PUB IRMA LILLE Vol No VII

Antoine Ayache

Description

Niveau: Supérieur, Master, Bac+5
PUB. IRMA, LILLE 2005 Vol. 64, No VII Hölderian random functions? Antoine Ayache Philippe Heinrich Laurence Marsalle Charles Suquet Laboratoire P. Painlevé, CNRS UMR 8524, Bât. M2, Université Lille 1, Cité Scientifique 59655 Villeneuve d'Ascq Cedex, France. Abstract Hölder regularity which plays a key rôle in fractal geometry raises an increasing interest in probability and statistics. In this paper we discuss various aspects of local and global regularity for stochastic processes and random fields. As a main result we show the invariability of the pointwise Hölder exponent of a continuous and nowhere differentiable random field which has stationary increments and satisfies a zero-one law. We also survey some recent uses of Hölder spaces in limit theorems for stochastic processes and statistics. Résumé La régularité hölderienne qui joue un rôle clé en géométrie fractale suscite un intérêt grandissant en porbabilités et statistique. Dans cette contribution nous discutons divers aspects de la régularité hölderienne lo- cale et globale pour les processus stochastiques et les champs aléatoires. Notre résultat principal est la constance temporelle et déterministe de l'exposant de Hölder ponctuel d'un champ aléatoire continu et nulle part différentiable, à accroissements indépendants et vérifiant une loi du zéro- un. Nous donnons aussi un panorama de quelques utilisations récentes des espaces de Hölder dans les théorèmes limites pour les processus stochas- tiques et en statistique.

• when ?n


• hölder spaces


• self-similarity param- eters


• bounded function


• tional central


• banach space


• hölder exponent


• continuous functions


• limit theorems

Sujets

• Suquet
• Ayache
• Bounded function
• Banach space

Informations

• Publié par: rasum
• Nombre de lectures: 97

Extrait

PUB.IRM o A,LILLE2005 Vol.64,NVII Hölderianrandomfunctions ∗ Antoine Ayache Philippe Heinrich Laurence Marsalle Charles Suquet LaboratoireP. Painlevé ,CNRSUMR8524, Bât.M2,UniversitéLille1,CitéScientifique 59655Villeneuved’AscqCedex,France. [e-mail] Abstract Hölderregularitywhichplaysakeyrôleinfractalgeometryraisesan increasinginterestinprobabilityandstatistics.Inthispaperwediscuss variousaspectsoflocalandglobalregularityforstochasticprocessesand randomfields.Asamainresultweshowtheinvariabilityofthepointwise Hölderexponentofacontinuousandnowheredifferentiablerandomfield whichhasstationaryincrementsandsatisfiesazero-onelaw.Wealso surveysomerecentusesofHölderspacesinlimittheoremsforstochastic processesandstatistics. Résumé Larégularitéhölderiennequijoueunrôlecléengéométriefractale susciteunintérêtgrandissantenporbabilitésetstatistique.Danscette contributionnousdiscutonsdiversaspectsdelarégularitéhölderiennelo- caleetglobalepourlesprocessusstochastiquesetleschampsaléatoires. Notrerésultatprincipalestlaconstancetemporelleetdéterministede l’exposantdeHölderponctueld’unchampaléatoirecontinuetnullepart différentiable,àaccroissementsindépendantsetvérifiantuneloiduzéro- un.Nousdonnonsaussiunpanoramadequelquesutilisationsrécentesdes espacesdeHölderdanslesthéorèmeslimitespourlesprocessusstochas- tiquesetenstatistique. Keywords:changepointdetection,empiricalprocess,epidemicmodel,func- tionalcentrallimittheorem,Hölderspace,localHölderexponent,multifrac- tionalBrownianmotion,pointwiseHölderexponent,self-normalization,zero- onelaw. MathematicsSubjectClassifications(2000):Primary60G17,Secondary 60-02,60B12,60F17,60F20,62G10. ∗ Preprint

