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$5-l{lí
, .l'', ,,;: :
, rir' ,.,,',i i: ,
I(Cluj-Napocn)' 'r"1 : i.ll11
:.þt ({,d)
Þu a,_Tgtiig qpace a,ndT
â, nonvoid ßubset ofT.
Á, funo¡fion / : y -
R is caJlecl Lipschit? onr
iftnei";iúrrl;ö";;;h rü;*""-
for
all
u, y e T.-4. numrrerr( >
0vcr.if-r,ing (r.1) is cailecr a r,i1tschi,r,z constatni ,fpr .f.ft is
easyto srorv that
i;trequantity illllr
deiinedby
,l
734 Mitcea lvan' 1
Dans le
ca,sde I'interpolation habituelle
d.utype
l,agrange,on a:
' : Pn:
iL(fru Qzt.. .¡frni ')¡ .i .
-Po
:
L(nrrfiz¡...¡
frr¡t i')¡
[P"; .] : lfrt.
fiz¡ . . .¡ û_n¡ ûi
.)tKN(E):
çE- ørX\-
az) ...
(q-..eo)
,,,(æ
-
nr)(u-
nz)...
(m -L ¡;¡)': EiTBLIOGRAPHIE
: .,i I
ú. M, Ivan, Opérateurs d.'interpolation dans'des esþacès abslrails, Mathematica,4, 7 (19?5), 19-28 2. M. Ivan, Dífférences d.iuisées génhalisóes et fonctionnelles d.e Ia forme simple, Mathematica,
9,
I(1980),55-58. : r': :
rr'3. Cui Ming-Gen, Zhang Mian, Deng zhoug Xing, Two-dimensi.onal rcproilucing
kunel
and.surface interpolation,
J.
Computational. Mathematics, 1r, 2 (1986), 177-18L, 4. M. Ivan, Sur une suite d'opérateurs d'interpolation et.rl'aþþronimation àans'ún ëspoce deHilbeú, Mathematica, 20,
7-2,
(7991), 43-46.Recu le 20 aott 1991:
'ri.s
'
Institutul Potitehnic Cluj-Napoaa, '
Sfr. (3400) CIuj-NcipocaC. Daicouiciu 15, Dept. I(ATEMATICA ROMANIA':
Selections associated',
:to
Mc$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-rrtipeclu{lty in
(.f ,8).suppose,.on,':the.qont/rarI,that
ih.ereeiists nreX
puchthal {(Ø.rÞn'r(rt),
Þi+c.g tllq.fg4cbigns 'F, -fil. are conl,ituous ónX
and'-Flr¿'f 'n'zLy; it fcillciws"tltat
phe5''.agregoñ
t¡"
closure' of Y.
Thereforer, e,\Y, irnplying
d(nr;:11);'y' 0for
aIIa
e
Y.lalring into
accorurb definition (1 .6) of P, and 1,he incqualii,yFr(øt)<.
;p(*r)r
1,here "exists an elemestr gi e
Y
'sti'oht̡at
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
sullspaceof Y ini'f,,ipo¡. ':
'; r.'r: | '.;ii.'i i
;.¡'ì.';i;'
r'':1''..t. 1:! ; :, " ,T
Cóstiéá
Tho
spaceLipoX is
cléfineclsimilarly:
(1.4).
I,iPoX: flî
e I',iPX tT(n): 0Ì'
lþ
anclll ll*
a,r'q nornrson
TliPoYLip"Y
are Banach spa,ceswlt'n
res-i'" 'j 'r j i '
theorem, er¡ery
/
e I-lipoY 4a,*.*
normslttr 3{ensions
âreeffectively
glven'
t'lt: t' ' -
(1.5)
-E'1(r):
sup{/(g) ll/ll"
d(n' v) t u eyI
ancl
çrro),,.,,
,,.
1,,,.
F,(,.;),=,,io,t{/(v)t:ll/llf $!r1,XlidìfiYl ,.' "'i "' ,,,,
a- %ir:;f;' o,
-u : r_,ip;y ,;tzLipoxtlre set-\,aluett operator
crefinecli . '1, ,': ili:
-" ' li'!:l
r,r'¡: ,.; j,.'
by
i:;f I
(1.7) ,ancl
]lf'
'i . Ii(#-).< P{r
).,'6''?'r(ø); m eX¡
T,
are tlta efiLensions of "t'gíben' tìY(I
.6)'timót''(1"6)j":r'. :;
ir 'anil, only
3' Ì'or
iJf
i's ll"fllran: I,
t'lueentremal'
ß a
.fa'ceof
tlt'e'ttue '\aû;il'l¡all o,f LiPo
anit bail
Bv¡ox í'Jsat
E(l)
'ltoitut oJ
v.
-
:
Selections asSociated,; to , .Mc$hane's extension
We shâll
sa,J¡that the set Y
2..L natural
questionis
wlrenJP"r
contiìiuorìs seläctions.ff
S :-{
function.s; A.* B is
calleda
selectí,on,for ¡S,if, s(ø)eS(ø), for.all neA.
