Quasilinear subdivision s hemeswith appli ations to ENO interpolationAlbert Cohen, Nira Dyn and Basarab MateiAbstra tWe analyze the onvergen e and smoothness of ertain lass of nonlinearsubdivision s hemes. We study the stability properties of these s hemes andapply this analysis to the spe i� lass based on ENO and weighted-ENOinterpolation te hniques. Our interest in these te hniques is motivated bytheir appli ation to signal and image pro essing.Introdu tionSubdivision s hemes are a powerful tool for the fast generation of urves andsurfa es in omputer-aided geometri design. In su h algorithms dis rete dataare re ursively generated from oarse to �ne s ales by means of lo al rules.The stability and the onvergen e of su h re�nement pro ess, as well as thesmoothness properties of its limit fun tion if it exists, have been the subje tof a tive resear h in re ent years. We refer to [6℄ and [19℄ for general surveyson subdivision algorithms, and e.g. to [14℄, [15℄, [22℄ for more spe ializedresults on their onvergen e and smoothness.An important motivation for the study of subdivision algorithms is theirrelation to multiresolution analysis and wavelets (see e.g.[11℄ or [13℄). In par-ti ular, the ontribution of a single wavelet oeÆ ient in the representationof a dis rete signal is pre isely obtained by applying a subdivision s hemefrom the s ale of the oeÆ ient up to the signal dis retization s ale. There-fore, understanding the stability and smoothness of subdivision algorithms isfundamental in the ontext of appli ations of wavelets to data ompressionor signal denoising, in whi h ertain oeÆ ients are quantized or dis arded.In all those instan es of subdivision s hemes whi h have been analyzedso far, the re�nement pro ess is based on linear rules. The present work is1

on erned with the situation where this rule is nonlinear in the sense thatthe re�nement operator depends itself on the data to be re�ned .Our main motivation for su h a study is the analysis of nonlinear mul-tiresolution representations introdu ed by Ami Harten [23℄ in the ontextof the numeri al simulation of onservation laws. As we re all in more de-tails, these representations are based on nonlinear re�nement rules whi hinvolve a data dependent sten il sele tion. The goal of this sten il sele tionis to make the re�nement pro ess more a urate in the presen e of isolatedsingularities su h as dis ontinuities. It is no surprise that these ideas havere ently been applied to image ompression. In this ontext, it is hopedthat a better adapted treatment of the singularities orresponding to edgesmight improve the sparsity of the multis ale representations of images, andin turn the rate/distortion performan e of ompression algorithms based onsu h representations (see [12℄, [17℄,[21℄ and [26℄ for several results whi h re-lates the sparsity of the representation to on rete rate/distortion bounds).Some �rst numeri al results, all based on tensor produ t te hniques, whi hdo on�rm this intuition are available in [2℄, [3℄, [8℄, and [9℄.From a mathemati al point of view, edges are indeed the main limitationto the performan e of wavelet based oding: this is re e ted by the poorde ay - O(N�1=2) - of the error of L2 best wavelet N -term approximationfor a \sket hy image fun tion" f = �, where is a bounded domain witha smooth boundary. This re e ts the fa t that this type of approximationessentially provide lo al isotropi re�nement near the edges. Improving onthis rate through a better hoi e of the representation has motivated there ent development of ridgelets in [4℄ and of urvelets in [5℄ whi h are basesand frames having some anisotropi features, resulting in the better rateO(N�1).Nonlinear multis ale representations are another possible tra k for su himprovements, provided that one an over ome two diÆ ulties: �rstly, fora proper anisotropi adaptation to the edges, it is ru ial to develop non-linear methods whi h are not based on tensor produ ts, and se ondly, oneneeds to ontrol the stability of these representations. This se ond pointis ru ial: sin e nonlinear multis ale representations annot be thought asde ompositions of the signal into a �xed wavelet basis, the error produ edby thresholding or quantizing the oeÆ ients is no more learly understood:su h perturbations might be greatly ampli�ed by the iteration of the non-linear re�nement rules involved in the predi tion pro ess. In order to solve2