3–IIV 1Introduction TheconceptofHölderregularityisquiteimportantinfractalgeometry,signal andimageprocessing,finance,statisticsandtelecommunications[8].Hölder exponentshavebeenusedfrequentlytomeasuretheroughnessofacurveor ofasurface[10];applicationsinsignalandimageprocessingarenumerousand includeinterpolation,segmentation[26]anddenoising[27].Theyareclosely relatedtootherfractalindicessuchasfractaldimensions,self-similarityparam- etersandmultifractalspectra(seee.g.[12,18,44,45]).Ontheotherhand,the Hölderspacesprovideafunctionalframeworkforlimittheoremsinthetheoryof stochasticprocesses.TheuseofHöldertopologiesleadstomorepreciseresults thantheclassicalframeworkofcontinuousfunctionsspaces. ThispaperdiscussbothusesofHölderregularityinthestudyofstochastic processes. 1.1Hölderexponents Whenlookingforrandomfieldsmodelingsomeroughness,itisquitenatural toinvestigatethepointwiseHölderregularityofvariousextensionsofthewell knownBrownianmotion. Recallthat { B H (t),t ∈ R d } ,thefractionalBrownianmotion(fBm)ofHurst parameterH ∈ (0,1)isthereal-valued,self-similarandstationaryincrements continuousGaussianfielddefinedforeveryt ∈ R d astheWienerintegral Ze it.ξ − 1 B H (t)= d | ξ | H+d/2 dW c (ξ),(1) R wheredW c isacomplex-valuedwhitenoise.Thisfieldwasfirstintroducedby Kolmogorov[20]forgeneratingGaussian“spirals”inaHilbertspace.Later, theseminalarticleofMandelbrotandVanNess[30]emphasizeditsrelevance forthemodelingofnaturalphenomena(hydrology,finance,...)andthusgreatly contributedtomakeitpopular.Sincethen,thisfieldturnsouttobeavery powerfultoolinmodeling.ThemonographofDoukhan,OppenheimandTaqqu [11]offersasystematictreatmentoffBm,aswellasanoverviewofdifferent areasofapplications. Thefield { B H (t),t ∈ R d } isanaturalgeneralizationoftheWienerprocess ( { B 1/2 (t),t ∈ R } isaWienerprocess)andsharesmanynicepropertieswithit. However,oneofthemainadvantagesoffBmwithrespecttoWienerprocess isthatitsincrementsarecorrelatedandcanevendisplaylongrangedepen- dence.Still,fBmisnotalwaysarealisticmodel.Indeed,itspointwiseHölder exponentremainsconstantallalongitstrajectorywhichcanbeaseriousdraw- backinseveralapplications(seeforexample[28,2,3,4]).Generallyspeaking, amultifractionalfieldisafieldwithcontinuoustrajectoriesthatextendsfBm andwhosepointwiseHölderexponentisallowedtochangefromonepointto another.Recallthat { α X (t),t ∈ T } thepointwiseHölderexponentofacontin- uousandnowheredifferentiablefield { X(t),t ∈ T } isdefinedforeveryt ∈ T

