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Measuring Functions Smoothness with Local Fractional
Derivatives
Kiran Kolwankar, Jacques Lévy Véhel
To cite this version:
Kiran Kolwankar, Jacques Lévy Véhel. Measuring Functions Smoothness with Local Fractional
Deriva-tives. Fractional Calculus and Applied Analysis, De Gruyter, 2001, 4 (3), pp.285-301. �inria-00578645�
Lo al Fra tionalDerivatives
K.M.KolwankarandJ.LevyVehel
ProjetFra tales,INRIARo quen ourt B.P.105,78153LeChesnayCedex,Fran e
and
Ir yn,B.P92101,44321Nantes, Fran e
email: Ja ques.LevyVehelinria.fr,Kiran.Kolwankarir yn.e -nantes.fr
Abstra t
We studyanotion of lo alfra tional di erentiation,obtained by lo- alizing the lassi al fra tional derivative. We show that it is strongly relatedwiththelo alHolderexponent,andgiveaninterpretationofthis resultintermsof2-mi rolo alanalysis.
1 Introdu tion
Measuringthelo alsmoothnessoffun tionsprovestobeanimportanttaskfor many appli ations in su h diverse elds as mathemati al analysis, signal and imagepro essingorgeophysi s. Dependingonthesituation,variousde nitions of lo al regularity havebeen proposed. The most often used is probably the onebased on Holderspa es in their various versions. Su h a hara terization is for instan e entral in multifra tal analysis, and is aninstrumental tool for image segmentationor denoising, and Internet traÆ hara terization. Other importantmeasuresoflo alregularityin lude(lo al)fra tionaldimensions(e.g. box, Hausdor or regularizationdimension), whi h have been used in various ontexts, su h as tribologyorimage lassi ation. In thispaper,weare inter-estedin omparingthe lassi alHolder hara terizations(andtheirre nements, see below), with yet another measure, based on the degreeof lo al fra tional di erentiability (LFD). This notion was introdu ed in [11℄ as an attempt to lo alizethe lassi alfra tionalderivative[18℄. In[11℄, itisfor instan eproved that Weierstrass fun tion W is lo ally fra tionally di erentiable at any point upto anorderwhi his pre iselythepointwiseHolderexponentof W. In this work,weprovidefurtherresultsinthisdire tion( orre tingalongthewaysome ina ura iesof[11℄). Inparti ular,weprovethat,forallfun tionsbelongingto alargefun tionalspa e,thedegreeofLFD oin ideswiththelo alHolder expo-nent. Furthermore,wegiveapre ise interpretationofthefra tionalderivative in termsof2-mi rolo alanalysis.
integrodi erentiationin lude[5,6,10,16,17℄,andwerefertheinterestedreader tothese papers.
The rest of this paper is organized as follows: In se tion 2, we re all the de nitionsofthelo alandpointwiseHolderexponentsandoftheLFD.Se tion 3provestheequalitybetweenthelo alHolderexponentandthedegreeofLFD. In se tion 4, we extend this result to a more pre ise one using 2-mi rolo al analysis. Finally, se tion5 ontainsexamplesonsimplefun tions,whi hallow to understandina on retewayhowLFDa tsonsignals.
2 Measures of lo al regularity
2.1 Pointwise Holder exponent
De nition1 Let be a positive real number whi h is not an integer, and x
0
2 R. A fun tion f : R ! R is in C x0
if there exists a polynomial P x
0 of degreeless than su hthat:
jf(x) P x0 (x)j jx x 0 j : (1)
When 2℄0;1[, this redu esto:
jf(x) f(x 0 )j jx x 0 j (2)
ThepointwiseHolderexponentoff atx 0 ,denoted p (x 0 ),isthesupremum ofthe -sforwhi h(1)holds. Extensiontohigherdimensionsisstraightforward, butwill notbe onsideredhere.
