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trIEI'UN D'ANAT,I-STì hIUIíI1¡ITQUì' ì]T NT fl{I.IOIìID

DII

I,'APPEOIilBIATIOÑ

'

'Iortre 21, N" P, 100¡1, pp,

l$5-l{lí

, .l'', ,,;: :

, ri

r' ,.,,',i i: ,

I

(Cluj-Napocn)' 'r"1 : i.ll11

:.þt ({,d)

Þu a,_Tgtiig qpace a,nd

T

â, nonvoid ßubset of

T.

Á, funo¡

fion / : y -

R is caJlecl Lipschit? on

r

if

tnei";iúrrl;ö";;;h rü;*""-

for

all

u, y e T.-4. numrrer

r( >

0vcr.if-r,ing (r.1) is cailecr a r,i1tschi,r,z constatni ,fpr .f.

ft is

easy

to srorv that

i;tre

quantity illllr

deiined

by

,l

734 Mitcea lvan' 1

Dans le

ca,s

de I'interpolation habituelle

d.u

type

l,agrange,

on a:

' : Pn:

iL(fru Qzt..

frni ')¡ .i .

-

Po

:

L(nrr

fiz¡...¡

frr¡

t i')¡

[P"; .] : lfrt.

fiz¡ . . û_n¡ û

i

.)t

KN(E):

çE

- ørX\-

az) .

..

(q

-..eo)

,,,

-

nr)(u

-

nz)

...

(m -L ¡;¡)

': EiTBLIOGRAPHIE

: .,i I

ú. M, Ivan, Opérateurs d.'interpolation dans'des esþacès abslrails, Mathematica,4, 7 (19?5), 19-28 2. M. Ivan, Dífférences d.iuisées génhalisóes et fonctionnelles d.e Ia forme simple, Mathematica,

9,

I(1980),55-58. : r': :

rr'

3. Cui Ming-Gen, Zhang Mian, Deng zhoug Xing, Two-dimensi.onal rcproilucing

kunel

and.

surface interpolation,

J.

Computational. Mathematics, 1r, 2 (1986), 177-18L, 4. M. Ivan, Sur une suite d'opérateurs d'interpolation et.rl'aþþronimation àans'ún ëspoce de

Hilbeú, Mathematica, 20,

7-2,

(7991), 43-46.

Recu le 20 aott 1991:

'ri.s

'

Institutul Potitehnic Cluj-Napoaa

, '

Sfr. (3400) CIuj-NcipocaC. Daicouiciu 15, Dept. I(ATEMATICA ROMANIA

':

(2)

Selections associated',

:to

Mc$hand'Ìl extension theorem 737

)bea

;,

,' '< ll¡'* '?ll.t

cl(r;

ro):'i o

I

,thb

betiuence ø e..f¡ancl:

al[

in

l-ripo.T and.

t}¡at n' e

P(l),;

2".

Io

pl,ove the seqqr-rrt

ipeclu{lty in

(.f ,8).suppose,.on,':the.qont/rarI,

that

ih.ere

eiists nreX

puch

thal {(Ø.rÞn'r(rt),

Þi+c.g tllq.fg4cbigns 'F, -fil. are conl,ituous ón

X

and'-F

lr¿'f 'n'zLy; it fcillciws"tltat

phe5''.agreg

t¡"

closure

' of Y.

Therefore

r, e,\Y, irnplying

d(nr;:11);'y' 0

for

aII

a

e

Y

.lalring into

accorurb definition (1 .6) of P, and 1,he incqualii,y

Fr(øt)<.

;p(*r)r

1,here "exists an elemestr gi e

Y

'sti'oh

t̡at

I

'

'l)

ì:

f(yt) *

ll"fllt d(æt'

y) /-l(nr)]'.,'; ' '';"

rìi'1

"::: r'' '

'ìi'

14 e I-.,ipo{": J,l],[r

the anuihilator

sullspace

of Y ini'f,,ipo¡. ':

'; r.'r

: | '.;ii.'i i

;.¡'ì.';

i;'

r'':1

''..t. 1:! ; :, " ,T

Cóstiéá

Tho

space

LipoX is

cléfinecl

similarly:

(1.4).