IV4–I sa on α X (t)=supα;limsup | X(t+h) − X(t) | =0.(2) h → 0 | h | α AparadigmaticexampleofamultifractionalfieldismultifractionalBrownian motion(mBm).Itwasintroducedindependentlyin[28]andin[7]butthe denominationmultifractionalBrownianmotionisduetoLévyVéhel.MBm isobtainedbysubstitutingtotheHurstparameterintheharmonizablerep- resentation(1)offBmacontinuousfunctiont 7→ H(t)withvaluesin(0,1). WhenthefunctionH( ∙ )issmoothenough(typicallywhenitisaC 1 function), thepointwiseHölderexponentofmBmsatisfiesforanyt ∈ T,almostsurely α mBm (t)=H(t),whichmeansthatitcanchangefromonepointtoanother. In[5]ithasbeenprovedthatwhentheincrementsofanyorderofmBmare stationarythefunctionH( ∙ )isconstant(i.e.mBmreducestoanfBm).More- over,noexampleofamultifractionalfieldwithstationaryincrementshasbeen constructedyet.Thisiswhyitseemsnaturaltowonderwhetherthereexistsa continuous,nowheredifferentiableandstationaryincrementsfield { X(t),t ∈ T } whosepointwiseHölderexponentchangesfromonepointtoanother.InSec- tion2weshowthattheanswerisnegativewhenweimposeinadditionto { X(t),t ∈ T } tosatisfyazero-onelaw. 1.2Hölderspacesasafunctionalframework Inmanysituationssomeuniformcontrolontheregularityisneeded.Forin- stanceletusconsiderthefollowingstatisticalproblem.Havingobservedtheran- domvariablesX 1 ,...,X n ,weneedtotestthenullhypothesisthatX 1 ,...,X n havethesameexpectationµ 0 againstthealternativeofachangefromµ 0 toµ 1 betweentheunknowninstantsk ∗ andm ∗ withgoingbacktoµ 0 afterm ∗ .This isknownastheepidemicmodel.Itisquitenatural,see[41]forastepbystep explanation,touseheretheweightedteststatistics   S(j) − S(i) − S(n)(t j − t i )  UI(n,a):= 1 ≤ i m < a j x ≤ n | t − t | a ij P whereS(n):= 1 ≤ i ≤ n X i ,t i :=i/nand0<a<1/2.Theasymptoticdistri- butionofUI(n,a)followsfromtheweakconvergenceofapartialsumsprocess ξ n insomeHölderspace H a (aprecisedefinitionofHölderspacesisgivenin Section3).Thepracticalinterestoftheexponentahereliesinthesensitivityof thetest.Detectingtheshortestepidemicsrequirestotakethebiggestpossible aandthisleadstoinvestigateweakconvergenceofξ n in H a . InSection4wesurveysomerecentadvancesintheasymptotictheoryof sequencesofstochasticprocessesconsideredasrandomelementsinsomeHölder space H .Theissuesaddressedmaybeclassifiedalongthefollowingtwomain directions. A)Classicallimittheoremsfornormalizedsumsb n − 1 S n ofindependentran- domelementsX i in H :lawsoflargenumber,centrallimittheorems,see e.g.[31],[32],[35],[40].

–IIV5 B)Weakconvergencein H ofsequencesofrandomelementsξ n oftheform t 7−→ ξ n (t)=G n (X 1 ,...,X n ,t), whereX 1 ,...,X n isusuallyasampleofi.i.d.randomvariablesorrandom elementsinsomeBanachspaceandG n afunctionsmoothenoughto ensurethemembershipin H ofξ n . ProblemA)isdirectlyconnectedtotheProbabilityTheoryinBanachSpaces. Itiswellknowninthisareathatthelimittheoremsforasequenceofrandom elementsb n − 1 S n insomeseparableBanachspace B involvethegeometryof B . Forinstance,iftheX i ’sarei.i.d.withnullexpectation,thenthesquareinte- grabilityof k X 1 k givestheasymptoticnormalityofn − 1/2 S n when B isoftype 2(e.g. B isaHilbertspaceoraL p spacewith2 ≤ p< ∞ ).Butif B =c 0 ,the classicalspaceofsequencesconvergingto0,wecanfindaboundedrandomele- mentX 1 inc 0 whichdoesnotsatisfytheCLT,i.e.thecorrespondingsequence n − 1/2 S n isnotasymptoticallyGaussian.Thismakeshopelesscharacterizing theCLTinageneral B intermsofintegrabilitypropertiesofX 1 only.Because alltheHölderspaces H underconsiderationherecontainasubspaceisomorphic toc 0 andareconcretefunctionspaces,theyprovideaninterestingframework tostudytheasymptoticbehaviorofb n − 1 S n inacontextwherethegeometryof theBanachspaceis“bad”. ProblemB)ismoreorientedtostatisticalapplications.Indeed,theweak D convergenceξ n −→ ξinsomefunctionspaceEmeans Eg(ξ n ) −−−→ Eg(ξ),(3) ∞→ n foreverycontinuousandboundedfunctiong:E → R .Bythecontinuous mappingtheorem,thisimpliesthatforeveryfunctionalf:E → R ,continuous withrespecttothestrongtopologyofE, f(ξ n ) n − → −− ∞ → f(ξ),indistribution.(4) Theclassicalfunctionalframeworksforsuchconvergenceξ n − D → ξaretheSko- rohodspacewhenξ n hasjumpsandsomespace C ofcontinuousfunctionswhen ξ n hascontinuouspaths.Theinterestinreplacing,wheneverpossible, C by H isthatthisstrenghteningofthetopologyonthepathsspaceenlargesthe setofcontinuousfunctionalsf.Usually,therandomfunctionsξ n sharemore smoothnessthantheirweaklimitξ.ForinstanceintheHölderianversionof theinvarianceprincipleforpartialsumsprocesses,thepathsofξ n arerandom polygonallines,whileξisaBrownianmotion.Insuchcasestheglobalsmooth- nessofξputanaturalboundinthechoiceofthe“best”space H .Theexample oftheBrownianmotionWshowsherethattheclassicalladderofHölderspaces H a isnotrichenough.Indeed H a isthespaceoffunctionsxwhoseincrements x(t+h) − x(t),h ≥ 0areO(h a )uniformlyint.DuetoLévy’sresultonW’s modulusofuniformcontinuity[25,Th.52,2],itseemsdesirabletoconsideralso