In
the follou'ing'theorems, we shall provit ttrò existence of continuous selectionslor
the operatorsD
anùPy.r in
theparticular
casex :n,
,wjtþ,the nsual
clistanceù(n,y): l" - yl. :
Ith,e lrflt"-z;l!,h\i,n*":!"#ni,;l:;,1,{;.fl;ri}:ri
horn
;iþoy * I,ipoIl by
er(l)=þr, /eLipoY,
äxteirsionrot
/
givenby
1r.O¡.1
'er(o,lI@)
: n
{!.ilsJ)* ll"/llr
la- tJl:
y-ela,bf,}
,, ,
i n: lnf {/(y)"-L llÍll"ln
, ql,: ye
Ie:bl} :
:
q..na@).1 I
rbowirrg that, er_is
positiveìy
hornogerreous.ir-'o prove
the continuity of
e; faTå>
0, l,ake ð:
e/3 anct,showthat
(2.3)' ,ler(il:azlill <Fr,
ì
'tr ' *lJ,/,
''rt
rs easy g eLipuY to
eheek:l,rrat suchrhar
lhe- ,ll/-,øll"< mi¡lirnat
ð.extensionF,
orJ
hasthe
form Er(u)',: l@) r ll/ll,(ø - ø), for
c,<&
':l(æ), : :"' for nela,b)
#her" ?, is
the maximalff a)
Othen138 Costicä . Ì.Mustäfa:
. ) A
sirbsetÂ'of LipoX
is callecl p:ronintCnølin LipoX if
every'Jfl elipoX
has
a
nearestpoint in Â, i,e. there
exists G eÀ
suchthat
ll.F- Glli':
= 4(4,
,l , A)¡where- ' i, ,r ' i I r; rl I 'i ¡
ct(,E,
^) : inf {lE - Ell'
: -ã e Â}.T)ne meþic;gtrojection,'
P¡:
I-,ipoX-2^'is
clefineclby
P,1(.EJ)
- {Gç l:,llF - Gll* :
d(F, Â)}.If
P,r(@ is:a singleton'for overy,F eLipoX
1;hen¡\
is oallecl a,Olt'aWs.- h,eaian,; .'fher'e,is
subsetof,:I-ripoX. , ! a
clos-ê?,relation between',1,he¡¡ i: ,' ''
exl,ensioni '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,onGeY+
'isa
best:E :'E
oi¡lll,
il
,tlùare ei;ni,st'sH
e -E('X'")
sueh,(2 2)
:$re
provo
forrnula{2.1): ff -reT',ipo4r 'thefl
fof'
' å,nitrt,. 7lfllu!)tlÙJu
'g',' e'I
r-,',t!'* g'I,
r' i l;il
,t' I]il"I'':':'l
lrithere
elisüs -E ëI-riþ'X
suchthatì
lowd ,ttlat E.
), E
eryr
. anet , ll-Þ, l" ll¿i : G eY: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 setPr,r(?) - E -
lq(FF) is
bou:rdod,con-vex
and
closecl,for
¿ny -F ê lrìpn.Y..Selections associated to extension theorem
selectioi. er(fl': f
r) lvhere -É'.is the rninimal
exten- R'em,at'lt2.
C]nesion of
/
ctefinecl. trYThis
ean be Provecl (2.10)(1.5)
is
also conlinuous- ancl posidirectly'or tahing intolaccount
rhêâiiJ
,Tpkìnginto, accoutt the'ineclualities (2.4)-(2.9)., it.follo¡vs that'
lV, -
Grll*: llErll' -
sup{lEr(n) - Hr(E)lll" - lll: r,E
en; û+a\ <e
i.e. '
lerl'h-
'e'r(ùl<". 'r'ri '
' '(
' :' ,,.,.1
which '
h9k1sr "ì for.all
'these/eI-,,ipo{, i
,...trv
one obtainslhe
followingrì ;
. -- :: ì , . t,
,.,
.,
5..Tlte
oge1'gtl,orE:.LipsI
=.2''tÞ
i¿hùo og:eneous
gitsen' bye(l) : í12)"(er(il'l'
er(/)),/eLipoY.'' i"i:r
1I
.''''''
' i
Proof.
Obviousl¡' l]nat ais
continuous ancipositivel¡'
homogeneous.On the olher harirl by
(2.10),it follows .. '',:' ;
'rrií
'r.
;' e(-1.)= -'e(f)¡J
eLipoY, ', :
::lpnlvìnS the
homogeneity qfl' e ; e(ql)= ø' e(f\ a'e'R, I
F L]uoY,'.3. Tbis
sectionis
concernedrvith
t,rc
existence'of"
selectionsfor
tbe)".sl{ttric..projeetionrPir
.in',the
eaße,X,='R¡
y1:',[a¡
.b],\Ì10e
[a,b].