Assaidabove,thisregularity hara terizationiswidelyusedbe auseithas dire tinterpretationsbothmathemati allyandinappli ations. Ithasbeenfor instan e used for spee h synthesis [3℄ and image analysis [14℄. However, the pointwiseHolderexponenthasalsoanumberofdrawba ks,amajoronebeing that it isnotstable under thea tion of(pseudo) di erentialoperators. Thus, for instan e, knowing the pointwiseHolder exponent of a fun tion at a point x
0
isnotsuÆ ienttopredi ttheHolderexponentofitsderivativeat thesame point,andthesamehappensfortheWeylfra tionalderivative(seebelow).
2.2 Lo al Holder exponent Thelo al Holderexponent
l
measures slightlydi erentfeatures as ompared to
p
. Itis de ned as follows: Let 2℄0;1[,R. One lassi allysaysthat f 2C l ()if: 9C:8x;y2: jf(x) f(y)j jx yj C Letnow: l ( f;x 0 ;)=supf :f 2C l ( B(x 0 ;))g
l 0
thede nitionofthelo alHolderexponent:
De nition2 Let f bea ontinuousfun tion. The lo al Holder exponent of f atx
0
isthe real number:
l (f;x 0 )=lim !0 l (f;x 0 ;)
This exponent is stable under pseudo-di erentiationor integration. More-over, it is easier to estimate than the pointwise Holder exponent. Its main drawba kisthatitisnotaspre iseasthepointwiseone[7℄.
Animportantdi eren ebetween p
and l
iswellillustratedontheexample of the hirp f(x)= jxj
os(1=jxj
), f(0) =0, with ; > 0. Inthis ase, at x=0, p = while l = 1+ . Thus,while p
issensitiveonlytowhathappens \at"0,
l
measuresalsothelo alos illatorybehaviorofthesignal\around"0. Weshallneedthefollowing hara terizationofthelo alHolderexponentin termsofwavelet oeÆ ients:
Proposition1 Let j;k = 2 j=2 (2 j x k) be an orthonormal basis of L 2 (R) anddenote the dis rete wavelet oeÆ ientsof f by
j;k ,i.e. j;k =2 j Z f(x) (2 j x k)dx
Then, the lo al Holder exponent of f atx is
l = lim !0 (supfs=9C ; 8 j;k B(x;); j j;k jC2 sj g) (3)
Note that, while neither p
nor l
yield omplete hara terization, it is possible to ombine theni e properties of ea h exponent: this is the topi of 2-mi rolo alanalysis,whi h weshallusein se tion4.
2.3 Lo al fra tional derivative
In this se tion, we re all the de nition of LFD introdu ed in [11℄ and make pre isethenotionofdegreeofLFD.Westartbybrie yre allingthede nition ofthe lassi alfra tionalderivativeinthe asewheretheorderisbetween0and 1:
De nition3 [18 ℄ The (Riemann-Liouville)fra tional derivative of afun tion f of orderq (0<q<1)isde nedas:
D q x f(x 0 ) = D q x+ f(x 0 ); x 0 >x; D q x f(x 0 ); x 0 <x: = 1 (1 q) ( d dx 0 R x 0 x f(t)(x 0 t) q dt; x 0 >x; d dx 0 R x x 0 f(t)(t x 0 ) q dt; x 0 <x: (4)
solutely ontinuous ([18℄, page 35). When x = 1, i.e. the integration is performedonasemi-in nitedomain, the orrespondingderivativesD
q 1
f are alledtheWeylfra tionalderivatives.
In[18℄,thee e toffra tionalintegration onglobalHolderspa esis investi-gatedin fulldetail. Ouraimhereisto obtainresultsforfra tionalderivatives andlo al/pointwiseexponents.
Note for further use the following lassi al property of the Weyl deriva-tive([18℄,theorem 7.1):
Proposition2 Letf belongtoL 1
(R ). Then,providedf issuÆ ientlysmooth:
\ D q 1 f(!)=(i!) q b f(!) where b
f denotesthe Fouriertransformoff.