6–IIV
� thespacesoffunctionsxsuchthatx(t+h) − x(t)=Oh 1/2 ln b (1/h).Inmore generality,thisleadstointroducealadderofHölderspaces H ρ ,wheremember- shipofxin H ρ isequivalenttotheuniformestimatex(t+h) − x(t)=O(ρ(h)), forsomeweightfunctionρ. ThisraisesathirdproblemwhichinsomesenseisalsopreliminarytoProb- lemA): C)GivenastochasticprocessX= { X(t),t ∈ T } ,findconditionsintermsof itsfinitedimensionaldistributionssothatXadmitsaversionwithpaths intheHölderspace H ρ . 2CriticalExponents 2.1Zero-OnelawsandExponents LetX= { X(t),t ∈ T } bearealprocesswith,say,separableandmetrictime setT.WecanviewXasarandomelementin R T endowedwithproductσ-field B ( R ) ⊗ T where B ( R )denotestheBorelσ-fieldof R .Thekindofzero-onelaw weshallfocusoncanbestatedasfollows: Definition1.WewillsaythatXsatisfiesazero-onelawifforeachmeasurable linearsubspaceVof R T , P(X ∈ V)=0or1.(5) ItisknownthatGaussian,stableandsomeinfinitelydivisible(without Gaussiancomponent)processessatisfy(5).Theirassociatedfinite-orderchaos processesdoaswell.Wereferthereadertothepaper[43]byRosinskiand Samorodnitskyandthereferencestherein.Oneclassicalapplicationofsucha zero-onelawistoestablishregularitypropertiesofpaths.Forinstance,neglect- ingmeasurabilityquestionsatfirstglance,Vcouldbe: 1.thespaceofboundedfunctionsonT, 2.thespaceofcontinuousfunctionsonT,ifTiscompact, 3.thespaceofuniformlycontinuousfunctionsonT, 4.thespaceofLipschitzfunctionsonT, 5.thespaceofa-HölderianfunctionsonT, 6.thespaceofabsolutelycontinuousfunctionsonT,ifTisanintervalin R . IfforsomecountabledensesubsetSofT,wehave
� P ∀ t ∈ T, ∃ (s n ) n ≥ 1 ⊂ S,s n → t,X(s n ) → X(t)=1, thenthemeasurabilityof { X ∈ V } ,fortheV’sdisplayedabove,canbeestab- lishedprovidedtheunderlyingprobabilityspace(Ω, F ,P)iscomplete(ascan

