Strictly and Uniformly Monotone Musielak-Orlicz Spaces and Applications to Best Appr~ximatiofl

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69, 173-187 (1992)
LANRUOJ FO NOITAMIXORPPA YROEHT
Strictly dna Uniformly enotonoM zcilrO-kaleisuM
secapS dna snoitacilppA ot tseB Appr~ximatiofl
W. CRUK
etutitsnI fo ,scitamehtaM ytisrevinit fo ,irazizoP Pommi, dnaloP
detacitrumnoC yb naiilr sukniP
devieceR enuJ 21, 1990; desiver lirpA 30, 1991
In siht repap lareves yticinotonom seitreporp fo eht grubmexuL mron ni
zcilrO-kaleisuM secaps )&L dna )p(,E revo cimotanon erusaem secaps era
deziretcarahc ni smret fo eht noitcnuf 4. roF )g(,L ti si devorp taht lla eseht
seitreporp edicnioc htiw eht etulosba ytiunitnoc fo eht mron dna ( > 0. emoS
snoitacilppa ot tseb noitamixorppa era ,nevig neve rof lareneg hcanaB .secittal
a‘ 1992 cimedacA ,sserP .cnI
.1 SEIRANIMILERP
teL X eb a hcanaB ecittal htiw a ecittal mron .jI.Ij gniwolloF ,2[
.pahC ,]VX eht mron .11 /I si dias ot eb ylmrofinu enotonom MU( j, fi ref
yreve E > 0 ereht stsixe )E(U > 0 hcus taht -fII jlg > 1 + )s(q revenehw
fr g E +X evitisop( enoc ni ,)X II l/f = ,I dna /I jlg 3 .E We lliw llac hcus a
ecaps a MU ecaps ”BMU“( ecaps ni .)]2[ We yas taht eht mron si yltcirts
enotonom )MTS( fi +flI llg > fI/ I/ rof lla evitagennon f; g XE htiw
jl >llg .0 nI siht esac ew llac X na MTS .ecaps We lliw osla yas taht X sah
eht MU ro MTS ,ytreporp .ylevitcepser nA tnelaviuqe noitidnoc rof X ot
eb na MTS ecaps si taht --f/l )lg < /lflI afrevenehw O>g dna .O#g
NOITISOPORP .1.1 ehT gniwollof stnemetats era .tnelaviuqe
)a( X si a MU .ecaps
)b( roF yreve O>E ereht stsixe a EM ,0( )1 hcus taht -fiI //g <
1 - )c(b revenehw f > g 2 ,0 II f II = ,1 dna llglj > .E
)c( roF lla evitagennon secneuqes ,),,f( ),g( ni X gnijfsitas j/ ilxf, = f
dna j/ ,f + ll,g + 1 ereht sdloh 11 l/,g +- .0
)d( roF lla secneuqes ,),,f( ),g( ni X gniyfsitas I/ lil,f = ,2 ,f 2 l,g 2 ,0
dira II ,f - /l,g +- 1 ereht sdloh lj ,g jI +- .0
ehT snoitidnoc )a( dna )b( nac eb desserpxe ni smret of a suludom of
mrofinu .yticinotonom ylemaN X si a MU ecaps ylesicerp nehw )8(~ =
371
29/5409-lao $3.00
thgirypoC ,% 1992 gk cimedacA .sserP .cnI
114, sthgir of noitcudarper ni yna mtof ixvreser
471 CRUK.W
+fll(fni llg - :1 J ,O>g llflI = ,1 /lgl( O> ,ro ,yltnelaviuqe =)s(6
)~2
,Oag>f:llg--fII-l(fni ,l=/If\l ,O>}s>llgll ebnactI.)l,O(Eserehw
deifirev taht rof hcus eht gniwollof seitilauqeni dloh :eurt
E
lE@
~~4(E)b6(~)/(1--6(~)).
