_(i-1))])/(x_(i)-x_(i-1))=],[=(f(x_(i))-f(x_(i-1)))/(x_(i)-x_(i-1))=1]:}\begin{aligned}
\frac{g_{2}\left(x_{i}\right)-g_{2}\left(x_{i-1}\right)}{x_{i}-x_{i-1}}> & \frac{f\left(x_{i}\right)-\left[2 f\left(x_{i-1}\right)-g_{1}\left(x_{i-1}\right)\right]}{x_{i}-x_{i-1}}= \\
& =\frac{f\left(x_{i}\right)-f\left(x_{i-1}\right)}{x_{i}-x_{i-1}}=1
\end{aligned}
so that ||g_(2)||_(L) > 1\left\|g_{2}\right\|_{L}>1, i.e. g_(2)!in B(Iipp_(0)M)g_{2} \notin B\left(\operatorname{Iip} p_{0} M\right).
In the remaining cases, i.e. f(x_(i)) < f(x_(i-1))f\left(x_{i}\right)<f\left(x_{i-1}\right) and g_(1)(x_(i)) > f(x_(i))g_{1}\left(x_{i}\right)>f\left(x_{i}\right) or f(x_(i)) < f(x_(i-1))f\left(x_{i}\right)<f\left(x_{i-1}\right) and g_(1)(x_(i)) < f(x_(i))g_{1}\left(x_{i}\right)<f\left(x_{i}\right), we have similarly ||g_(2)||_(L) > 1\left\|g_{2}\right\|_{L}>1, respectively ||g_(1)||_(I) > 1\left\|g_{1}\right\|_{I}>1. The obtained contradictions show that ff must be an extreme point of B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right).
3.2 Remark. Taking in (16) all possible signs, it follows that the unit ball of the space Lip _(0)M{ }_{0} M has exactly 2^(n+1)2^{n+1} extreme points.
3.3 COROLLARY. Let fin Lip[a,b],M={x_(0),x_(1),dots,x_(n)}\mathrm{f} \in \operatorname{\operatorname {Lip}}[\mathrm{a}, \mathrm{b}], \mathrm{M}=\left\{\mathrm{x}_{0}, \mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right\}, a=x_(0) < x_(1) < dots < x_(n)=ba=x_{0}<x_{1}<\ldots<x_{n}=b, and f(x_(0))=0f\left(x_{0}\right)=0. If
In the remaining cases, i.e. f(x_(i)) < f(x_(i-1))f\left(x_{i}\right)<f\left(x_{i-1}\right) and g_(1)(x_(i)) > f(x_(i))g_{1}\left(x_{i}\right)>f\left(x_{i}\right) or f(x_(i)) < f(x_(i-1))f\left(x_{i}\right)<f\left(x_{i-1}\right) and g_(1)(x_(i)) < f(x_(i))g_{1}\left(x_{i}\right)<f\left(x_{i}\right), we have similarly ||g_(2)||_(L) > 1\left\|g_{2}\right\|_{L}>1, respectively ||g_(1)||_(I) > 1\left\|g_{1}\right\|_{I}>1. The obtained contradictions show that ff must be an extreme point of B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right).
3.2 Remark. Taking in (16) all possible signs, it follows that the unit ball of the space Lip _(0)M{ }_{0} M has exactly 2^(n+1)2^{n+1} extreme points.
3.3 COROLLARY. Let fin Lip[a,b],M={x_(0),x_(1),dots,x_(n)}\mathrm{f} \in \operatorname{\operatorname {Lip}}[\mathrm{a}, \mathrm{b}], \mathrm{M}=\left\{\mathrm{x}_{0}, \mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right\}, a=x_(0) < x_(1) < dots < x_(n)=ba=x_{0}<x_{1}<\ldots<x_{n}=b, and f(x_(0))=0f\left(x_{0}\right)=0. If
||f||_(L)=I=|[x_(k),x_(k+1);f]|\|f\|_{L}=I=\left|\left[x_{k}, x_{k+1} ; f\right]\right|
for all k=0,1,dots,n-1k=0,1, \ldots, n-1, then ff is an extreme point of the unit ball of Lip_(o)[a,b]\operatorname{Lip}_{\mathrm{o}}[\mathrm{a}, \mathrm{b}].