Thesametypeofpropertyholds forwavelet oeÆ ients:
Proposition3 [15℄Let j;k =2 j=2 (2 j x k)beanorthonormalbasisofL 2 (R) with intheS hwartz lass,anddenotethedis retewavelet oeÆ ientsoff by
j;k
. Then,the wavelet of oeÆ ientsof D q
1
f (inanother waveletbasis) are
d j;k =2 jq j;k
It is an easy onsequen e of this Proposition and Proposition 1 that the Weylderivativeoforderqde reasesthelo alHolderexponentbyexa tlyq. In ontrast,nosu h propertyholdsfor
p .
Themainmotivationforintrodu inglo alfra tionalderivativesistotryand remedyto twosometimesundesirable propertiesof fra tionalderivatives: Non lo alityandthebehaviourwithrespe tto onstants. Asforthe rstpoint,itis learfrom thede nition thatthe fra tionalderivativeofafun tion f depends onthevaluesoff onthewhole interval[x
0
;x℄. These ondfeature isalso well-known. Forinstan e, the fra tionalderivativeof order qfrom the rightof the fun tion f(x)=x p (x>0; p> 1)is: D q 0+ x p = (p+1) (p q+1) x p q (5)
Substituting p = 0 for a onstant fun tion in the above formula, one gets D
q 0+
1=1= (1 q) x q
, i.e. the fra tionalderivative ofa onstant is notzero in general. In parti ular,the fra tionalderivativeof afun tion hanges ifone adds a onstantto this fun tion. Thus, thefra tional derivativeof afun tion dependsonthe hoi eoftheorigin,whereastheusualnotionofdi erentiability is a lo al on ept independent of the origin. The aim of the lo al fra tional derivativeistomodifyinasimplewaytheusualfra tionalderivativetoobtain lo alityandtranslationinvarian e.
study the di erentiability of f. One rst subtra ts the valueof f at x. This washes out the e e t of a onstant term. Se ond, one introdu es a limit, as shownbelow,toobtainalo alquantity.
De nition4 Thelo alfra tionalderivativeoforderq(0<q<1)ofafun tion f 2C 0 :R!R is de nedas D q f(x)= lim x 0 !x D q x (f(x 0 ) f(x)) (6)
if the limitexistsinR[f1g.
Asanexample,takeagainf(x)=x p
(x>0; p> 1). One omputeseasily thatD
q
f(0)equals0ifq<p,1ifq>p,and (q+1)ifq=p. Althoughinthis ase,thelimitsallexistandthere isaqwherethelimitis nite andnonzero, thisisnotthegeneralsituation. Thus,asemphasizedin theremarkbelow,we arenotingeneralinterestedin thevalueofD
q
f, butinthe riti alq.
This on eptof lo alfra tional derivativehasbeenused to studythelo al fra tional di erentiability of nowhere di erentiable fun tions [11℄. Equations involvingthese lo al fra tional derivativeshavebeen studied [2, 12℄ and have foundtobeusefulin studyingphenomenainfra talspa eortime.
With this notionof LFD,it is naturalto de ne the riti al order of lo al fra tionaldi erentiability,ordegreeofLFD,asthelargestvalueforwhi h the LFD exists. ThenextPropositionshowsthat thisisawellde nednotion.
De nitionand Proposition1 ThedegreeofLFDofthe ontinuousfun tion f atx isde nedas:
q
(x)=supfq2[0;1℄:D q
f(x)existsatx andis nite g:
Proof:
LetE betheset:
E=fq2[0;1℄:D q
f(x)existsatx andis nite g:
AllweneedtoproveisthatE isnonempty,sothatq
(x)iswellde nedasthe supremumofasubset of[0;1℄. Note that D
0 x
f =f. Sin ewearedealingwith ontinuousfun tions,wegetthat02E. Thusq
(x)exists andisnonnegative. Remark nallythat,ifq>0belongstoE,then learlyallq
0
2[0;q℄alsobelong toE,asiseasilyseenfrom de nition4. ThusE isalwaysasegment.
Remark: Ingeneral,D q
f(x)willbezeroforq<q
andin niteforq>q
when the limit exists. Thus, q
may be understood as a ut-o value, mu h in the samewayasHolderexponentsorfra tionaldimensions. Atthe ut-o ,D
q f(x) mayornotbe nitenonzero.