)&(6-l
,ylraelC hcae MU ecaps si na MTS .ecaps nI MU secaps eht mron si
redro suounitnoc EX( ;)A( 0 Q ,x OJ seilpmi liaxll ( +- )0 dna yllacinotonom
etelpmoc EX( ;)B( 0 .J,x< dna ,pus Jx]l < oc+ ylpmi taht ,pus ,)XE,x
,Z[ .pahC ,VX meroehT .122 nI rehto sdrow X si a BK ecaps [ ,21 .pahC ,X
.tceS .14.4 teL su tniop tuo taht X si a BK ecaps fi dna ylno fi ti si redro
suounitnoc dna yllacinotonom etelpmoc rof secneuqes .ylno
roF ,elpmaxe secaps-,L htiw 1 < p < cc+ era MU ,secaps tub eht ecaps
,L si ton neve na MTS .ecaps ehT melborp si woh ot hsiugnitsid MTS ro
MU zcilrO-kaleisuM .secaps
ehT remrof stluser gninrecnoc eht noitaziretcarahc of eht MTS dna eht
MU ytreporp ni zcilrO secaps nac eb dnuof ni [ ,1 meroehT ;4.4 ,4
meroehT ;14 ees osla ,S[ meroehT .143 nI [ ]4,1 emos rehto yticinotonom
seitreporp era .deziretcarahc nI ,4[ 15 eht esac of eht zcilrO mron si osla
.deredisnoc
ehT tsrif mia of siht repap si ot wohs taht a zcilrO-kaleisuM ecaps revo
cimotanon erusaem ecaps dewodne htiw eht grubmexuL mron si rehtie a
MU dna ecneh na MTS ,ecaps ro ti si ton neve na MTS .ecaps nI eht esac
of ylerup cimota erusaem secaps ereht tsixe MTS zcilrO-kaleisuM secaps
hcihw era ton MU secaps .]91[ ruO hcaorppa si aiv snoitaziretcarahc of
MU dna MTS zcilrO-kaleisuM secaps )p(,L ni smret of eht raludom (+I ).
dna neht ni smret of eht noitcnuf 4 .ylno ecneH ti ,swollof ni eht cimotanon
,esac taht zcilrO-kaleisuM secaps era MU secaps :yltnelaviuqe( MTS
)secaps ylesicerp nehw 0~4 dna yeht evah redro suounitnoc )ecittal(
.mron ruO dnoces mia si ot ylppa eseht stluser ot emos tseb noitamixorppa
.smelborp
ehT MU dna MTS seitreporp nac eb deweiv sa eht mrofinu ytidnutor
)RU( dna eht ytidnutor )R( detcirtser ot eht evitisop enoc +X ni ,X -cepser
.ylevit nI ygolana ot etaidemretni ytidnutor seitreporp rof RU dna R
(WUR, CWUR, ,RUL ;RH cf. ,6[ ,32 )141 emos etaidemretni seitreporp
rof MU dna MTS era denifed .woleb ehT noitaler neewteb eht MU dna
RU ytreporp sa llew sa neewteb eht MTS dna R ytreporp swollof
yletaidemmi from snoitisoporP ,2.1 ,3.1 dna eht snoitinifed .evoba
NOITISOPORP .2.1 ehT gniwollof stnemetats era .tnelaviuqe
)a( X si a RU ,ecaps ,.e.i rof hcae E > 0 ereht stsixe a )c(8 E ,0( )1 hcus
taht f(lI + 112/)g < 1 )E(?C- revenehw II f )II llgll < 1 dna II f - l/g > .E
ZCILR4KALEISUM SECAPS 17.5
:fqorP )a( * .)b( fI )b( si ton deifsitas neht ereht stsixe na ls > 0 hcus
taht rof yreve ey E ,0( )1 eno nac dnif f, g E X gniyfsitas j/f j/ = ,I !lgil > la
il<j/Tfl/
htiw Txam .r+l<llgTflI teL .)lr+l(/)gI?If(=Tf We evah
II+f--fii .ls>)~+1(/1s2~ htiW siht ~E dna O>)rs&=~ from )a( ti
swollof taht -f(l/ /12/)+f+ = ,),s(d-l>))~~(d+l(/l a .noitcidartnoc
)b( 3 .)a( fI ,ton tuohtiw ssol of ,ytilareneg ereht tsixe 1a. > 0 dna
secneuqes ,),f( ),g( of eno-mron snoitcnuf hcus taht -,,fI/ /l,g ,> ;E
dna ,a = jl ,f + 2/ll,,g > 1 - .n/l teL ,h = ,,f( i .),,a2(/),,g nehT li,hlI = 1 dna