3.4 Remark. If ff is an extreme point of B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right) and the set MM is as in Corollary 3.3 , then the unique Lipschitz extension FF of f,F(x)=s u p{f(x_(i))-|x-x_(i)|:i=0,1,dots,n}=i n f{f^(')(x_(i))+|x-x_(i)|:i=0,1,dots,n},x in[a,b]f, F(x)=\sup \left\{f\left(x_{i}\right)-\left|x-x_{i}\right|: i=0,1, \ldots, n\right\} =\inf \left\{f^{\prime}\left(x_{i}\right)+\left|x-x_{i}\right|: i=0,1, \ldots, n\right\}, x \in[a, b], is an extreme point of B(Lip_(0)[a,b])B\left(\operatorname{Lip}_{0}[a, b]\right).
3.4 Remark. If ff is an extreme point of B(Lip_(0)M)B\left(\operatorname{Lip}_{0} M\right) and the set MM is as in Corollary 3.3 , then the unique Lipschitz extension FF of f,F(x)=s u p{f(x_(i))-|x-x_(i)|:i=0,1,dots,n}=i n f{f^(')(x_(i))+|x-x_(i)|:i=0,1,dots,n},x in[a,b]f, F(x)=\sup \left\{f\left(x_{i}\right)-\left|x-x_{i}\right|: i=0,1, \ldots, n\right\} =\inf \left\{f^{\prime}\left(x_{i}\right)+\left|x-x_{i}\right|: i=0,1, \ldots, n\right\}, x \in[a, b], is an extreme point of B(Lip_(0)[a,b])B\left(\operatorname{Lip}_{0}[a, b]\right).
4. Best approximation of Lipschitz functions.
Let
M={x_(0),x_(1),dots,x_(n)}quad,quad a <= x_(0) < x_(1) < dots < x_(n) <= bM=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\} \quad, \quad a \leq x_{0}<x_{1}<\ldots<x_{n} \leq b
and let
(17) quadLip_(0)[a,b]={f,f:[a,b]rarr R,f:}\quad \operatorname{Lip}_{0}[a, b]=\left\{f, f:[a, b] \rightarrow R, f\right. is Lipschitz and {:f(x_(0))=0}\left.f\left(x_{0}\right)=0\right\}. Let also
(17) quadLip_(0)[a,b]={f,f:[a,b]rarr R,f:}\quad \operatorname{Lip}_{0}[a, b]=\left\{f, f:[a, b] \rightarrow R, f\right. is Lipschitz and {:f(x_(0))=0}\left.f\left(x_{0}\right)=0\right\}. Let also
M^(_|_)={f in Lip_(o)[a,b];f|_(M)=0}M^{\perp}=\left\{f \in \operatorname{Li} p_{o}[a, b] ;\left.f\right|_{M}=0\right\}
Since every function f inLip_(0)Mf \in \operatorname{Lip}_{0} M has at least one extension F inLip_(0)[a,b]F \in \operatorname{Lip}_{0}[a, b] it follows that every function f inLip_(0)[a,b]f \in \operatorname{Lip}_{0}[a, b] has a best approximation (nearest point) in M^(_|_)M^{\perp}, i.e. there exists E_(0)inM^(_|_)\mathrm{E}_{0} \in \mathbb{M}^{\perp}. such that ||f-g_(0)||_(L)=i n f{||P-g||_(L):g inM^(_|_)}\left\|\mathrm{f}-\mathrm{g}_{0}\right\|_{\mathrm{L}}=\inf \left\{\|\mathrm{P}-\mathrm{g}\|_{\mathrm{L}}: g \in \mathbb{M}^{\perp}\right\}. It Was shown ( see [5] ) that g_(0)inM^(_|_)g_{0} \in M^{\perp} is an element of best approximation for f inLip_(0)[a,b]f \in \operatorname{Lip}_{0}[a, b] by elements from M^(_|_)M^{\perp} if and only if C_(0)=f-FC_{0}=f-F, for F in V(f)F \in V(f).