this problem, we essentially need to understand the behavior of the nonlinearsubdivision s hemes orresponding to these iterative re�nements.The obje tive of the present paper is to provide appropriate tools foranalyzing the smoothness and stability of quasilinear subdivision s hemes,and apply these tools in the parti ular ase of the essentially non os illatory(ENO) re�nements introdu ed in [24℄.The results of this paper represents the �rst step in the study of nonlinearmultis ale representations. Using these results, our next perspe tive, is theanalysis of data ompression algorithms based on su h nonlinear representa-tions.Our work is organized as follows. A qui k overview of the framework in-trodu ed in [23℄ is given in Se tion 1, together with several relevant examplesof quasilinear s hemes. In Se tion 2 and 3, we prove several results on ern-ing the smoothness and stability analysis of quasilinear subdivision s hemes,in the uniform and H�older metri . In Se tion 4, we apply these results tothe parti ular example of the four points ENO and WENO re�nement rules.Finally, an appendix is devoted to the generalization of the results of Se tion2 and 3, to other smoothness and error measures, su h as Lp, Sobolev orBesov norms.Motivation and Ba kgroundThe framework introdu ed by A. Harten [23℄ for the dis rete multiresolutionrepresentations of data is based on two inters ales dis rete operators: theproje tion and the predi tion operators .The proje tion operator P jj�1 a ts from �ne to oarse level of resolution.This operator extra ts from vj; the data string at the level j of dis retization,the dis rete information at the oarser level of resolution, j � 1; i.e. vj�1:The predi tion operator P j�1j ; a ts from oarse to �ne level of resolution.It yields an approximation of the the dis rete ve tor vj from the proje tedve tor vj�1. These two operators should in addition satisfy the propertyP jj�1P j�1j = I; (1)i.e., the proje tion operator is a left inverse to the predi tion operator.3

The approximation build by P jj�1 is de�ned as followsv̂j := P jj�1vj�1:This gives the redundant representation of the ve tor vj by its approximationv̂j and the predi tion error ej := v̂j � vj:From (1) , we have that P j�1j is onto, and that the predi tion error belongsto the �nite dimensional spa e W j�1; de�ned as the null spa e of the pro-je tion operator. Therefore by de omposing ej in terms of a basis of W j�1,we an eliminate the redundant information in ej. We denote by dj�1 the oordinate ve tor of the error ve tor in this basis of W j�1. In analogy withthe wavelet terminology we all dj�1 the detail ve tor. Sin e v̂j = P jj�1vj an be equivalently hara terized by (vj�1; dj�1): By iteration we obtain aone to one orresponden e between vj and its multiresolution representation(v0; d0; : : : ; dj�1).If both dis rete operators, proje tion and predi tion are linear, thenthe orresponding multiresolution transform is equivalent to a biorthogonalwavelet transform.Some of the predi tion operators proposed by Harten [23℄ are nonlinearlydata dependent sin e they are based on essentially non-os illatory (ENO)predi tion te hniques. By using them, the orresponding multiresolutiontransforms annot be thought as a hange of basis, whi h make the analysisof these transforms more diÆ ult.The representations introdu ed by Harten are formulated for spe i� types of dis retization, often used in omputational appli ations (e.g., thepoint values and the ell averages dis retization). The sele tion of the dis- retization depends on the problem under onsideration, e.g., for the imagemodelisation by square integrable fun tions, an appropriate hoi e of thedis retization is by the ell averages (instead of point values dis retization,whi h does not make sense in this ase). In the following, we brie y evokethe nonlinear predi tion operators based on ENO, in the point value and ellaverages ontext.Example 1. Point Value MultiresolutionIn this setting, we interpretate the dis rete ve tor vj = (vjk)k2ZZ as the4