,,YL + .1 roF ,z = ,f( - ),cc2(/),g ew evah I/!,z/I .2/*&a gniylppA )b( htiw
2/i&=& ereht stsixe )s(8 O> hcus taht rof ,k dna ,z nevig ,evoba ew evah
Fxam ,,h/l T jI,z > 1 + .)s(8 nO eht rehto dnah ,Z/II //,z- = ,!o/l +- 1 dna siht
noitcidartnoc sehsinif eht .foorp
Remark. gnitoneD eht suludom of mrofinu yticinotonom yb dna
)E(+II
eht suludom of mrofinu ytidnutor gnitluser from )b( yb )e(q ti swollof taht
.E<)E(+~<)E(~<O roF )u(,L ew evah 0=)s(!r tub .&=)E(+V
NOITISOPORP .3.1 ehT gniwollof snoitidnoc era .tnelaviuqe
)a( X si dnutor ,)R( ,.e.i fI( + llg ,2= f/I sI/ /IgIj d 1 ylpmi f = .g
)b( roF hcae oreznon ,S xam i jl f f llh > // f I! revenehw lihl~ > .0
ehT elpmis foorp si .dettimo gnitcelloC lla eseht stcaf ti swollof taht
=RU MU .fc( ,2[ elpmaxE ,4 .pahC ,VX ,)]41 MU + ,MTS >=RU
R * .MTS
We llac a hcanaB ecittal X yllacol ylmrofinu enotonom )MUL( fi
htiw)NEz~(O~,g>f IIfII=l, ylpmil+j/,g-fII otdiassiX.O+ll,grC
eb ylkaew ylmrofinu enotonom ,)MUW( fi rof hcae evitisop lanoitcnuf
lj,f
I E X* eht( hcanaB laud ot )X dna lla secneuqes ,f > ,g > 0 htiw // = 1
eht noitidnoc jl ,f - llirg +- 1 seilpmi ),g(1 +- .0 We llac X ylkaew ylmrofinu
enotonom ni eht dnoces esnes )MUWC( .fc( eht CWUR ytreporp ni [ :)141
fi rof f,, B ,g 2 0 htiw 11 II,,f = 1 dna evitisop slanoitcnuf I E )*X(S eht( tinu
erehps ni )*X eht noitidnoc ,,f(Z - ),g +- 1 seilpmi taht I/ /j,,g +- .0 -azilacoL
snoit of eseht owt seitreporp ew( redisnoc tnatsnoc secneuqes ,,f )f= dael
ot eht stpecnoc of WLUM dna CWLUM ,secaps ylevitcepser ees( ,6[ :32
]41 rof eht evitcepser ytidnutor seitreporp dna rehtruf .)secnerefer ,yllaniF
ew yas taht a hcanaB ecittal X sessessop eht +H ytreporp na( ygolana ot
eht zseiR-nodaR ytreporp ;H ,.g.e ,32[ ,)]41 if.f b ,f > ,0 j/f j/ = ,I dna eht
671 W. CRUK
kaew ecnegrevnoc f, +-- f ylpmi eht fo’ecnegrevnoc 11 f - f,ll ot .orez ecniS
X si a hcanaB ,ecittal ni eseht snoitinifed eno nac laed yltnelaviuqe htiw eht
slanoitcmif 1 from eht elohw ecaps X* ro ,)*X(S .ylevitcepser
ehT gniwollof snoitacilpmi era :tnedive MU 3 MUL * WLUM,
MUL *CWLUM + +H dna MTS ,.e.i( ,)MTS+H MU * WUM *
WLUM + ,MTS MU -= CWUM * .MULWC
We evael eht yduts of eseht seitreporp rof hcanaB secittal ot rehtona
.noisacco teL su tniop tuo ylno taht eht +H ytreporp of hcanaB secittal
syawla sdleiy eht redro ytiunitnoc ni .X ,revoeroM a hcanaB ecittal X
si a CWLUM ecaps ylesicerp nehw ti si na MTS ecaps htiw na redro
suounitnoc .mron
teL ,T( ,C )p eb a ,etinif-o etelpmoc ,)laivirt-non( evitisop erusaem ecaps
dna d(r, :)t R + x T + ,1 eb a noitcnuf hcus taht rof .e.a-LF t E ,T ,O& )t = ,0
(d ,. )t si laivirt-non suounitnoc( ta orez htiw oreznon ,)seulav ,xevnoc dna
.csl revoeroM tel ,r(Sq ). eb ,elbarusaem rof lla r > .0 zcilrO-kaleisuM
secaps )p(+Z ,12[ ,51 ,61 17 tsisnoc of lla elbarusaem-p snoitcnuf j t-T ?i
hcus taht a&.J=)fro(,Z t(fI <&)t cc+ rof emos 0>10 gnidneped( no