V7–II bealwaysassumed).Indeed,theevents { X ∈ V } maythenbeexpressed,upto negligeablesets,asonesinvolvingonlytherestrictionofXtoS.Thefollowing resultillustrateshowusefulthiszero-onelawcanbe. Theorem1.AssumethatTisanopensubsetof R d (d ∈ N ).LetX= { X(t),t ∈ T } beacontinuous,nowheredifferentiableprocesssatisfyingthe zero-onelaw(5).Then,forallt ∈ T,thepointwiseHölderexponentofXat tisalmostsurelydeterministic.Inotherwords,forallt ∈ T,thereexistsa numberH(t) ∈ [0,1]suchthat
� Pα X (t)=H(t)=1. Proof.ThisresulthasalreadybeenestablishedbyAyacheandTaqquforGaus- sianprocesses,see[6].Theirproofisbasedonthesamekey-argument(zero-one law),buttheonewegivehereusesitmoredirectlyandexplicitely. WesetS= Q d .LettbesomearbitrarypointoftheopensetTandchoose η>0suchthattheballB(t,η)beinT.SinceXhasalmostallcontinuousand nowheredifferentiablepathsonT,weknowthatα X (t,ω)belongsto[0,1]for almostallω ∈ Ω.Wecanthusdefine  u ∗ (t):=supu ∈ R ;P(α X (t) ≤ u)=0,  u ∗ (t):=infu ∈ R ;P(α X (t) ≤ u)=1. Bydefinition,theinterval[u ∗ (t),u ∗ (t)]isthesupportofthedistributionfunction ofα X (t).Toprovethatα X (t)isalmostsurelydeterministic,weonlyneedto checkthatu ∗ (t) ≤ u ∗ (t).Letu<u ∗ (t),wethushaveP(α X (t)>u)>0.On theevent { α X (t)>u } ,wehavelimsup h→0 | h | −u | X(t+h) − X(t) | =0which impliesthath 7→ h − u (X(t+h) − X(t))isboundedonthecountablebounded subset { h ∈ S; | h | <η } .Itfollows,byinclusion,that  0<P � α X (t)>u
≤ P   sup | X(t+h) − X(t) | < ∞   . | h | <η | h |u S ∈ h Weshallprovethatthislastprobabilityisequalto1,usingthezero-onelaw (5).Tothisend,notethattheevent    | X(t+h) − X(t) |   ∞ <pus   | h | <η | h |u   S ∈ h canclearlybewritten { X ∈ V } forsomelinearsubspaceV,whichis B ( R ) ⊗ T - measurablesinceitinvolvesonlycountablemanyprojectionsx 7→ x(t)from R T to R .Thezero-onelawensuresconsequentlythat !  Psup | X(t+h) − X(t) | < ∞ =1, | h | <η | h |u

IIV8– whereweskippedtherestrictionh ∈ Sinthesupremumthankstothecontinuity ofX.Butnow,asimpleinclusionofeventsyields  Plimsup | X(t+h) − X(t) | < ∞ =1, h→0 | h | u whichcanbereadasP(u ≤ α X (t))=1or,equivalently,asP(α X (t)<u)=0. Thismeansthatu ≤ u ∗ (t)whichgivesu ∗ (t) ≤ u ∗ (t),sinceuisarbitraryin ( −∞ ,u ∗ (t)).Besides,bydefinitionu ∗ (t) ≤ u ∗ (t),whenceu ∗ (t)=u ∗ (t).We callH(t)thiscommonvalue.Wejusthaveprovedthatthedistributionfunction ofα X (t)jumpsfrom0to1atH(t)inotherwordsP(α X (t)=H(t))=1. Remark1.Otherexponentsmaybedefinedtocharacterizetheregularityof afunction,forinstancethelocalHölderexponentattimet )( α˜ X (t)=supα; ∃ η>0sup | X(u) − X(v) | < ∞ , α u,v ∈ B(t,η) | u − v | whereB(t,η)denotestheopenballcenteredattandofradiusη,andtheglobal HölderexponentonacompactsetK ⊂ T  | X(u) − X(v) |  β X =supβ; u, s v u ∈ p K | u − v |β < ∞ . WhenXisacontinuousnowheredifferentiableprocess,satisfyingazero-one law,thesamepropertyasinTheorem1holds.Moreprecisely,forallcompact subsetKofT,β X isalmostsurelydeterministic,andforallt ∈ T,α˜ X (t) isalmostsurelydeterministic.Theproofofbothresultsisthesameasfor Theorem1,theonlychangeconcernsthemeasurablesubspaceof R T involved inthezero-onelaw.Inthecaseofα˜ X (t),weuse )( V ˜ := [ x ∈ R T ;sup | x(u) − x(v) | < ∞ , n ∈ N ∗ u,v ∈ B(t,1/n) ∩ S | u − v | α whereSisacountabledensesubsetofT,andinthecaseofβ X ,wedefine  W:=x ∈ R T ;sup | x(u) − x(v) | < ∞ . u,v ∈ K ∩ S | u − v | α 2.2ProcesseswithStationaryIncrements Throughoutallthisparagraph,T,thesetoftimes,canbetakentobeequalto anynon-emptyopensubsetof R d (d ∈ N ). LetX= { X(t),t ∈ T } denoteacontinuousnowheredifferentiableprocess, forwhichazero-onelawholds(see(5)).ThankstoSubsection2.1,weknow thatthepointwiseHölderexponentofXattisdeterministic,butdependson t.Now,weassumebesidesthatXhasstationaryincrements.Thismeansthat