Remark: Thede nition ofq
aneasily beextended tofun tions inL 2
,andto some lassesdistributions,asforinstan ehomogeneousones.
exponents
It is intuitively learthat thenotionsof degreeofLFD andHolder exponents mustberelatedinsomeway. Theaimofthisse tionistoprove,viaelementary means,that,foralarge lassoffun tions,q
indeed oin ideswith l
. Froman intuitivepointofview,thefa tthatitisthelo alexponentthat omesintoplay ratherthanthepointwiseonestemsfromthefa tthatLFDstartsby integrat-ing thefun tion aroundthe pointof interest, so that thebehaviorin awhole neighborhood is important. Thus, for instan e, if f has a strong os illatory behavioraround0,likethe hirp,this will have onsequen es onD
q 0
f through theintegrationin(4). Also,
l
behaveswellunderpseudo-di erentiation,while
p
doesnot.
Westartbyprovingasimplepropositionaboutlo alHolderexponents. Let g :!R bein C
(), >0. whereR is openand x 0 2. Forx 2 de ne g + (x)= g(x) g(x 0 ) x>x 0 0 xx 0 (7) and g (x)= g(x) g(x 0 ) x<x 0 0 xx 0 : (8) Let + = fx 2 : x x 0 g and = fx 2 : x x 0 g Also de ne g R : + ! R to be the restri tion of g on + and g L : ! R to be the restri tionof g on . Theproposition statesthat thelo al Holderexponent of g is exa tly the minimumof theexponents of g
+
and g . This will result from twobasi lemmas. The rstoneis lemma1.1 from[18℄. Inournotation, itreads: Lemma1 l (g;x 0 ;)minf l (g R ;x 0 ;); l (g L ;x 0 ;)g8s. t. B(x 0 ;).
Sin ethisistrueforall,itimpliesthat
l (g;x 0 )minf l (g R ;x 0 ); l (g L ;x 0 )g (9) Lemma2 l (g + ;x 0 )= l (g R ;x 0 )and l (g ;x 0 )= l (g L ;x 0 ).
Proof: We onsideronlythe rst ase,i.e., l (g + ;x 0 )= l (g R ;x 0 ). These ond followssimilarly. Theinequality
sup jg R (x) g R (y)j jx yj sup jg + (x) g + (y)j jx yj
holdsbe ausethesupremumontheleftistakenonasubintervalofthedomain for thesupremumonthe right. This implies
l (g + ;x 0 ) l (g R ;x 0 ). Forthe
0 0 g + (x)=0,g + (y)=g(y) g(x 0 )andjx yj>jx 0 yj. Asa onsequen e: jg + (x) g + (y)j jx yj jg(x 0 ) g(y)j jx 0 yj :
Thisinturnimplies l (g + ;x 0 ) l (g R ;x 0 ). Proposition4 l (g;x 0 )=minf l (g + ;x 0 ); l (g ;x 0 )g. Proof:
Frominequality9andlemma2itfollowsthat l (g;x 0 )minf l (g + ;x 0 ); l (g ;x 0 )g. Inordertoprovethe onverseinequality,weprove
l (g + ;x 0 ) l (g;x 0 ). sup B(x0;) jg + (x) g + (y)j jx yj = sup [x0;x0+) jg + (x) g + (y)j jx yj = sup [x0;x0+) jg(x) g(y)j jx yj sup B(x 0 ;) jg(x) g(y)j jx yj :
Thisimpliesthat l (g + ;x 0 ) l (g;x 0 ). Similarly, l (g ;x 0 ) l (g;x 0 ),hen e theresult.