f .) nehW dewodne htiw eht grubmexuL mron .11 dII ti semoceb a hcanaB
ecittal rednu eht larutan gniredro ,.g.e( ,)]12,42[ erehw 11 f dII = >l{fni :0
)A/fJZ < 1 .} hT e f noitcnu 4 si dias ot yfsitas a ,A noitidnoc 4( E )>A fi ereht
tsixe a tes oT of orez ,erusaem a tnatsnoc >K ,0 dna na elbargetni -non(
)evitagen noitcnuf ,h hcus taht rof lla t E ,,T\T dna r > 0 ereht sdloh
,r@ )t d ,r(dK )t + .)t(h
nI eht gniwollof ew lliw ,etirw rof ,trohs 4 O> ro 4 < ,oc+ fi rof .e.a-p
r E T eht noitcnuf (d ,. )t si yltcirts evitisop tpecxe( )orez ro semussa etinif
seulav ,ylno .ylevitcepser teL )p(;L = {f e :)p(,L fI I 2 ,f OJ * /I ll,f d }OJ
eb a ecapsbus of snoitcnuf htiw redro suounitnoc mron dna =)p(,E
fi E :)L&q- )fu(,Z < oc rof lla xc > .}O nehT )p(,E c )p(;L c )&L sa desolc
slaedi ees( ,42[ .p ;71 ;21 .)152 fI 4 < a+ neht )p(,E si repus redro esned
ni )p(,L dna )p(:L )&E= ,42[ .p .191 )u&L sah na redro suounitnoc
mron ylesicerp nehw )p(,L .)@L= ylraelC eht mron ni )p(,E si redro
.suounitnoc ecniS 4 si suounitnoc ta ,orez )L&L si elbasopmoced .fc( ]7[
rof .)yhpargoilbib ,oslA fi 4 < oo+ neht )&E si elbasopmoced sa .llew nI
ralucitrap siht snaem taht ereht stsixe na gnisaercni ylimaf },,T{ c C of stes
of etinif erusaem htiw ,Jl ,,T P= T hcus taht ,1 )F(,LE ro ,1 E ,)&E
,ylevitcepser rof lla n E N ees( [ ,21 .pahC ,VI .tceS .)13
We llac eht raludom &,Z ). ylmrofinu& enotonom ylmrofinu( )enotonom
fi rof hcae E > 0 ereht stsixe na )E(]Y > 0 hcus taht f 2 g > 0 ni )u&L htiw
)fJZ = 1 dna )g(,Z > E .pser( htiw II f cII = 1 dna @llg/l 2 )E ylpmi taht
fJZ - )g Q 1 .)E(]Y- oslA (,Z ). si dias ot eb yltcirts& enotonom yltcirts(
,)enotonom fi f 2 g 2 0 ni )p(,L htiw )f(,Z = 1 dna )g(bZ > 0 .pser(
11 f I) o = 1 dna II ,/lg > )0 ylpmi fJZ - )g < .1 We lliw yltrohs etirw taht
JZ )= si ,MU& ,MU ,MTS& dna MTS ,raludom .ylevitcepser ylraelC hcae
ZCILRO-KALEISUM SECAPS
MU raludom si MU& dna erofereht .MTS& oslA hcae MTS raiudom
(41 ’ ) si .MTS-$ ,revoeroM ew evah eht gniwollof .noitisoporp
NOITISOPORP .4.1 ehT raludom JI ). si s?)awfa .MU%q y!ineuqeszroC eht
snoiton fo MUbq dna MTSbq sraludom edicnioc .)yrartibra-L/(
.foorP teL E > 0 eb .yrartibra oT evorp eht tsrif trap of eht noitisoporp
ew llahs evorp taht )E(S = ~)2/~( si a doog .eciohc teL f 3 g 2 0 eb hcus taht
=)f(,I
1 dna .&>)g(,I teL =A et( :T .}.Z/)t(“@d)r(g nehT =)~JIQE
)A?.Tlg&I+).lgJI ylpmi .@>I.,-,lg(& ,txeN 3)5-f&~-)f&~
4df
1 A.’T )g-f(($t-, ).,..I 3 )A.7ff(~J)2'!&-1(-),~.lf(,r 2 x!Q%J(
lg(,I T )A 3 2)w = ,)E& jt(f)2/,&-l(<)t(g-)t(fecnis no ,A\‘T dna eht
tsrif noitressa si .devorp ehT dnoces eno swollof yletaidemmi ecnis hcae
MU& raludom si .MTS&
NOITISOPORP 1.5. fI )L&L si na MTS ecaps neht 5q > .0 ehT emas si eurt
rof )p(,E reitezrehw 5q < .cc+
foorP We deecorp yb a .noitcidartnoc teL CA { Et :T )O>r(3
,r& )t = 0 j, )A(p > ,0 dna =B .A\T ecniS T si ton na mota ew nac
emussa taht )B(u, > .0 erehT stsixe a elbarusaem rotceles )r(g E )t(I =