; rr",!l,A.rßelection,,fi :
IiipoX,-+Yr
of .P'r.is,,rcallgcl
,:gùd,itipe .mgdwl-oYt,
toJi{llf
srf-,ip5,Y 'anctìG eI¡,
, ,;Tn¡onnru Py
¡
Þos a, h,omogeneous, add,i-r;'i j .- . : I
6. 'Ilta
metri,c projectiont anil
conl,inuous seiec|tiott' pP:I-aof¡
."..;',' ,...-' i-iiì',
pox - LipnY is the restricl,ion olterator given by r(I) : f : IlipoI,.; I,jpo;T
riSt,he identity map.
Thenp(X) : (I -
e"r)(tr').¡.Ì] ,. u(I
l") çI'or(P).
1:.' ,:li'"r; ,.:'j" t)
t:40 Costicã
< llj -.sll"'
,lI:t, follows :that ;th'e. differencê
r
-nÌ,i-iG,
has the;(ll,fll"'L'
lls llr)i._1'.ii,r ':,¡i : ,'ti l' , i¡ ;:
. 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"
il''r":
jrj;r:l t¡"'i"tti't''¡" :t'!l-
(2.7)
lr.r(m)-
Hr(Ü)ltll(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.
Iìeasoningliko,ip
Case 4" one obtains(2.8) , lE,(,") :
H'p?t)l5
(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'
læ* !/ -
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 otrtainsL¡r )i.;.'''
' ¡l
r, :l
(2.5)
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,
showingthat ¿ is
linóar.The
continuity of
ø.'rvas prrovecì.in
Corollary5. ì
,(b) The application p
: I-'ipoYÌ yt
clefinecl,for F
eLipoX, by
,i.. r'. ìi,j'r , ;, .,..,.'...(3.5) p(P):I - e(I'l") -l¡ - (U2)(ILlIr),
where-
F' I,
a¡ethe
ext¿nsionsgiven by (1.5),
(1.6),is
linea,r' ancl con- tinuous.(c)I-ret ',,i1, i, : ,, .r.
(3:Ð
' ì. .Ì ,$
t,'=.
' {e(F,lÐ: ,,i -E.e },ipoX}.. r, i,,_ Th"
linea,rityof
e implies tþq,tï/
i,sa
subspaceof lipoX. By
(8,3)and.
the
eclualityll¿("1)oll'': llni"ll" ' :' ';
it follo#i l,h;t
l,Iz c](er Pur. ' ' ¡
,r I.or
-27 eIripoX 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 sumof,the
subspaces projection operator on7r
is -+G, where
G enr
iis
thear and. continuous operator
fromir.'r; The linearitv is *oioru,. To prove the continuity
supposethat
tt)À
î,x in Lipox,"i.e.
llE,- rlÞ* o',-iné"-*,
,-'-"'.^*'*"
:
G"lu):0t &ela,bl
: r:\:) - tl(ø),
ø1a :1"(æI-I"(b)¡ u>b
742 Costiiá '
Indeecl'
trr'-- a(-l'¡r¡'': (1/2)(Il :'l1r)'*
(r/2)(-n'Ir)
eset ?r.r(F) is
convex.I
li:: '.11; i,..:i;,i:rr,i .,:l
Pyr(f); 'since
thej
e' 1)(I),
r| / I , 1 ', r' ,; .
Obviously the
selectionp 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,
sinceforGeYi, (G):
{G} and2(G):G
' By
(3.2),the i¿ì¿ãt¿"'2
is,aclditivemo
Y-rlnd Th!!re+ 6.il;
provetl.
It is
easily seen .ühatthe lrornel of Pvr'r ,
,ìKer vr :
{.F eLiPoX,
o eP"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'lowingo" r ní,iiiul. *t'¿. tn'ue :'
' la)
T.ha 'en1a'itsíotu' olterutoriÐ
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
sutrrpooalv or the
subspaoa K:erPrt
Swc-'h,. th,atetut¡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
corryilemeìúedin
r-,ipn-x'Proaf.
(a) lYe
sho-wthat 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'
rìwhere
ext¡enþ,I extcnqio4s'of./. sitlenb)'
(1'5) and(t'6)'
islinear
sse'lecl,ionofl' .:.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¡rnll¡t
'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'obtainecllot *,9>b' '
Oase2. 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 ineclualityholds for û <'ü <!l <b'
''Case
3. n <ü <b
1Y,.In this
caselU*(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¡natI* -lil
implies Gn*
G, showing.tha,ttþe
proie-ction.ro""rfõ"- oo yt is
continuous^and
óonsequenttylipoX is the direct
*i* ãt Ii and I[. Corollary ? is
completely proved':. '
Rerna,rks3; (a)lfn the'considered
caSll (:.4= 4, :^Í : tq,
b1,.yoe/
e LipoY,l
+9.'tr-
lact, et(f)<e(l) <
X
0å:
jection Pyt
aïø,.Iiqe.4,r ¿nQ-^ gingle.v-alued'I44, 10
m e
lu¡bl
. '.: I