point values of a ontinuous fun tion v on the grid �j := (2�jk)k2ZZ, i.e.vjk := v(2�jk). This suggest the hoi e for P jj�1 as the simple downsamplingoperator. For the predi tion operator, we noti e that the ve tor v̂j should oin ide with vj on the oarse grid; then building predi tion operator an beviewed as an interpolation problem. The details are de�ned by the restri tionof the interpolation error ej on �j�1 := �j n �j�1; i.e.dj�1 := (vjk � v̂jk)k2�j�1 :In the sequel, we present an important lass of lo al predi tors obtained byLagrange interpolation.At s ale j we want to predi t for ea h k 2 ZZ the value v̂j2k+1 from thevalues (vj�1l )l2ZZ. To su h a k we asso iate a predi tion sten il of length MSr(k) := f(k � r)2�j+1; : : : ; (k � r +M � 1)2�j+1g;with r an integer representing the position of the sten il with respe t to k.Using the values (v( )) 2Sr(k); we de�ne pr 2 �M as the unique polynomialof degree M whi h interpolates the values of v on Sr(k): We then de�ne thepredi ted value v̂j2k+1;r := pr((2k + 1)2�j+1):Note that M is exa tly the order of a ura y of the predi tion. If the pa-rameter r is �xed independently of the data, we obtain a linear predi tionoperator, and the multiresolution transform is then equivalent to a biorthog-onal interpolatory wavelet transform, for whi h the dual s aling fun tion isthe Dira distribution.The goal of ENO interpolation is to obtain a better adapted predi tionnear the singularities of the data. The idea is to sele t by some pres ribednumeri al riterion the polynomial pr whi h is the least os illatory in theneighborhood of k.We give below the formulae for the third order a urate predi tion (M =4). The predi ted values v̂j2k+1;r using Sr(k); r = 0; 1; 2; are obtained respe -tively by 8>><>>: v̂j2k+1;0 := 516vj�1k + 1516vj�1k+1 � 516vj�1k+2 + 116vj�1k+3;v̂j2k+1;1 := � 116vj�1k�1 + 916vj�1k + 916vj�1k+1 � 116vj�1k+2;v̂j2k+1;2 := 116vj�1k�2 � 516vj�1k�1 + 1516vj�1k + 516vj�1k+1: (2)5

In the ase of predi tion by the value of the unique ubi polynomial thatinterpolates vj on the entered sten il, the orresponding multiresolutiontransform is equivalent to the Dubu {Deslaurier interpolatory wavelet trans-form (see [16℄ and [20℄). For the properties of the interpolant as well as forthe smoothness of the limit of this iterative pro ess, we refer the reader to[14℄, [20℄ and [16℄.Example 2. Cell average multiresolutionIn the ell average ontext, IR is partitioned in disjointed dyadi ells �j :=f�jk = [k2�j; (k+1)2�j)gk2ZZ: In this ontext, the dis rete ve tor vj is viewedas the average �2j R�jk v(t) dt�k2�j of a lo ally integrable fun tion.As in the point values setting, this suggests to take for P jj�1 the averag-ing operator. The onstru tion of the predi tion operator is similar to thepredi tion in the point values setting. To ea h �j�1k ; we asso iate a sten il of ellsSr(k) := f[(k�r)2�j+1; (k�r+1)2�j+1℄; : : : ; [(k�r+M�1)2�j+1; (k�r+M)2�j+1℄g:Using the averages within the sten il Sr(k); we de�ne qr 2 �M�1 as theunique polynomial of degree M � 1 whi h interpolates these averages.We then de�ne the predi ted averages as those of qr on the half intervals[2�j+12k; 2�j+1(2k + 1)℄ and [2�j+1(2k + 1); 2�j+1(2k + 2)℄.Noti e that by using the averages of a lo al integrable fun tion we anobtain the point values of its primitive fun tion. This interpretation allowsto obtain the polynomial used to make the predi tion in ell averages ontextthrough a derivation of the predi tion polynomial used in the point valuessetting for the primitive fun tion.The multiresolution de omposition based on ell averages is equivalentto the biorthogonal wavelet transform, for whi h the dual s aling fun tion isthe box fun tion, [18℄.We an also make the same remarks on erning the possibility of usingENO-type re onstru tions. In the ase of two order a urate predi tion basedon Lagrange interpolation, the predi ted averages v̂j2k;r using Sr(k); r =0; 1; 2; are given by8>><>>: v̂j2k;0 := 118 vj�1k � 12vj�1k+1 + 18vj�1k+2;v̂j2k;1 := 18vj�1k�1 + vj�1k � 18vj�1k+1;v̂j2k;2 := �18vj�1k�2 + 12vj�1k�1 + 58vj�1k : (3)6