Y{ > :0 ,r(d )t = ,iO erehw I E .A ecniS j&L si elbasopmoced tuohtiw ssol
of ytilareneg ew nac emussa taht ,lg E .)p(,L teL h E )p(,L eb hcus taht
l=,ll,lhlI dna .O>h enifeD .Alg+Blh=f ,OZ.lgg>fylraelC ,O#,Lg
dna =1 Ilhl,ll,< fll ybll <),lh(,I=),lg-f(,I=)f(,I .1 yltneuqesnoC
1 = fI/ =$Il II f - ,lj,lg =( ,)~llelhlI a .noitcidartnoc roF eht ecaps )u&E
ew deecorp .ylsuogolana nI siht esac ew ylppa eht ytilibasopmoced ’ro
.)P(,E
.krameR tI nac eb devorp taht eht MTS ytreporp of eht )p(,L ecaps
seilpmi taht 4 < .a+ ,sihT ,revewoh lliw eb ylticilpmi deniatnoc ni 4 E .IA
2. YLTCIRTS DNA YLMROFINU ENOTONOM ZCILRO-KALEISUM SECAPS
llaceR emos cisab stcaf gninrecnoc snoitaler neewteb eht raludom ).(dI
dna eht grubmexuL mron II /I. 6 ni eht zchrO-kaleisuM ecaps )c&L dna ni
eht ecaps .)p(,E erehwyrevE woleb p si demussa ot eb cimotanon eht(
ylerup cimota esac si deredisnoc ni .)]91[
NOITISOPORP 1.2 kizduH( .)]ol[ ehT gniwollof stlfeznetats era -aviuqe
.tnel
)a( Q seifsitas eht ,A .noitidnoc
jb( fI/ =dlj 1 seilpmi =)fJZ .1
871 W. CRUK
(cl IIfnI14 i+ 1 implies &(fJ + .1
)d( )L&L seod ton niatnoc na cirtemosi ypoc fo .,I
)e( )u&L seod ton niatnoc a ecittal cirtemosi ypoc fo .,I
)f( )p(bL = ,)p(;L ,.e.i eht mron 11.II it si redro suounitnoc no .)&L
.skrameR ehT sfoorp of eht secnelaviuqe )a( o )b( o )d( nac eb dnuof
ni ]9[ eht( noitpmussa no eht dednetxe ytiunitnoc of eht noitcnuf (4 ,. t)
ni ]9[ nac eb ;)deppord ees osla ]7[ erehw 1~ si demussa .elbarapes morF
,21[ .pahC ,X .tceS 14 ti swollof taht )f( + .)e( oslA )a( + )f( ecnis )&E c
)p(;L c )L&L dna )a( seilpmi taht =)U&E )c&L .]51[ ehT noitacilpmi
)b(*)c( si laivirt dna )c(*)b( swollof from ,9[ ammeL .15.1 ,txeN
)d( 3 )e( si laivirt dna from eht foorp of meroehT 1.1 ni ]ol[ noitacilpmi(
)d( * ,)a( .p ,)16 ti swollof taht eht evitcepser yrtemosi detcurtsnoc ni ]ol[
si osla a ecittal msihpromomoh os taht )e( * .)a(
AMMEL .2.2 ehT gniwollof snoitidnoc era deifsitas rof .)L&E
)a( t,ll,hlI 1 seilpmi ),h(,Z + ,1 revenehw 60 l,hI ~fdnaf< .)p(,E
)b( )&E seod ton niatnoc na cirtemosi ypoc fo .,I
nI ralucitrap ni ,)u&E fi II f dII = 1 neht )fJI = .1
JoorP teL su tsrif evorp eht tsal .noitressa ylraelC JZ ). si etinif dna
xevnoc no .)p&E ecniS )zZJZ d ,jlklI d 1 ti tsum eb suounitnoc no .)p(,E
ecneH eht derised ytreporp .swollof
gniylppA eht emas hcaorppa sa ni ,ol[ ammeL ,15.1 ew niatbo htiw
~,=l/llkIld )11~~( taht ~),h(~Z*),a-2(+),h2(~Z)l-,a(~),h,a(~Z=l
,a( - )1 )f2(,Z + 2( - ),xt .),h(,Z ecniS f E ,)p(,E )f2(,Z < .IK+ gnitteL
n + + oc ti si nees taht ti tonnac eb taht ),,,k(,Z < lt < 1 rof yna ecneuqesbus
.)kn( ecneH )a( .swollof
ylraelC )u&E sah redro suounitnoc mron os yb eutriv of a lareneg tluser
ees( ,21[ .pahC ,X .tceS )11.4 eht noitidnoc )b( si .deifsitas
AMMEL .3.2 teL $ > .0 emussA taht f 3 ,g > ,0 II f 4II G ,1 erehw rehtie
f, ,g E ,))L&E ro f, E,,g )L&L dna $I E .,A nehT ),g(,Z +- 0 seilpmi taht
lIg,II,+O.