I,iPoX

: flî

e I',iP

X tT(n): 0Ì'

ancl

ll ll*

a,r'q nornrs

on

TliPoY

Lip"Y

are Banach spa,ces

wlt'n

res-

i'" 'j 'r j i '

theorem, er¡ery

/

e I-lipoY 4a,*.

*

norm

slttr 3{ensions

âre

effectively

glven

'

t'lt: t' ' -

(1.5)

-E'1(r)

:

sup

{/(g) ll/ll"

d(n' v) t u e

yI

ancl

çrro),,.,,

,,.

1,,,.

F,(,.;),=,,io,t

{/(v)t:ll/llf $!r1,XlidìfiYl ,.' "'i "' ,,,,

a- %ir:;f;' o,

-u : r_,ip;y ,;tzLipox

tlre set-\,aluett operator

crefinecl

i . '1, ,': ili:

-" ' li'!:l

r,r'¡: ,.; j,.'

by

i:;f I

(1.7) ,ancl

]lf'

'i . Ii(#-).< P{r

).,'6''?'r(ø); m e

T,

are tlta efiLensions of "t'gíben' tìY

(I

.6)'timót''(1"6)

j":r'. :;

ir '

anil, only

3' Ì'or

iJ

f

i's ll"fllran

: I,

t'lue

entremal'

ß a

.fa'ce

of

tlt'e

'ttue '\aû;il'l¡all o,f LiPo

anit bail

Bv¡ox í'J

sat

E(l)

'ltoitut oJ

v.

-

:

(3)

Selections asSociated,; to , .Mc$hane's extension

We shâll

sa,J¡

that the set Y

2..L natural

question

is

wlren

JP"r

contiìiuorìs seläctions.

ff

S :

-{

function.s; A.* B is

called

a

selectí,on,for ¡S,if, s(ø)

eS(ø), for.all neA.

In

the follou'ing'theorems, we shall provit ttrò existence of continuous selections

lor

the operators

D

anù

Py.r in

the

particular

case

x :n,

,wjtþ,

the nsual

clistance

ù(n,y): l" - yl. :

I

th,e lrflt"-z;l!,h\i,n*":!"#ni,;l:;,1,{;.fl;ri}:ri

horn

;

iþoy * I,ipoIl by

er(l)=þr, /eLipoY,

äxteirsionrot

/

given

by

1r.O¡.

1

'

er(o,lI@)

: n

{!.ilsJ)

* ll"/llr

la

- tJl:

y-e

la,bf,}

,, ,

i n: lnf {/(y)"-L llÍll"ln

, ql,: y

e

Ie:

bl} :

:

q..na@)

.1 I

rbowirrg that, er_is

positiveìy

hornogerreous.

ir-'o prove

the continuity of

e; faT

å>

0, l,ake ð

:

e/3 anct,show

that

(2.3)' ,ler(il:azlill <Fr,

ì

'tr ' *lJ,/,

'

'rt

rs easy g e

LipuY to

eheek:l,rrat such

rhar

lhe- ,ll/

-,øll"< mi¡lirnat

ð.extension

F,

or

J

has

the

form Er(u)'

,: l@) r ll/ll,(ø - ø), for

c,

<&

':l(æ), : :"' for nela,b)

#her" ?, is

the maximal

ff a)

Othen

138 Costicä . Ì.Mustäfa:

. ) A

sirbset

Â'of LipoX

is callecl p:ronintCnøl

in LipoX if

every'Jfl e

lipoX

has

a

nearest

point in Â, i,e. there

exists G e

À

such

that

ll.F

- Glli':

= 4(4,

,l , A)¡where

- ' i, ,r ' i I r; rl I 'i ¡

ct(,E,

^) : inf {lE - Ell'

: e Â}.

T)ne meþic;gtrojection,'

P¡:

I-,ipoX

-2^'is

clefinecl

by

P,1(.EJ)

- {Gç l:,llF - Gll* :

d(F, Â)}.