9–IIV (X(s 2 ) − X(s 1 ),...,X(s n ) − X(s 1 ))and(X(s 2 +t) − X(s 1 +t),...,X(s n +t) − X(s 1 +t))areidenticallydistributedforanys 1 ,...,s n ,s 1 +t,...,s n +t ∈ T andanyintegern ≥ 2.SincethepointwiseHölderexponentisdefinedbymeans ofincrements,wecanshowthatitdoesn’tdependanymoreont. Theorem2.LetX= { X(t),t ∈ T } beacontinuousnowheredifferentiable process,withstationaryincrements.Weassumethatazero-onelawholdsfor X.ThenthereexistsH ∈ [0,1]suchthatforallt ∈ T P(α X (t)=H)=1. Proof.Theschemeoftheproofisthefollowing:sinceTisanon-emptyset,it containsatleastoneelement,thatwewilldenote0forthesakeofsimplicity. UsinganequivalentdefinitionofthepointwiseHölderexponent,weprovethat α X (t)andα X (0)havethesamelaw,forallt ∈ T.ThankstoTheorem1,we knowthatthereexistsH=H(0) ∈ [0,1]suchthatthelawofα X (0)isaDirac massatpointH.Consequently,forallt ∈ T,thelawofα X (t)isaDiracmass atpointH. AsintheproofofTheorem1,SdenotesacountabledensesubsetofT.Lett beafixedpointofT.ItcanbeeasilyshownthatthepointwiseHölderexponent ofXattisgivenby: α X (t)=liminf log | X(t+h) − X(t) | , h → 0 log | h | withtheusualconventionthatlog0= −∞ .Thisdefinitionreadsas α X (t)=supinf log | X(t+h) − X(t) | R>0 |h|<R log | h | log | X(t+h) − X(t) | = n su ∈ N p | h | i < nf 1/n log | h | , S ∈ h thelastequalitycomingfromthemonotonicityoftheinfimumwithrespect toRandfromthecontinuityofthepathsofX.Toobtaintheidentityinlaw betweenα X (t)andα X (0),weintroduceanincreasingsequence(S k ) k ≥ 1 offinite setssuchthat ∪ k ≥ 1 S k =S.Then,notethatforu ∈ R   [\\ { α X (t) ≤ u } = ↓↓↑ min log | X(t+h) − X(t) | <u+ 1 , n ∈ N m ∈ N k ∈ N  | h h | ∈ < S 1/n log | h | m  k sothat,foreveryt ∈ Tandu ∈ R ,bysequentialmonotoniccontinuityofP,  P(α X (t) ≤ u)=limlimlimP  min log | X(t+h) − X(t) | <u+ 1  . nm k| h | <1/n log | h | m S ∈ h k Theeventmentionedinthelastprobabilityinvolvesafinitenumberofincre- mentsofX,allofthembasedonpointt.Thestationarityoftheincrements