:foorP emussA ot eht ,yrartnoc taht ,10 = (I ,/l,g 3 E rof emos E > ,0
erehw tuohtiw ssol of ytilareneg n E IW. gniylppA ammeL 2.2 ew niatbo taht
1 = JM&Z < kl( - )1 Jg&Z + 2( - Q/l J&Z htiw Jg(,Z +- 0 dna ,,a
.dednuob yB eht noitpmussa ,,,g + 0 .e.a no T htiw 0 < ,,g J< IffE )p(,E
neht )f2JZ < .oc+ ,oslA fi fE )L&L from eht noitpmussa d E ,A ew evah
)y(,Z < .oc+ ,ecneH gniylppa eht eugsebeL meroeht ti swollof taht
ZCILRO-KALEISUM SECAPS 1~3
),,g2($1 + .0 ,woN yb eutriv of eht ytilauqeni ,evoba ew teg a -cidartnoc
,noit os eht foorp si detelpmoc .fc( .)]31[
ecniS 4 E ,A seilpmi 4 < ,GC+ from snoitisoporP 1.2 dna 5.1 ti swollof
YRALLOROC .4.2 If )L&L si a MU ro na MTS ecaps :lrezlt
)a( <%q<o .a+
)b( .zA~d
Remark. fI ~!q seod ton dneped no t E T neht )b( S= .)a(
We nigeb htiw snoitaziretcarahc of MTS dna MU seitreporp of l&L
dna )p(,E ni smret of eht evitcepser yticinotonom seitreporp of .(,I .)
of
NorTIsOPoh .5.2 ehT gniwollof sriap ,ja(( ))b( stlfemetats era
.tnelaviuqe
)a( )p(+L si na MTS ecaps )&L( si a MU .)ecaps
)b( )i( (,I ). si MTS raludom ,pser( ).(,I si MU .jradzdom
)ii( fEL&"), f// 4I/ = 1 ylpmi taht )f(aI = 1 .pser( ,i^;J E ,)p(BL
I/ @llL. 7 l WG taht ‘,(f,,j + .)1
nI weiv of noitisoporP 1.2 dna eht ,snoitinifed eht foorp si .tnedive
NOITISOPORP .6.2 ehT gnivlolfof sriap ,)a(( ))b( fo stnemetats era
tnelaviuqe rof eht ecaps .)p(,E
)a( )p(,E si na MTS ecaps )p(,E( si na MUL .)ecaps
)b( ($I ). si MTS no )p(,E raludom .pser( .(@I j si MUL no )p&E
.)raiudom
JoorP oT evorp )a( .s= )b( ew esu eht evitcepser snoitinifed dna taht
glihll Q 5 < 1 seilpmi )h(,I < .4 oT evorp )b( .z= )a( ew ylppa osla
ammeL .2.2 teL su tniop tuo taht .($I ) si MUL no )p&E fi rof hcae af. 0
ni )&E htiw f/I ol( = 1 nevig E > 0 ereht stsixe 6 > 0 hcus taht -f&Z )g <
6-l frevenehw O>g> dna .&3,ljgI/
,woN ew nac evorp ruo niam stluser of eht .repap
MEROEHT .7.2 roF a, cimotanon eht gn$wollof stnemetats era .tnefa~tiuqe
)a( )p(,L si na MTS .ecaps
)0( )p(,k si a MU .ecaps
)c( )i( .0>5q
)ii( 4 E d z ,,>ltnelaviuqe( I/ \I. d si redro suounitnoc j.