If

P,r(@ is:a singleton'for overy,F e

LipoX

1;hen

¡\

is oallecl a,Olt'aWs.- h,eaian,

; .'fher'e,is

subset

of,:I-ripoX. , ! a

clos-ê?,relation between',1,he

¡¡ i: ,' ''

exl,ension

i 'l :tit ì

opefål,torl i-D'and!

the

projection oÉerator

-R"r,

ilustrated"

in tho foüo'wiig

theorem:

':1,' ,' t't';

: l:ìj ,t.

r.Ì; 'ri .

':,.- " .,' : ',,.,,.'i .,

(2.1) d(?, vr) =

ll-E'lrll,"; ,

,istyueJoralllrelripu{,; ' ,,r:, ,'l ,,, ,:'i

3"

A

fwncùí,on

GeY+

'is

a

best

:E :'E

oi¡lll,

il

,tlùare ei;ni,st's

H

e -E('X'

")

sueh,

(2 2)

:$re

provo

forrnula

{2.1): ff -reT',ipo4r 'thefl

fof'

' å,nitrt,. 7l

fllu!)tlÙJu

'g',' e'

I

r-,',t!'

* g'I,

r' i l;il

,t' I]il"I'':':'l

lrithere

elisüs -E ë

I-riþ'X

such

thatì

lowd ,ttlat E.

), E

er

yr

. anet , ll-Þ, l" ll¿i : G e

Y:r]: d(T, Y")' showihg,that'

' tr. "t .1, '

I

trow

from [6],

Lernma L,

p.223 unil

:.

àssertion

l-o follorrs flom

3".

Now,

by

Theorern 2, a", tlne set

Pr,r(?) - E -

lq(F

F) is

bou:rdod,

con-vex

and

closecl,

for

¿ny -F ê lrìpn.Y..

(4)

Selections associated to extension theorem

selectioi. er(fl': f

r) lvhere -É'.

is the rninimal

exten- R'em,at'lt

2.

C]ne

sion of

/

ctefinecl. trY

This

ean be Provecl (2.10)

(1.5)

is

also conlinuous- ancl posi

directly'or tahing intolaccount

rhê

âiiJ

,Tpkìng

into, accoutt the'ineclualities (2.4)-(2.9)., it.follo¡vs that'

lV, -

Grll*

: llErll' -

sup

{lEr(n) - Hr(E)lll" - lll: r,E

e

n; û+a\ <e

i.e. '

lerl'h

-

'e'r(ùl

<". 'r'ri '

' '

(

' :' ,,.,.1

which '

h9k1s

r "ì for.all

'these

/eI-,,ipo{, i

,...

trv

one obtains

lhe

following

;

. -- :: ì , . t,

,.,

.,

5.

.Tlte

oge1'gtl,or

E:.LipsI

=.2''tÞ

i¿hùo og:eneous

gitsen' by

e(l) : í12)"(er(il'l'

er(/)),

/eLipoY.'' i"i:r

1

I

.'

'''''

' i

Proof

.

Obviousl¡' l]nat a

is

continuous anci

positivel¡'

homogeneous.

On the olher harirl by

(2.10),

it follows .. '',:' ;

'rr

'r

.

;' e(-1.)

= -'e(f)¡J

e

LipoY, ', :

::

lpnlvìnS the

homogeneity qfl' e ; e(ql)

= ø' e(f\ a'e'R, I

F L]uoY,'.

3. Tbis

section

is

concerned

rvith

t,

rc

existence

'of"

selections

for

tbe)".sl{ttric..projeetion

rPir

.in',

the

eaße

,X,='R¡

y1

:',[a¡

.b],\Ì10

e

[a,

b].

; rr",!l,A.rßelection,,fi :

IiipoX,-+Yr

of .

P'r.is,,rcallgcl

,:gùd,itipe .mgdwl-o

Yt,

toJi{llf

srf-,ip5,Y 'anctìG e

I¡,

, ,;

Tn¡onnru Py

¡

Þos a, h,omogeneous, add,i-

r;'i j .- . : I

6. 'Ilta

metri,c projection

t anil

conl,inuous seiec|tiott' p

P:I-aof¡

."..