Theorem1 Letf be a ontinuousfun tion inL 2 . Then q (f;x 0 )= l (f;x 0 ). Proof: De ningf as in(7),(8),wewrite D q x 0 (f(x) f(x 0 )) = D q x 0 (f + +f ) = D q x 0 + (f + (x)+f (x)); xx 0 ; D q x0 (f + (x)+f (x)); xx 0 : = 1 (1 q) d dx R x x0 f + (t)(x t) q dt; xx 0 ; d dx R x 0 x f (t)(t x) q dt; xx 0 ; = 1 (1 q) d dx R x 1 f + (t)(x t) q dt; xx 0 ; d dx R 1 x f (t)(t x) q dt; xx 0 ; D q 1 f + xx 0 ; D q +1 f xx 0 ;
Fromthede nitionsoff +
andf ,itis learthat D q 1 f + =0whenx<x 0 andD q +1 f =0whenx>x 0 . Therefore,ifwewriteg(x)=D q x0 (f(x) f(x 0 )), wehavethatg
,asgivenbyde nitions (7)and(8),arealsoequalto D q 1
f
. We havethus repla ed our derivativesby Weylones, for whi h weknowthat
Proposition4, l (g;x 0 ) = minf l (g + ;x 0 ); l (g ;x 0 )g = minf l (f + ;x 0 ) q; l (f ;x 0 ) qg = minf l (f + ;x 0 ); l (f ;x 0 )g q = l (f;x 0 ) q:
Thusthe lo alHolderexponentofthe fun tion x!D q x 0 (f(x) f(x 0 ))is non negativei q< l (f;x 0
). In onsequen ethe riti alvalueq
(x 0
)su hthatthe limitD
q f(x
0
)existsandis niteforq<q
(x 0
)butnotforq>q (x 0 )isexa tly l (f;x 0 ).
4 Fra tional Di erentiation and2Mi rolo al Anal-ysis
Inthisse tion,weprovideanewinterpretationoffra tionalderivativeinterms of 2-mi rolo alanalysis. Thiswill allowin parti ularto understand theresult of the previousse tion in amoretransparent way. We rst re all somebasi fa ts about2-mi rolo alanalysis.
4.1 2-mi rolo al spa es
Thede nitionof2-mi rolo alspa es[1℄isbasedonaLittlewoodPaleyanalysis. A Littlewood Paley analysis is aspatially lo alized lterbank. One may also understand itas anintermediatebetween adis reteand a ontinuous wavelet analysis. Morepre isely,letS(R ) betheS hwartzspa eandde ne:
'2S(R )= b ' ()=1;kk< 1 2 b ' ()=0;kk>1: and ' j (x)=2 j '(2 j x): Onehas b ' j ()=' (2b j ): Thef' j
gseta tsaslowpass lterbank,whi hleadsnaturallytotheasso iated band pass lterbank:
j =' j+1 ' j :
De nition5 Let u2S (R) . The Littlewood Paley Analysisof u isthe setof distributions: S 0 u = 'u j u = j u Onehas: u=S 0 u+ 1 X j=0 j u:
We annowde ne thetwomi rolo alspa esC s;s 0 x 0 . De nition6 A distribution u2S 0
(R ) belongs tothe 2-mi rolo al spa eC s;s
0 x
0 if thereexistsapositive onstant su hthat,for allj:
jS 0 u(x)j (1+jx x 0 j) s 0 j j u(x)j 2 js (1+2 j jx x 0 j) s 0
The2-mi rolo alspa esarerelatedtothepointwiseHolderspa esthrough:
Theorem2 [9 ℄8x 0 2R, 8s>0: C s x0 C s; s x0 C s;s 0 x 0 C s x 0 ;8s+s 0 >0
Foragivenf,wemayasso iatetoea hpointx 0
its2-mi rolo aldomain,i.e. thesubsetofRR of ouples(s;s
0 )su hthatf 2C s;s 0 x 0
. Itiseasytoshowthat f 2C s;s 0 x0 impliesthatf 2C s ;s 0 + x0
forallpositive. Thisindu esaparti ular shapeforthefrontierofthe2-mi rolo aldomain:
De nitionand Proposition2 2-mi rolo al frontierparameterization[7 ℄ Letf :R!R, and S(s 0 ;x) = sup n s:f 2C s;s 0 x o
The 2-mi rolo al frontieristhe set ofpoints
(f;x 0 ) = f(S(s 0 );s 0 ))g The fun tion S(:;x 0
)isde reasing and onvex. Moreover,one has,for all pos-itive : S(+ ;x 0 )S(;x 0 )
0
frontier. As said in se tion 2, the 2-mi rolo al spa es generalize the Holder spa esandallowtore-interpretboth
l and
p :
Proposition5 [7 ℄Forall x,wehave :
l
(x)=S(0;x)
and, providedsup >0 S()>0, p (x)= 0 (x) where 0
(x)isthe uniquevalue forwhi h
S( 0 ;x)= 0 Inotherwords, l
isobtainedastheinterse tion betweenthe2-mi rolo al frontier and the s-axis, while
p
is the interse tion between the 2-mi rolo al frontierandthelines
0
= s,providedsup >0
S()>0. Thislastrelationholds if f has some minimum overall regularity, i.e. for instan e f belongs to the globalHolderspa eC
!