In both types of dis retization, the details are de�ned as the predi tion errorat the odd samples.Weighted-ENO interpolation . The weighted-ENO (WENO) interpo-lation developed in [7℄ is based on the ENO idea. In this te hnique, in on-trast to ENO interpolation whi h uses only one of the andidate sten ils tomake the predi tion, one onsider a onvex ombination of the polynomialsasso iated to these sten ils, i.e.̂vk := M�1Xr=0 �rv̂rk;with �r � 0 and PM�1r=0 �r = 1. In ENO interpolation, a small round-o� errorperturbation of the data an result in hanging the sele ted sten ils. Thissituation is avoided in WENO interpolation whi h provides with a smoothtransition between the sten ils. A possible form of the weights is given in [7℄by �r := arPM�1l=0 al ; r = 0; : : : ;M � 1;where ar := dr(�+ br)2 ; and br := M�1Xl=1 2�j(2l�1) Z�jk ��lpr(x)�lx �2 dx: (4)The dr are �xed positive onstants. The br are de�ned by the sum of thesquares of L2 norms for all the derivatives of the interpolation polynomial prover the interval �jk. The fa tor 2�j(2l�1) is introdu ed to remove any leveldependen y on the derivatives. Here � is introdu ed in order to avoid thedenominator to vanish, br are the so alled \smoothness indi ators" of thesten il Sr(k) : if the fun tion v(x) is smooth inside the sten il Sr(k), thenbr � O(2�2j), else if the fun tion has a dis ontinuity inside the sten il Sr(k),then br � O(1).The rational form of the weights is hosen in order to emulate the ENOidea and to be omputationally eÆ ient. If the sten il Sr is lo ated in asmooth region, the smoothness indi ator br is lose to 0 and then the weight�r is lose to 1. In ontrast, if the sten il ontains a singularities the smooth-ness indi ator br is larger and the weight �r is loser to 0.7

In the ase of four point interpolatory s hemes, we ompute the predi tedvalue as a onvex ombination of the predi ted values by the three sten ils,as follows v̂j2k+1 := �0v̂j2k+1;0 + �1v̂j2k+1;1 + �2v̂j2k+1;2 (5)where �0; �1 and �2 represent the weights asso iated to the right, enteredand left sten il, respe tively. More pre iselyv̂j2k+1 := �216 vj�1k�2 � 5�2+�116 vj�1k�1 + (1 + 5�2+2�18 )vj�1k+(1 + 5�0+2�18 )vj�1k+1 � 5�0+�116 vj�1k+2 + �016 vj�1k+3; (6)The weights asso iated to the three sten ils are de�ned as in [25℄ and [7℄. Inthis ase (M = 3), (4) gives8><>: b0 := 0;0(vjk+2 � 2vjk+1 + vjk)2 + 0;1(vjk+2 � 4vjk+1 + 3vjk)2;b1 := 1;0(vjk+1 � 2vjk + vjk�1)2 + 1;1(vjk+1 � vjk�1)2;b2 := 2;0(vjk � 2vjk�1 + vjk�2)2 + 2;1(3vjk � 4vjk�1 + vjk�2)2; (7)where i;j; i = 0; 1; 2; j = 0; 1; are �xed positive onstants. Some possible hoi es of the onstants are suggested in [7℄.As we already explained, stability of the multiresolution transform isa key issue in appli ations where some oeÆ ients are dis arded (su h as ompression or denoising). In this paper, we limit our study to the nonlinearsubdivision s heme orresponding to the iterative appli ation of predi tionoperator, from oarse to �ne s ales, without adding any details. To beginwith, we give some basi notations and de�nitions and re all some propertiesof the subdivision operators.A subdivision s heme de�nes a fun tion ( alled the limit fun tion) as thelimit of a subdivision pro ess in whi h an initial �nite set of points, alledthe ontrol points, is re ursively re�ned.De�nition 1 A data dependent subdivision rule is an operator valued fun -tion S whi h asso iates to ea h v 2 `1(ZZ) a linear operatorS(v) : `1(ZZ)! `1(ZZ);de�ned by a rule of the type(S(v)w)k :=Xl ak;l(v)wl; (8)8