081 W.KURC
:JoorP noitacilpmI )b( .s= )a( si raelc dna )a( + )c( yb yralloroC 4.2 dna
noitisoporP .5.2
oT teg a)c( )b( ew lliw evorp taht eht evitcepser snoitidnoc )ii(-)i()b(
from noitisoporP 5.2 era .deifsitas ,tsriF ni weiv of noitisoporP 1.2 eht -noc
noitid )ii()b( .swollof ,txeN ni weiv of noitisoporP ,4.1 eht raludom &Z ). si
syawla ;MU-$ ,.e.i rof hcae is. O> ereht stsixe ),E(,S O> hcus taht
ia-ld)g--f(6Z revenehw ,,&,>)g(,Z ,O>gaf .l=)f(,Z oT wohs
eht MU of JZ ). ti seciffus ot evorp taht )g(+Z 2 is dna )f(,Z = 1 nac eb
decalper yb 11 llg 4( B 1s dna 11 f 11 b = ,1 .ylevitcepser
nI eht tsal esac noitisoporP 1.2 noitidnoc( ))b( nac eb .deilppa nI eht
tsrif eno noitisoporP 2.2 ,seilppa os taht rof yreve E > 0 ereht stsixe )E(Y > 0
hcus taht ,l/g/ > E seilpmi )g(,Z Z .)E(]Y ,ecneH htiw ,E = ,)s(q ew teg j&(6 =
))E(]Y(,~ hcus taht )s(86-l<)g-f&Z ,O>g>frevenehw ,l=61lflI dna
,/lgll .EB ,woN from noitisoporP 5.2 ti swollof taht )L&L si a MU ecaps
dna eht foorp si .dehsinif
MEROEHT .8.2 roF cimotanon-u, dna 4 < cc+ eht gniwollof era
.tnelaviuqe
)a( )u&E si na MTS .ecaps
)b( )p(,E si na MUL .ecaps
lc( .0>4
:foorP noitacilpmI )b( 3 )a( si raelc dna )c(*)a( swollof from
noitisoporP .5.1 oT evorp )b(*)c( ew tsrif evorp taht 4 O> seilpmi taht
(,Z ). si MUL .raludom nI weiv of noitisoporP ,4.1 (6Z ). si syawla MU&
dna ecneh .MUL-IQ morF ammeL 2.2 dna ammeL 3.2 ti swollof taht JZ )-
si yllautca MUL .raludom gniylppA niaga ammeL )a(2.2 ew teg taht )p(,E
si na MUL ecaps sa .derised
.krameR nI lareneg ew tonnac tcepxe taht )u(,E si a MU ecaps fi 4 > 0
.ylno nI tcaf from ,S[ .p 1357 ti si nwonk taht fi 4 $ ,d neht )u&E sniatnoc
na cihpromosi ypoc of .Oc ,ylraelC ni siht esac )u&E si a reporp laedi ni
.)&L morF ,21[ .pahC ,X .tceS 34.4 ti swollof won taht )U&E si ton a BK
ecaps ni( tcaf .)ecaps-B nO eht rehto ,dnah sa saw ydaerla ,denoitnem hcae
MU hcanaB ecittal si a BK .ecaps eroferehT )L&E tonnac eb a MU .ecaps
.3 SNOITACILPPA OT TSEB NOITAMIXORPPA
teL X eb a demron ,ecittal f~ ,X dna cK X a xevnoc .tesbus ,etoneD
=)f(KP EU( :K llu-flI ,Ehfni= (I f - h II .> llaceR taht JZ si .cimotanon
)A( ehT stluser woleb etacidni emos ygolana neewteb eht elor of tcirts
ytixevnoc ni eht yticinu melborp ni tseb noitamixorppa rof demron ,secaps
ZCILRO-KALEISUM SECAPS 181
dna eht MTS ytreporp ni hcus smelborp rof demron .secittal We lliw laed
htiw detanimod tseb ,noitamixorppa ,.e.i tseb noitamixorppa htiw tcepser
ot K rednu eht noitpmussa taht j-3 K .)S>K(
NOITISOPORP .1.3 ehT gniwollof stzremetats era .tnelaviuqe
)a( X si na MTS .ecaps
)b( roF lla f E X dna redro slavretni ,a[ b] c X gniyfsitas f 3 ,a[ ]ii
f( < ,a[ )]b ereht sdloh ))f(,6,,cP(draC d .1
jc( roF Na f E X dna lla secittalbus cK X hcus taht f b K <f( )K
ereht sdloh (draC (KP f )) < ,1 ,.e.i eht detanimod tseb noitamixorppa htiw
tcepser ot secittalbus ni X si .euqinu
foorP )a( * .)c( fI ton ereht tsixe u, w E ,K II of ,’N hcus taht //f--u// =
//f 11’I:;- ..,fni= II f - .l/h ecniS K si a ecittalbus u v 1~ E .K ecniS f 9 ,K
0 6 u-f v '~1 d f - u, os u v w E .)f(KP ecniS M # ,VI ew evah rehtie
u<u v ’tl ro u<v v .~1 nI eht tsrif esac j/f IIu-- = u-f/I v /jw = -!,:-fII
U( v -’ti ,jl)u a noitcidartnoc ot eht MTS ytreporp of .X ehT dnoces esac
snur ylsuogolana dna erofereht )c( .swollof