forsomepositive!.
Finally,wementionthefollowing ru ialpropertyof2-mi rolo al spa es.
Proposition6 f 2C s;s 0 x0 i df dx 2C s 1;s 0 x0
In fa t, more is true, as pseudo-di erential operators may be onsidered insteadofplaindi erentials. Weshalldealwithaversionofthisresultbelow.
4.2 Fra tional Derivative as 2-mi rolo al frontier shifting Itiswell-knownthattheWeylfra tionalderivative,beingapseudo-di erential operator,amountstoahorizontaltranslationofthe2-mi rolo aldomain. Inthis se tionweshowthat thisalsoholdsunder ertain onditionsforthe Riemann-Liouville fra tional derivative. This allows to understand the results of the previous se tion in a more general frame. We start by a Lemma (re all the de nitions off
frompreviousse tion).
Lemma3 Letf bea ontinuousnowheredi erentiablefun tion. DenoteS() (resp. S
+
(), S ()) the frontier of the 2-mi rolo al domain of f (resp. f + , f )atx. Then: 8; S()=min(S + ();S ())
Sin ef =f +
+f ,wehavethatS()min(S +
();S ()). Forthereverse inequality,note that,foranarbitraryfun tion g, uttingg intog
+
andg will at most introdu eadis ontinuityin thederivative ofg. Sin eweare dealing herewithanowhere di erentiablef,lumping togetherf
+
and f at x annot in reasetheregularity.
Theorem3 Let f be a ontinuous nowhere di erentiable fun tion in L 2 (R ). Then f 2C s;s 0 x0 i D q x0 f(x)2C s q;s 0 x0 . Proof: Write D q x0 f(x)= 1 (1 q) d dx (I q 1 + f + +I q 1 f ); (10) where I q 1 + f + = Z x 1 f + (t)(x t) q dt and I q 1 f = Z 1 x f (t)(t x) q dt: I q 1
are byde nition the Weylfra tional integraloperators. Itis well-known and easy to see that the Weyl fra tional integral of order q 1 shifts the 2-mi rolo alfrontierby1 qtowardstherightalongthes-axis. Foraproof,note forinstan ethat,forg2L
2 , \ I q 1 + g(!)=(i!) 1 q ^ g (seetheorem7.1in[18℄). As a onsequen e,j j I q 1 + gj 2 j(q 1) j j gj.
Sin ethederivativeof rstorder shiftsthefrontierto theleftby1, weget thattheoperator(d=dx)I
q 1
shiftsthefrontierbyqtowardstheleft. Fromthis it is lear that f 2 C s ;s 0 x0 i (d=dx)I q f (x) 2C s q;s 0 x0
. Hen e the result followsfrom Lemma3.
5 Examples
In this se tion we onsider two examples and show that the riti al order is equalto thelo al Holderexponent in these ases. Of ourse, this is simplya onsequen eof theorem 1, but makingthe dire t omputation is enlightening andallowstounderstandmore on retelytheme hanismsofLFD.