where the oeÆ ients ak;l(v) are zero if jk � 2lj > M for some �xed M > 0.We de�ne the asso iated quasilinear subdivision s heme as the re ursivea tion of the quasilinear rule Sv := S(v)v on an initial set of data v0, a - ording to vj := Svj�1 = S(vj�1)vj�1; j � 1: (9)In the above de�nition, M typi ally represents the size of the sten il used inthe subdivision rule. For linear subdivision s hemes, the oeÆ ients ak;l donot depend on the data v, i.e. S(v) = S a �xed operator. For linear anduniform subdivision s hemes, these oeÆ ients have the form ak;l = ak�2l.The analysis of a subdivision s heme onsists of establishing onditionsfor the onvergen e of the s heme, and in hara terizing the smoothness aswell as the order of approximation of the set of limit fun tions. We refer thereader to [6℄,[19℄ and [22℄ for a general survey on this subje t, in the linearand uniform ase.De�nition 2 A subdivision s heme, generating re ursively the data fvj : j 2ZZ+g; is alled uniformly onvergent if, for every set of initial ontrol pointsv0 2 `1(ZZ), there exists a ontinuous fun tion f 2 C(IR), alled the limitfun tion, su h that limj!+1 supk2ZZ jvjk � f(2�jk)j = 0; (10)and that f is non-trivial at least for one initial data v0.We also asso iate a fun tion f j to the data vj as the pie ewise aÆne inter-polation to f(2�jk; vj) : j 2 ZZ+g: Thusf j(x) := Xk2ZZ vjk'(2jx� k); (11)where '(x) := maxf1�jxj; 0g is the hat fun tion. It is lear that the uniform onvergen e of the subdivision s heme is equivalent tolimj!+1 kf j � fkL1 = 0:The limit fun tion f is denoted by S1v0. The following de�nition plays animportant role in the analysis of subdivision s hemes.9

De�nition 3 Let N � 0 be a �xed integer. The data dependent subdivisionrule has the property of polynomial reprodu tion of order N if for all u 2`1(ZZ) and P 2 �N there exists eP 2 �N with P � eP 2 �N�1 su h thatS(u)p = ep where p and ep are de�ned by pk = P (k) and epk = eP (k2 ).In parti ular, the ENO and WENO s hemes dis ussed in the previousse tion satisfy su h a property up to the order M for point values and M �1for ell averages. We re all the de�nition of the n-th order forward �nitedi�eren e operator, (�nv)k = nXm=0 (�1)m�nm�vk+m: (12)For the �rst order �nite di�eren e we omit the supers ript 1. In the ase oflinear subdivision s heme, using a formalism based on Laurent polynomials[19℄, it has been proved that if the subdivision s heme has the property ofpolynomial reprodu tion up to the order N , then there exist similar s hemesfor the di�eren es of order n := 1; � � � ; N + 1Sn : `1(ZZ)! `1(ZZ); �n(Sv) = Sn(�nv):The onvergen e and smoothness properties of a subdivision s hemes arethen studied through the ontra tion properties of the s hemes Sn. Morepre isely, denoting by �1(A) the spe tral radius of an operator A in `1, theuniform onvergen e of the linear subdivision is equivalent to the property�1(S1) < 1. Moreover, if for some m 2 f1; � � � ; N + 1g, we have �1(Sm) <2�m+1, then the limit fun tion is in Cs for all s < s� = � log �1(Sm)log 2 (andtherefore m� 1 times di�erentiable sin e s� > m� 1).In order to study quasilinear subdivision s hemes, we need to introdu esome additional de�nitions. We start with the boundedness property.De�nition 4 A data dependent subdivision rule is alled bounded if thereexists a onstant B > 0 su h that for all v 2 `1(ZZ),kS(v)k`1 � B; (13)where the norm stands for the `1 operator norm.10