)b( 3 .)a( gnideecorP yb a ,noitcidartnoc ereht tsixe f> O>g hcus
taht O#g dna 11 f - llg = fI/ //. enifeD =K ,OC .]g nehT :0 og .)f(KP aI
weiv of )b( ew teg a .noitcidartnoc
)c( * .)b( sihT si ,raelc ecnis redro slavretni era .secittalbus
morF smeroehT 7.2 dna 8.2 ti swollof won
NOITISOPORP .2.3 ehT detanimod tseb noitamixorppa ni )&L )p(,E(
htiw 4 < -I- a ) htiw tcepser ot secittalbus si euqinu fi dna ylno ft # > 0 dna
,AESq .pser( .)O>$f
gninrecnoC eht ecnetsixe of tseb noitamixorppa tel su tniop tuo taht ni
)A&L htiw 4 E ,,A ro )p(,E htiw 4 < ,cz‘+ ew evah )f(lb,aCP # .0 .deednI
rednu eht snoitpmussa )p(,L .pser( ))p(+E sah redro suounitnoc .mron
,suhT yltnelaviuqe ,21[ .pahC ,X .tceS ,14 lla redro slavretm era ylkaew
.tcapmoc woN ti seciffus ot eton taht eht mron si ylkaew .csl nI tcaf ew
evah a elttil .erom
NOITISOPORP .3.3 roF eht zcilrO-kaleisuM ecaps )D&L eht gniwollof
stnemetats era .tnelaviuqe
)a( .,f-d
)b( ehT detanimod tseb noitamixorppa htiw tcepser ot desolc secittalbus
syawla sah a .noitulos
,revoeroM eht noitidnoc )b( llits sdloh eurt rof )p(,E revenehw 4 < CZ,+
281 CRUK.W
foorP )a( 3 .)b( teL f > K dna ~,h K eb a gniziminim :ecneuqes
=d ,.,fni +,,mil=,llh-flI m+ 11 f - jl,zI .d ecniS K si a ecittalbus ,,u =
=;V I ,h E .K revoeroM 0 <f - ~,u. 9 f - ,,zl seilpmi taht ,U si gniziminim
.ecneuqes ecniS ereht stsixe u = ,V ,,u 6 f ew evah 0 d u - gniylppA
.01,42
taht ,dud si tnelaviuqe ot eht redro ytiunitnoc of yna ecittal ,mron
noitisoporP ,1.2 ew edulcnoc taht - ,,u segrevnoc ni mron ot .orez ecniS
24
K si mron desolc u E .K oslA d = 11 f - u 4II = ,mil 11 f - ,u II .$ sihT sdleiy taht
PK(flZ125.
)b( 3 .)a( tI seciffus ot ylppa noitisoporP 1.2 dna eht emehcs of
foorp of eht noitacilpmi *)ii( )iii( ni meroehT 01 ni .]02[ ,ylemaN
gnimussa taht 4 $ ,ZA ,.e.i 11 dII. si ton redro ,suounitnoc ereht stsixe a
ecneuqes f,l hcus taht 0 d f, OJ dna ,fni 11 ,f II ( > .0 gnicalpeR fi yrassecen f,,
yb 1( + )~/l f, ew niatbo II f,ll d > /I ,A ,+ .dII ,nehT rof eht ecittalbus
=K ,},,f{ ~0 ,K =)0(KP .@ nO eht rehto dnah K si )mron( .desolc
,deednI tel rof a tnemom II f,,- gll, g+,,fnehT.K$gdnaK~,,frofO+
ni erusaem dna yltneuqesnoc g = .0 suhT II f,J ( + 0 hcihw si .elbissopmi
,eroferehT ni weiv of eht ruo ,snoitpmussa )O(kP # .@ sihT noitcidartnoc
sehsinif eht .foorp
gnitcelloC snoitisoporP ,2.3 ,3.3 dna smeroehT ,7.2 8.2 ew niatbo
MEROEHT .4.3 ehT gniwollof era tnelaviuqe rof zcilrO-kaleisuM secaps
htiw eht grubmexuL .mron
)a( ehT detanimod tseb noitamixorppa ni )u(,L ni( )u(,E htiw
4 C- )CC-I- htiw tcepser ot desolc secittalbus stsixe dna si .euqinu
)b( )u(,L .pser( )L&E htiw 4 < )j(c+ si na 44TS .ecaps
)c( 4 > 0 dna 4 E ,A .pser( 4 > .)0
nI tcaf a regnorts tluser naht noitisoporP 3.3 nac eb detats sa a yralloroc
from meroehT 6 ni .]3[
MEROEHT .5.3 roF zcilrO-kaleisuM secaps eht gniwollof stnemetats era
.tnelaviuqe
)b( roF lla desolc raenil secittalbus cK ,)L&L #)f(KP 0 rof lla
f E I&L
foorP )b( * .)a( sihT swollof ni weiv of noitisoporP .3.3 ehT -acilpmi
noit )a( * )b( swollof from meroehT 6 ni ,]3[ hcihw setats taht rof ylkaew
yllaitneuqes etelpmoc hcanaB secittal hcae desolc raenil ecittalbus si
.lanimixorp llaceR [ ,21 .pahC ,X .tceS 14.4 taht ylkaew yllaitneuqes etelpmoc
hcanaB secittal edicnioc htiw BK secittal noitceS( .)1 nI weiv of -isoporP