Our rstexampleisthefun tion de nedby: f(x)= 8 < : a n x+b n 1 n n x 1 n a n x+ n 1 n x 1 n + n 0 otherwise ; (11) wheren2N, n =n ,a n =n b n =n n 1 and n =n +n 1 , with >2and 2(0;1). Wehavetoevaluate
1 (1 q) d dx Z x 0 f(t) (x t) q dt = 1 (1 q) ( d dx G(x)+ d dx H(x)) where,for 1 n+1 +(n+1) <x 1 n +n , G(x)= Z 1 n+1 +(n+1) 0 f(t) (x t) q dt (12) and H(x)= ( R x 1 n n f(t) (x t) q dt 1 n n <x 1 n +n 0 1 n+1 +(n+1) <x 1 n n (13) Consider G(x) = 1 X j=n+1 Z 1 j +j 1 j j f(t) (x t) q dt: Z 1 j +j 1 j j f(t) (x t) q dt = Z 1 j 1 j j a j t+b j (x t) q dt+ Z 1 j +j 1 j a j t+ j (x t) q dt = j (1 q)(2 q) ((x 1 j +j ) 2 q +(x 1 j j ) 2 q 2(x 1 j ) 2 q ) Thereforewehave dG dx = 1 X j=n+1 j 1 q ((x 1 j +j ) 1 q +(x 1 j j ) 1 q 2(x 1 j ) 1 q ) = q 1 X j=n+1 j (x 1 j +j ) 1 q (j ) 2 ; (14)
abovesumisboundedbyj
+1+q
andthereforethesum onvergesuniformly andgoestozerowithxatleastwhenq< + 2. InordertoevaluateH(x) wehaveto onsidertwo ases: 1=n n
x1=nand1=nx<1=n+n
. Inthe rst asewehave
H(x) = Z x 1 n n a n t+b n (x t) q dt = n (x 1 n +n ) 2 q (1 q)(2 q) :
Thereforeinthis asewehave dH dx = n (x 1 n +n ) 1 q (1 q) : (15)
Inthese ond aseweget
H(x) = Z 1 n 1 n n a n t+b n (x t) q dt+ Z x 1 n a n t+ n (x t) q dt = n (1 q)(2 q) (x 1 n +n ) 2 q 2(x 1 n ) 2 q :
Inthis asewehave dH dx = n (1 q) (x 1 n +n ) 1 q 2(x 1 n ) 1 q : (16)
Nowwe ansubstitutex=1=n+n
(2[ 1;1℄)inequations(15)and(16). and he k for the behavior asx approa hes zero (n ! 1). This showsthat dH=dxis oftheorderofn
q
,giving = asa riti alorder.
Ontheotherhand,itisnothardtoprovethatthepointwiseexponentis , whilethelo aloneis = ,asexpe ted.
5.2 IFS
Self-similar fun tions,as onsidered in [8℄, orFra tal InterpolationsFun tions (see [4℄) provide awhole lassof fun tions forwhi h theequality betweenthe degreeofLFDandthelo alHolderexponentisinterestingto he k. Contrarily to the aseabove, the graphsof su h fun tions are, under some assumptions, nowheredi erentiable,and possess amultifra talstru ture. Without entering into details, let us re all that their pointwise Holder exponent varies dis on-tinuouslyeverywhere,while
l
is onstantwith, for allx, l (x)=min y p (y). Furthermore,the levelsets of the fun tion x !
p
(x) are alldense orempty, sothat,in everyneighbourhood ofanyx,there isay where
p
(y)= l
. This is pre iselytheme hanismthatmakesq
and
l
oin ide inthis ase: indeed, simplebut tedious omputations showthat, atea h pointx, q
isobtainedas theliminfof
p
Wehaveelu idatedthemeaningofthedegreeoflo alfra tionaldi erentiability as anequivalentto the lo al Holder exponent, and given aninterpretation of fra tional di erentiation in terms of 2-mi rolo al analysis. This provides yet another link between the elds offra tional integration and Holderregularity analysis (see also [19℄). An easy extension of our work is to the ase where the order of di erentiation is larger than one. Finally, this paper has dealt ex lusively with theoreti al onsiderations. In [13℄, we study the question of estimatingsomelo alregularitymeasuresonsampledata.
Wethank the refereesfor pointing outsome referen esand improving the manus ript.
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