;',' ,...-' i-iiì',

pox - LipnY is the restricl,ion olterator given by r(I) : f : IlipoI,.; I,jpo;T

riS

t,he identity map.

Then

p(X) : (I -

e"r)(tr').

¡.Ì] ,. u(I

l") ç

I'or(P).

1:.' ,:li'"r; ,.:'j" t)

t:40 Costicã

< llj -.sll"'

,l

I:t, follows :that ;th'e. differencê

r

-nÌ,

i-iG,

has the;

(ll,fll"'L'

lls llr)i

._1'.ii,r ':,¡i : ,'ti l' , ;:

. j ì, :..,i¡"jj 1 :

!,!

Cnse

3". o'4''à,3/'<

It;

m* tÌ.Iir

this'caiie

(2;6)

l'rruþb)'--

H,\tùl ::t(/ - øxd'll-Ì(/ I sx3/)l

<

: ì , 1. .':

< tl/- sll''ln'-Yl <(el})'lt-Yl'

'.": i r:: È.'.liì ¡r,::1. '. ¡' :il:'':"+ .i' '.it 'ti¿'r"':'

"''.!,iti ::::i ,l':;..ji -0a60:4.?:r,,;ü'3n 4þ:Ã'U'

ln'thiSti'og'Ëc"

i

l''r":

jrj;r:l t¡"'i

"tti't''¡" :t'!l-

(2.7)

lr.r(m)

-

Hr(Ü)l

tll(Ï'r'ø)(')'t (/ -

gxb)

- ' . "''ti,

j,{

!:i:

,)

ii

r

: ,

,

' '., ,r:(llf ll"''llglii)(r ''Þ)l!(.ú/; ûll'+l'ttlll""-¡b'li"'il''.''

, ' lF -,b)S

z

: ll/ - 'olly!.y,'.='Yl.l\2e13)' ln - vl'

Cuse 5". u,

;¡1.,rl f l,:b.

Iìeasoning

liko,ip

Case 4" one obtains

(2.8) , lE,(,") :

H'p?t)l

5

(2'[3) ",lm

-

Yl'

,.,...,,,.,cq,[r,,F?;.,.q.,n1c\<þ<y,'

rn'this

gâse

:;.,',.,,,.-r,.,,..',.rr",',..,',,t1,, jì:r

(2.e)

lr.r(æ)

-

Hr(y)l

: lff '-

s)("-)

I ('llTllr'- [ts[l")ft'' a\'-"''

I <.ll-f -

g'llvln

-

bl

*lll/ll"'-'[lgllvl'

* !/ -

al

-bl<

<3'll/ - gllv|n'tl l(38i+il* -tyl : e'ln -Yl'

Cuse

2". n,y 1!r

,n*,U,frclsoninp lã,(n) - H,(Y)l <"

lilie

Case 1o . ; .,r.!..ioue otrtains

L¡r )i.;.'''

' ¡l

r, :l

(2.5)

(5)

for any /eL,,ipoY. Obviou,sly

e(ol

*

þg)(n)

: q.'e(l)@) +

p 'e(g)10t¡

for all ne& and all f,geI-'ipuY, a.,þeÈ,

showing

that ¿ is

linóar.

The

continuity of

ø.'rvas prrovecì.

in

Corollary

5. ì

,

(b) The application p

: I-'ipoY

Ì yt

clefinecl,

for F

e

LipoX, by

,i.. r'. ìi,j'r , ;, .,..,.'...

(3.5) p(P):I - e(I'l") -l¡ - (U2)(ILlIr),

where-

F' I,

a¡e

the

ext¿nsions

given by (1.5),

(1.6),

is

linea,r' ancl con- tinuous.

(c)I-ret ',,i1, i, : ,, .r.

(3:Ð

' ì. ,

$

t,'

=.

' {e(F,lÐ: ,,i -E.e },ipoX}.. r, i,,

_ Th"

linea,rity

of

e implies tþq,t

ï/

i,s

a

subspace

of lipoX. By

(8,3)

and.

the

ecluality

ll¿("1)oll'': llni"ll" ' :' ';

it follo#i l,h;t

l,Iz c

](er Pur. ' ' ¡

,

r I.or

-27 e

IripoX define ' i.