Clearly, this property an also be expressed by saying that the oeÆ ientsfak;l(v)g are bounded independently of k, l and v. In the following, we alwaysassume that the rules that we study are bounded.We have already remarked that, in the WENO te hnique, the transitionbetween two sten ils is made in a ontinuous way. This property is ru ialin the study of the stability of quasilinear subdivision s hemes. This notionis expressed in the next de�nition.De�nition 5 A data dependent subdivision rule is alled ontinuously de-pendent on the data if for every v; w 2 `1(ZZ), the asso iated operators S(v)and S(w) satisfy kS(v)� S(w)k`1 � Ckv � wk`1; (14)where C dependents in a non-de reasing way on maxfkvk`1; kwk`1g.The fa t that the onstant C might grow with kvk`1 and kwk`1 is en oun-tered in the pra ti al examples that we have in mind su h as WENO inter-polation.We �nally introdu e the notion of joint spe tral radius asso iated to adata dependent subdivision rule.De�nition 6 The joint spe tral radius of a data dependent subdivision ruleS is the number�1(S) := lim supj!1 sup(u0;u1;:::;uj�1)2(`1(ZZ))j kS(uj�1) � � �S(u0)k 1j̀1:In other words, �1(S) is the in�mum of all � > 0 for whi h there existsC > 0 su h that for all arbitrary (uj)j�0 in `1 and v 2 `1 one haskS(uj�1) � � �S(u0)vk`1 � C�jkvk`1; (15)for all j � 0. Note that in the ase of linear subdivision s hemes, this isexa tly the spe tral radius of S in `1.Convergen e and Smoothness analysisIn this se tion, we provide suÆ ient onditions for the onvergen e of quasi-linear subdivision s hemes and for the smoothness of the limit fun tion. In11

fa t the results in this se tion, but not those of the next se tion, apply to awider lass of subdivision s hemes than the lass of quasilinear subdivisions hemes. In this lass, a s heme is de�ned by a data dependent rule ; S; andby given sequen e of data ful : l 2 ZZ+g and initial data v0 a ording tovj := S(uj�1) � � �S(u0)v0: As in the linear ase, the results of this se tion areobtained through the study of the asso iated s hemes for the di�eren es. Theexisten e of the s heme for the di�eren es is obtained by using the propertyof polynomial reprodu tion of the data dependent rule. This result is givenin the next proposition.Proposition 1 Let S be a data dependent subdivision rule whi h reprodu espolynomials up to degree N . Then for 1 � n � N + 1 there exists datadependent subdivision rule Sn with the property that for all v; w 2 `1,�nS(v)w := Sn(v)�nw:Proof :Let 1 � n � N + 1 and let u := S(v)w. Combining (12) and (8), we obtain(�nu)k = nXm=0(�1)m�nm� Xl s:t: jk+m�2lj�M ak+m;l(v)wl: (16)Therefore, (�nu)k an be written as a linear ombination of the wl(�nu)k =Xl bk;l(v)wl; (17)where bk;l(v) := Pnm=0(�1)m�nm�ak+m;l(v). Note that bk;l(v) is zero for l <(k �M)=2 and l > (k + n +M)=2. For ea h �xed k we thus have a �niteve tor (bk;l(v))l2Ek with Ek := fl : (k �M)=2 � l � (k + n +M)=2g.Sin e the rule reprodu es polynomials of degree up to N , we haveXjk�2lj�M ak;l(v)lm = Pm(k); 0 � m � n� 1; (18)with Pm 2 �m. Applying the n-th order �nite di�eren e operator �n on thisidentity, we obtain Xl bk;l(v)lm = 0; m = 0; : : : ; n� 1: (19)12