Taking into account the precedings results it follows :
4.1 THEOREIM. Let f in Lipp_(0)[a,b]f \in \operatorname{Lip} p_{0}[a, b] and M={x_(0),x_(1),dots,x_(n)}M=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}, a <= x_(0) < x_(1) < dots < x_(n) <= ba \leqslant x_{0}<x_{1}<\ldots<x_{n} \leqslant b.
(a) If the relation (15) holds for an index k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\} then all the best approximation elements for ff in M^(_|_)M^{\perp} vanish on the interval [x_(k),x_(k+1)]\left[\mathrm{x}_{\mathrm{k}}, \mathrm{x}_{\mathrm{k}+1}\right];
(b) If x_(0)=a,x_(n)=bx_{0}=a, x_{n}=b and ||f||_(L)=|[x_(0),x_(n);f]|\|f\|_{L}=\left|\left[x_{0}, x_{n} ; f\right]\right|, then 0 is the only best approximation element for ff in MM;
(c) If ||f||_(L)=|[x_(k),x_(k+1);f]|\|f\|_{L}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|, for all k=0,1,dots,n-1k=0,1, \ldots, n-1, then 0 is the only best approximation olement for it in u^(_|_)u^{\perp}.
4.1 THEOREIM. Let f in Lipp_(0)[a,b]f \in \operatorname{Lip} p_{0}[a, b] and M={x_(0),x_(1),dots,x_(n)}M=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}, a <= x_(0) < x_(1) < dots < x_(n) <= ba \leqslant x_{0}<x_{1}<\ldots<x_{n} \leqslant b.
(a) If the relation (15) holds for an index k in{0,1,dots,n-1}k \in\{0,1, \ldots, n-1\} then all the best approximation elements for ff in M^(_|_)M^{\perp} vanish on the interval [x_(k),x_(k+1)]\left[\mathrm{x}_{\mathrm{k}}, \mathrm{x}_{\mathrm{k}+1}\right];
(b) If x_(0)=a,x_(n)=bx_{0}=a, x_{n}=b and ||f||_(L)=|[x_(0),x_(n);f]|\|f\|_{L}=\left|\left[x_{0}, x_{n} ; f\right]\right|, then 0 is the only best approximation element for ff in MM;
(c) If ||f||_(L)=|[x_(k),x_(k+1);f]|\|f\|_{L}=\left|\left[x_{k}, x_{k+1} ; f\right]\right|, for all k=0,1,dots,n-1k=0,1, \ldots, n-1, then 0 is the only best approximation olement for it in u^(_|_)u^{\perp}.
Proof. Assertion (a) follows from Corollary 2.3 , taking into account that the best approximation elements g_(0)g_{0} for ff in ii ^(_|_){ }^{\perp} have the form
g_(0)=I-E quad,quad F inV(P)g_{0}=I-E \quad, \quad F \in \mathbb{V}(P)
Assertions (b) and (c) follow from Corollary 2.4 .
4.2 Remark. Let MM be the set (8) and let
4.2 Remark. Let MM be the set (8) and let
Lip_(0)[a,b]={f:[a,b]rarr R,f(x_(0))=0,f" is Lipschitz "}\operatorname{Lip}_{0}[a, b]=\left\{f:[a, b] \rightarrow R, f\left(x_{0}\right)=0, f \text { is Lipschitz }\right\}
If f!in W(f)f \notin W(f), then, by Theorem 2.1, it follws
||f||_(L) > ||f|_(M)||_(L)\|f\|_{L}>\left\|\left.f\right|_{M}\right\|_{L}
If, further
(i)
f(x) <= F_(1)(x)quad,quad x in[a,b]f(x) \leqslant F_{1}(x) \quad, \quad x \in[a, b]
where F_(1)F_{1} is given by (5) (with A=MA=M ), then all bect approximation elements of ff in M^(_|_)M^{\perp} are non-positive, and if
(i)
f(x) <= F_(1)(x)quad,quad x in[a,b]f(x) \leqslant F_{1}(x) \quad, \quad x \in[a, b]
where F_(1)F_{1} is given by (5) (with A=MA=M ), then all bect approximation elements of ff in M^(_|_)M^{\perp} are non-positive, and if
{:(ii)f(x) >= F_(2)(x)quad","quad x in[a","b]:}\begin{equation*}
f(x) \geqslant F_{2}(x) \quad, \quad x \in[a, b] \tag{ii}
\end{equation*}
with F_(2)F_{2} given by (6) (with A=MA=M ), then all the elements of best approximation for ff in M^(_|_)M^{\perp} are non-negative.