VII–10 impliesthatwecanreplacethemwiththeanalogueincrements,basedonpoint 0.Itfollowsthatforallu ∈ R P(α X (0) ≤ u)=P(α X (t) ≤ u), whichmeansthatα X (t)andα X (0)havethesamelaw,forallt ∈ T.Wealready knowthatthepointwiseHölderexponentofXatpoint0isdeterministic,so thatthereexistsH ∈ [0,1]suchthatP(α X (0)=H)=1.Theequalityinlaw betweenα X (t)andα X (0)thusleadstoP(α X (t)=H)=1,forallt ∈ T. 3Hölderspaces LetusintroducetheHölderspacesbyaninformaldescriptionofthemostfa- miliarcase.Forfixed0<a<1, H a isthesetoffunctionsx:[0,1] → R such that | x(t) − x(s) |≤ K | t − s | a forsomeconstantKdependingonlyonxand a.ThebestconstantKinthisuniformestimatedefinesasemi-normonthe vectorspace H a .Byadding | x(0) | tothissemi-normweobtainanorm k x k a whichmakes H a anonseparableBanachspace.Clearlyif0<a<b<1, H b istopologicallyembeddedin H a andalltheseHölderspacesaretopologically embeddedintheclassicalBanachspace C ofcontinuousfunctions[0,1] → R . Toremedythenonseparabilitydrawbackof H a ,oneintroducesitssubspace H a,o offunctionsxsuchthat | x(t) − x(s) | =o( | t − s | a )uniformly.Thissubspace isclosed(hencealsoaBanachspaceforthesamenorm k x k a )andseparable. Oneinterestingfeatureofthespaces H a,o istheexistenceofabasisof triangularfunctions,see[9].Itisconvenienttowritethisbasisasatriangular arrayoffunctions,indexedbythedyadicnumbers.LetusdenotebyD j theset ofdyadicnumbersin[0,1]oflevelj,i.e.  D 0 = { 0,1 } ,D j =(2l − 1)2 −j ;1 ≤ l ≤ 2 j−1 ,j ≥ 1. Writeforr ∈ D j ,j ≥ 0, r − :=r − 2 − j ,r + :=r+2 − j . Forr ∈ D j ,j ≥ 1,thetriangularFaber-SchauderfunctionsΛ r arecontinuous, piecewiseaffinewithsupport[r − ,r + ]andtakingthevalue1atr:   2 j (t − r − )ift ∈ (r − ,r]; Λ r (t)=  2 j (r + − t)ift ∈ (r,r + ]; 0else. Whenj=0,wejusttaketherestrictionto[0,1]intheaboveformula,so Λ 0 (t)=1 − t,Λ 1 (t)=t,t ∈ [0,1]. Thesequence { Λ r ;r ∈ D j ,j ≥ 0 } isaSchauderbasisof C .Eachx ∈C hasa uniqueexpansion ∞ XX x=λ r (x)Λ r ,(6) j=0r ∈ D j

VII–11 withuniformconvergenceon[0,1].TheSchauderscalarcoefficientsλ r (x)are givenby x(r + )+x(r − ) λ r (x)=x(r) − ,r ∈ D j ,j ≥ 1,(7) 2 andinthespecialcasej=0by λ 0 (x)=x(0),λ 1 (x)=x(1).(8) n P Thepartialsum j=0 intheseries(6)givesthelinearinterpolationofxbya polygonallinebetweenthedyadicpointsoflevelatmostn. Λ r (t) ✻ 1 ❈✄ ❈✄ ❈✄ ❈✄ ❈✄ ❈✄ ❈✄ ❈✄ ❈✄ ❈✄✲ 0r − rr + t ✲✛ j− 2 Figure1:TheFaber-SchaudertriangularfunctionΛ r Ciesielski[9]provedthat { Λ r ;r ∈ D j ,j ≥ 0 } isalsoaSchauderbasisofeach space H a,o (hencetheconvergence(6)holdsinthe H a topologywhenx ∈H a,o ) andthatthenorm k x k a isequivalenttothefollowingsequencenorm: k x k s a eq :=sup2 ja max | λ r (x) | . j≥0 r∈D j Thisequivalenceofnormsprovidesaveryconvenientdiscretizationprocedure todealwithHölderspacesandisextendedinRačkauskasandSuquet[34]to themoregeneralsettingofHölderspacesofBanachspacevaluedfunctionsx, withamodulusofcontinuitycontrolledbysomeweightfunctionρ. Let( B , kk )beaseparableBanachspace.Wewrite C B fortheBanach spaceofcontinuousfunctionsx:[0,1] → B endowedwiththesupremumnorm k x k ∞ :=sup {k x(t) k ;t ∈ [0,1] } .Letρbearealvaluednondecreasingfunction on[0,1],nullandrightcontinuousat0.Put ω(x,δ):=sup k x(t) − x(s) k . ρ s,t ∈ [0,1], ρ(t − s) 0<t−s<δ