Therefore, for ea h k (bk;l(v))l2Ek is orthogonal to the ve tors (lm)l2Ek form = 0; : : : ; n � 1. It follows that (bk;l(v))l2Ek an be written in terms of abasis of the orthogonal omplement of spanf(lm)l2Ek jm = 0; : : : ; n� 1g. Anatural hoi e for this basis is given bye0(l) := �nl �(�1)n+l; if l = 0; : : : n;e0(l) := 0; if l =2 f0; : : : ; ng;and taking eq(l) := e0(l�q) with (k�M)=2 � q � (k�n+M)=2. Therefore,we have bk;l(v) := X(k�M)=2�q�(k�n+M)=2 �k;q(v)eq(l); (20)from whi h we derive a subdivision rule for the n-the order di�eren es of thetype (�nu)k =Xl bk;l(v)wl = Xjk�2qj�M �k;q(v)(�nw)q: (21)Noti e from the above proof that the the sten ils used in Sn are alwayssmaller than those used in S. Moreover, if S is bounded (resp. ontinuouslydependent on the data), then Sn is also bounded (resp. ontinuously depen-dent on the data). The next result gives a relation between the joint spe tralradius of these s hemes.Proposition 2 For all n = 0; � � � ; N , one has �1(Sn+1) � �1(Sn)=2.Proof:We shall prove that �1(S1) � �1(S)=2, and the general result will follow byindu tion. Let � > �1(S1), and C > 0 su h that for all sequen e (ul)l�0 in`1 and v 2 `1 one haskS1(uj�1) � � �S1(u0)�vk`1 � C�jk�vk`1; (22)for all j > 0. De�ning wj := S(uj�1) � � �S(u0)v; (23)it follows that k�wjk`1 � C�jk�vk`1: (24)13

We use the relationkwjk`1 = supk2ZZ supfjwjl j : l 2 [2jk; 2j(k + 1))g; (25)and exploit the fa t that the s heme is lo al. The values of wjl for l 2[2jk; 2j(k + 1)) only depend on those of vl for jl � kj � M . For a �xed k,we de�ne ev by evl = vl if jl � kj � M and evl = 0 otherwise, and we letewj := S(uj�1) � � �S(u0)ev. It follows that wjl = ewjl for l 2 [2jk; 2j(k + 1)) andthat ewjl = 0 for jl � 2jkj > 2j2M . In turn, we obtain thatsupl2[2jk;2j(k+1)) jwjl j = supl2[2jk;2j(k+1)) j ewjl j� Pjl�2jkj<2j2M j� ewjl j� C2jk� ewjl k`1� C(2�)jk�evk`1� 2C(2�)jkvk`1It follows that kwjk`1 � 2C(2�)jkvk`1, and thus �1(S) � 2�. Letting �tendto �1(S1), we obtain the laimed result.Note that onvergen e of the subdivision s heme implies �(S) � 1 sin eotherwise S1v0 = 0 for all initial data v0. Therefore, the above result showsthat we always have �1(Sn) � 2�n: (26)We are now ready to establish a suÆ ient ondition for the onvergen eof quasilinear subdivision s hemes and for the Cs smoothness of the limitfun tion with s < 1.Theorem 1 Let S be a data dependent subdivision rule whi h reprodu es onstants. If the rule for the di�eren es satis�es �1(S1) < 1; then the quasi-linear subdivision s heme based on S is uniformly onvergent and the limitfun tion S1v0 is Cs for all s < � log �1(S1)log 2 .Proof:Let � be su h that �1(S1) < � < 1. There exists a onstant C su h that forall initial data v0 2 `1 and j � 0,k�vjk`1 � C�jk�v0k`1: (27)14

Observe thatkf j+1 � f jkL1 � supk2ZZ jvj+12k � vjkj; jvj+12k+1 � vjk + vjk+12 j: (28)We now write vj+12k � vjk = Xl2Fk k;lvjl (29)and vj+12k+1 � vjk + vjk+12 = Xl2Fk dk;lvjl (30)where Fk := fl ; jk � lj � Mg, k;l := a2k;l � Æ(k � l) and dk;l := a2k+1;l �Æ(k�l)+Æ(k+1�l)2 Sin e our s heme reprodu es onstants, the ve tors ( k;l)l2Fkand (dk;l)l2Fk are orthogonal to the onstant ve tor. By the same reasoningas in the proof of Proposition 1, we on lude that both vj+12k �vjk and vj+12k+1�vjk+vjk+12 are linear ombinations of the �nite di�eren es �vjl for l = k �M; � � � ; k +M � 1 From this it follows thatkf j+1 � f jkL1 � Ck�vjk`1 � C�jk�v0k`1: (31)Therefore the sequen e fj onverges uniformly to a ontinuous limit f =S1v0. We also see thatkfkL1 � kf 0kL1 +Pj�0 kf j+1 � f jkL1� C(kv0k`1 + k�v0k`1) � Ckv0k`1:In order to prove that f 2 Cs it suÆ es to evaluate jf(x) � f(y)j forjx� yj � 1. Let j be su h that 2�j�1 < jx� yj � 2�j. We then writejf(x)� f(y)j � jf(x)� f j(x)j+ jf(y)� f j(y)j+ jf j(x)� f j(y)j� 2kf � f jkL1 + jf j(x)� f j(y)j� C�jk�v0k`1 + 2�jk(f j)0kL1� C�jk�v0k`1 + k�vjk`1� C�jk�v0k`1� Cjx� yjsk�v0k`1;with s := � log(�)= log 2. This on ludes the proof.In the following, we give suÆ ient onditions for the Cs smoothness of thelimit fun tion for s � 1: 15