Obviously, there exist functions f inLip_(o)[a,b]f \in \operatorname{Lip}_{o}[a, b] verifying the conditions (i) and (ii) and which are not fixed points of WW.
For example, takin _(G)g inLip_(o)[a,b]{ }_{G} g \in \operatorname{Lip}_{o}[a, b], the function
f(x)=s u p{g(x_(k))-L_(i)|x-x_(k)|:x_(k)in M,k=0,1,dots,n}f(x)=\sup \left\{g\left(x_{k}\right)-L_{i}\left|x-x_{k}\right|: x_{k} \in M, k=0,1, \ldots, n\right\}
with L > ||g|_(M)||_(L)L>\left\|\left.g\right|_{\mathbb{M}}\right\|_{L}, verifies the condition (i) and the function
h(x)=i n f{g(x_(k))+I.|x-x_(k)|:x_(k)in h,k=0,1,dots,n}h(x)=\inf \left\{g\left(x_{k}\right)+I .\left|x-x_{k}\right|: x_{k} \in h, k=0,1, \ldots, n\right\}
with L > ||B|_(||)||_(I)L>\left\|\left.B\right|_{\|}\right\|_{I} verifies the condition h(x) <= I_(2)(x)h(x) \leqslant I_{2}(x), x in[a,b]x \in[a, b] (i.e. condition (ii) ) .
1 COBZAS, S. , MUSTATA, C. , Norm Preserving Extension of Convex Iipschitz Functions, J.A.T. 24 (1978), 236 - 244.
2 DUTHAS, O.B. , Ohebyshev approximation with a null space , Proc. Amer. Math. Soc. 41 (1973), 557-558.
3 JOLINSON, J. A., Banach Spaces of Lipschitz Functions and Vector-Valued Lipschitz Functions, Trans. Amer. Math. Soc. 148 (1970), 147-169.
4 Wic SHATL, E.J., Extension of range of functions, Bull. Amer. Jath. Soc. 40 (1934), 837-842.
5 USTATA, C., Best Approximation and Unique Intiension of Lipsch itz Functions , J.A.T. 19 (1977), 222-230.
6 R.OI, A.I., Extreme Points and Iinear Isometries of Banach Space of Lijschitz Functions, Canad. J. of Math. 20 (1968), 1150-1164.
7 SIMGR, I., Cea nai bună aproximare in spatii vectoriale normate prin elemente din subspatii vectoriale, Edit.
Acad. R.S. România, Bucureşti, 1967 .
3 JOLINSON, J. A., Banach Spaces of Lipschitz Functions and Vector-Valued Lipschitz Functions, Trans. Amer. Math. Soc. 148 (1970), 147-169.
4 Wic SHATL, E.J., Extension of range of functions, Bull. Amer. Jath. Soc. 40 (1934), 837-842.
5 USTATA, C., Best Approximation and Unique Intiension of Lipsch itz Functions , J.A.T. 19 (1977), 222-230.
6 R.OI, A.I., Extreme Points and Iinear Isometries of Banach Space of Lijschitz Functions, Canad. J. of Math. 20 (1968), 1150-1164.
7 SIMGR, I., Cea nai bună aproximare in spatii vectoriale normate prin elemente din subspatii vectoriale, Edit.
Acad. R.S. România, Bucureşti, 1967 .