aSpaceaSpaceaSpaceANDSpaceandSpaceandSpaceandSpaceandSpaceandSpaceandSpaceandcommaSpaceAperiodSpaceApplySpaceapplySpaceassumptionbeSpacebeSpacebeSpacebetween

JraseSpaceconsistingSpacecontainedSpacecontrary

definedSpacediagonal

equalSpaceequalitySpaceeverySpaceeverySpaceexistenceSpaceexistenceSpaceexists

familySpacefollowingSpaceForSpaceForSpaceforSpaceforSpacefor

haveSpacehave

IfSpaceIfSpaceIfSpaceimpliesSpaceinSpaceincludingSpaceisSpaceIt

JENKINSSpaceJperiodSpacejustifyKperiod

LemmaSpaceLemmaSpaceLetSpaceletSpacelimitSpacelimitSpacelimitscolonSpacelimitsperiod

moduleSpacemuch

epsilondoubleprime(b)<xiprime(b),Spacexiprime(b)>Rezetan+etaperiod

bbalpha0bprimegammaJmujn,Spacexi2gammajcdotminusinftyn=1epsilon>0,Spacegammajminus1gamma(b)gamma(c)Rjprime,Spaceb<bprime,Spacexi2>xi*alpha0=minusinftycommax0>minusinfty,Spacealpha0>minusinftyperiodSpacealpha0>minusinftyperiodSpacegamma(bprime)periodSpacej=1,Spaceldots,Spacencomma

V(c*),Spacegamma1,Spaceldots,Spacegamman,Spacegamman+1=gamma(b)

iotar(a0, bprime)minuspiminus1xidoubleprime(bprime)leqmu(a0, b)minuspiminus1xiprime(b),Spacemu(b, bprime)leqoverlinemu(b, bprime)leqpiminus1(xiprime(bprime)minusxidoubleprime(b))+2

u(a0, b)minusfracpi1xiprime(b)leqmu(a0, c*)minusfracpi1xin(c*)+2minusnf(frac3eta)comma

limrightarrowinfty(mu(a0, b)minusfracpi1xiprime(b))=limbrightarrowinfty(mu(a0, b)minusfracpi1xikappa(b))period

varlimsuprightarrowinfty(mu(a0, b)minusfracpi1xin(b))=varlimsupbrightarrowinfty(mu(a0, b)minusfracpi1xiprime(b))

u(a0, bprime)minusmu(a0, b)=mu(b, bprime)leqoverlinemu(b, bprime)leqpiminus1(xidoubleprime(bprime)minusxiprime(b))period

varliminfrightarrowinfty(mu(a0, b)minusfracpi1xin(b))=varliminfbrightarrowinfty(mu(a0, b)minusfracpi1xiprime(b))comma

varlimsuprightarrowinfty(mu(a0, b)minusfracpi1xidoubleprime(b))leqvarliminfbrightarrowinfty(mu(a0, b)minusfracpi1xiprime(b))period

varlimsuprightarrowinfty(mu(a0, b)minusfracpi1xiprime(b)minus2)leqvarliminfbrightarrowinfty(mu(a0, b)minusfracpi1xim(b))period

overlineu(c*, b)=sumj=1n+1mujleqfracpi1(xiprime(b)minusxidoubleprime(c*))+2minussumj=1nf(xijdoubleprimeminusxijprime)comma

now

obtainSpaceobtainSpaceofSpaceofSpaceofSpaceofSpaceofSpaceofSpaceofSpaceOIlIAWA

16singleendquartationperiodSpace17periodSpace18periodSpace3commaSpace3commaSpace48SpaceleftparilrightparSpaceleftpar22rightparSpaceleftpar22rightparSpaceleftpar24rightparcommaSpaceleftpar27rightparSpaceleftpar27rightparSpaceleftpar27rightpar

ProofSpaceprove

rectangle

3atisfySpaceshowSpaceshowSpacesoSpace^soSpacesuchSpacesufficesSpacesufficient

takeSpacethatSpacethatSpacethatSpacethatcommaSpaceTheSpacetheSpacetheSpacetheSpacetheSpacetheSpacetheSpacetheSpacetheSpacetheSpacetheSpacetheSpacetheSpacethererhereforeSpacethereforeSpacethisSpacethoseSpacetoSpacetoSpacetoSpacetoSpacetoSpacetoSpacetoSpacetrivially

taualidity

WeSpaceWeSpaceweSpace^weSpacewhichSpacewhichSpacewillSpacewithSpacewith