,

(33)

"o :r:,,T;'--;,:,':: n.*'' ' | " 'i

"

extension theorem 743

and j'

,

: f@) - F(b), lar s>

b

(3'8) i' : . H(a) -F(r), tor uela,bf : n@), lot u<'a

"

: I(b), for u>

b.

e(Fl;)

'w'

clirect sum

of,the

subspaces projection operator on

7r

is -+

G, where

G e

nr

i

is

the

ar and. continuous operator

from

ir.'r; The linearitv is *oioru,. To prove the continuity

suppose

that

tt)À

î,x in Lipox,"i.e.

llE,

- rlÞ* o',-iné"-*,

,

-'-"'.^*'*"

:

G"lu):0t &ela,bl

: r:\:) - tl(ø),

ø

1a :1"(æI-I"(b)¡ u>b

742 Costiiá '

Indeecl'

trr'-- a(-l'¡r¡'': (1/2)(Il :'l1r)'*

(r/2)(-n

'Ir)

e

set ?r.r(F) is

convex.

I

li:: '.11; i,..:i;,i:rr,i .,:l

Pyr(f); 'since

the

j

e' 1)(I),

r

| / I , 1 ', r' ,; .

Obviously the

selection

p is

fg.ntinuopq ana ,

p,\url 7, n"I + e("{ lr) :

ø(-F

*

q(

lf lr)) :

, .i ' lrr

:

for aii o,,.;li,

'sliowrpg ünut 7t is'þor.noþenqbup':

Now \ ,,,. ,i

ì

(3.2) p(I + G):t +G - e((I +

G)1")

: r +G - e(7ttl"\:

. t ., ' l;

- I -¿(Fl") *

G

:

1)@)

+

G

: P(Il I r(G)'

'':, ltl.

"

rl ' !:

" - '1 ""

fnr aJl-I7el-,irr^;g*r.äç eYL,

sincefor

GeYi, (G):

{G} and

2(G):G

' By

(3.2),

the i¿ì¿ãt¿"'2

is,aclditive

mo

Y-r

lnd Th!!re+ 6.il;

provetl.

It is

easily seen .ühat

the lrornel of Pvr'r ,

Ker vr :

{.F e

LiPoX,

o e

P"r(f)}

..,. ,t : .. :rr i I 't'l

v"tifiu* the equality ir: 'r' : " I :'i i¡ l'r"i

I

(3.3) I(er P'r

=:lI tlipoX:ll/[l' : [lFl"ll"]'

Ccr,rcir,r.¡r,v.-l. n'or

X : ß, y ='lui úl'

and'

noe'lø.;'bl

tlta fol'lowing

o" r ní,iiiul. *t'¿. tn'ue :'

' la)

T.ha 'en1a'itsíotu' olterutori

Ð

ltas a'Úínaur qii'd' 'comtimuow;s sal'aation;

ìfrl ,fn,

,n¿n¿'r.1rro¡ibtto*

po¡li;as,a

lirieur:6,nd,l co:ntinuoq,ti 'selèatiom',

(c) lltere

en¡sti.

i

sutrrpooa

lv or the

subspaoa K:er

Prt

Swc-'h,. th,at

etut¡llelripoXcan,beuq'iq¡'gtyrepres.entod'i'tt't'l¡efo1ryn'-:HtG^t¡ti'tlt"

H".u+l,G; Íï;

i..':."

tt;";;lie;ít ii' ¿t

corryilemeìúed

in

r-,ipn-x'

Proaf.

(a) lYe

sho-w

that the application

e

:'LipoY ;

'I-,.4¡oX ldefi-

ned.llY , ,, : ,i.

.!.

r . ...,,:..r.

__,

1..,.,

.,\

i' ,' :r'ì:

'r:'

(3.4) e{l):(1/2)(:IÌ1 f F')'

'"t I' r'

where

ext¡enþ,I extcnqio4s'of./. sitlen

b)'

(1'5) and

(t'6)'

is

linear

sse'lecl,ionof

l' .:.t:' : :';""

'

\\¡riting exPlicitlY

¿

we fin'[

dì'at

:i

a(/Xø)

: l@)¡ for

m 1ffit ,: ' ¡tr.. ,lì '. ^- -1: ì

"i:

rr:i''''] -/(æ)¡ lori'øe.fu'ltf'

\' ':"f(b),

fo¡r

nll¡t

'

(6)

11 Selections associated to.: McShane's theorem 746 REFERENCES

2. Azipser, J., Géher, L., Er/r.nsi.ort of Fttnctions salisfyi.rtg ct Lipscþilz.con¿ifio¿s..Áôtanlatl.

Acad. Sci. IJurrgar 6 (1955), 213-220.

3. Deutsch, F.., Wu Li, ttlg-I.tg Patl<, Tklze îølrnsio.rrs and Conlinuotts Seleelio¡ts fot Metríc Projections. J.4.1'. 64 (1991), 55-68.

4, Fakhoury, H., SéIections linéaires associées au théorème de Hahn-Butaclr, J. F¡nct. Analysis

tI (7972),436-452.

5. lvlc shane; E, J., Ertens[on of range of functions, Bu]], Amer. l\,Iath. soc. d0 (1gs4), Bgz_g42.

6. Mustãla' C., BesI Approaimalton and Unique Extension of Lipschilz ¡.4ttitioní,'J.A.T. fg (ts??), 222-230.

7. Mustãfa, C.,

^[ - ideals in metric spaccs, "Babeç-Bolyai,, University, Fac. of Ì\{ath. and Physics, Research seminars, seminaron Math. Anal., preprint Nr. z, 1ggg, 65-?4.

8. Rudin W.¡ Fi¡nclional Analysis, l{ccraw-Ililt 19?3.

Receiled 15.V.1992 Instilulul ile CeIcuI

Oficittl Poçtal 1

c.P. 6s 3400 'CIuj-Napoca

Iì.omqnia

i l ::, r.i

:l(t,--- ilj(ø) \ (r,, -rltEit

< lln'"

- rW'@ - ol-'"'

ii

Thd

same ìnèquzility is'obtainecl

lot *,9>b' '

Oase

2. a 1ø <b

<11.

In this

case

,

':'

IU^(n)

- U*(v)l :

lu*(tJ)l

:

lE*(s)

- E(v) (r(b) - lfl"fb))l:.

: l(I* - E)(y) - (F,-]7x¿,)l ( ll7, - Ill*lv

... bl < llF,,

-rll"'' ln - vl

The

same inecluality

holds for û <'ü <!l <b'

''

Case

3. n <ü <b

1Y,.

In this

case

lU*(n) - U,,(u)1: lr*(a) -F(r) - (T,(a\ -l¡(ø)) -r"(tt) *r(u) I

+I*(b) -r(¿')l < l(lî,, _ P)(r) -(t,-I)(aJl + l(r'-rxb) - - (I"- fl)(a)l <

ll'F"

-Ill,' ln _ El+ llÍ',,-rll*'Q - a)

1

(

2lln'"

-111"'ln -Yl.

. It follows that

lU,(r) -,Ar@)l < 2lE' -Fll' ' lc - Yl, , o

.

for all

ü¡ U eJ?,

imptying llU'llx <LllE, -Fll' - 0':

'

It

toltóws l¡nat

I* -lil

implies Gn

*

G, showing.tha,t

tþe

proie-ction

.ro""rfõ"- oo yt is

continuous^

and

óonsequentty

lipoX is the direct

*i* ãt Ii and I[. Corollary ? is

completely proved'

:. '

Rerna,rks

3; (a)lfn the'considered

caSll (:.4

= 4, :^Í : tq,

b1,.yoe

/

e LipoY,

l

+

9.'tr-

lact, et(f)

<e(l) <

X

0

å:

jection Pyt

aïø,.Iiqe.4,r ¿nQ-^ gingle.v-alued'

I44, 10

m e

lu¡bl

. '.: I

Referințe

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