Theorem 2 Let S be a data dependent subdivision rule whi h reprodu espolynomials up to degree N . If the rule for the di�eren es satis�es �1(Sn+1) <2�n for some n 2 f0; � � �Ng, then the quasilinear subdivision s heme basedon S is uniformly onvergent and the limit fun tion S1v0 is Cs for alls < � log �1(Sn+1)log 2 .Proof:Noti e that by Proposition 2, the assumption that �1(Sn+1) < 2�n impliesthat �1(Sm+1) < 2�m for m = 0; 1; � � � ; n. In parti ular �1(S1) < 1 and thes heme is onvergent by Theorem 1.We shall use indu tion on n to prove Cs smoothness. For n = 0, theresult is proved by Theorem 1. For n = 1, we let f = S1v0 and we assumethat �1(S2) < 1=2. Introdu ingwj := 2j�vj = 2jS1(vj�1)S1(vj�2) � � �S1(v0)�v0; (32)we have �wj := 2j�vj = 2jS2(vj�1)S1(vj�2) � � �S2(v0)�2v0; (33)and therefore if � is su h that 2�1(S2) < � < 1, thenk�wjk`1 := 2j�vj = C�jk�2v0k`1: (34)We obtain as in Theorem 1 that wj uniformly onverges to a ontinuousfun tion g namely limj!1 supk jwjk � g(2�jk)j = 0:Introdu ing the fun tion e' := �[0;1℄ and the fun tionsgj := Xk2ZZwjk e'(2j � �k); (35)one easily he k that gj = ddxf j, where f j is the aÆne fun tion de�ned by(11), i.e. Z ba gj(x)dx = f j(a)� f j(b); (36)for all a and b. We know that limj!1 kf j � fkL1 = 0, and we also havelimj!1 kgj � gkL1 = 0. It follows thatZ ba g(x)dx = f(a)� f(b); (37)16

for all a and b. Therefore, f is di�erentiable with f 0 = g. Moreover, as inTheorem 1, we obtain that g 2 Ct for all t < � log 2�1(S2)log 2 < �1 � log �1(S2)log 2 .Therefore f 2 Cs for all s < � log �1(S2)log 2 . Iterating this argument for n > 1,we obtain the general result.Stability analysisIn this se tion, we study the stability of quasilinear subdivision s hemes, e.g.properties of the typekS1v0 � S1ev0kL1 � Ckv0 � ev0k`1: (38)In the linear ase, this is a simple onsequen e of onvergen e, namely ofkS1v0kL1 � Ckv0k`1. In the nonlinear ase, it requires a more spe i� study.In our study of stability we need the additional assumption that thereexists a linear left inverse operator of the subdivision operator ( alled re-stri tion or proje tion operator by Harten). More pre isely we assume thatthere exists oeÆ ients ( l)jlj<P with Pjlj<P l = 1 su h thatvj�1k := Xjlj<P lvj2k�l; (39)whenever vj := Svj�1:In many interesting ase of linear or nonlinear subdivision algorithms,su h an operator exists. In the point-value ontext l = Æ0;l, and in the ell-averages ontext 0 = �1 = 1=2, l = 0 otherwise. In the followingwe always assume the existen e of a restri tion operator of the form (39) .In the next result we obtain the existen e of a similar left-inverse for thesubdivision s hemes Sn asso iated to the �nite di�eren es.Proposition 3 Let S be a data dependent subdivision rule whi h reprodu espolynomials of degree N . Then, for n = 1; � � � ; N +1 there exists oeÆ ients( nl )jlj<P with Pjlj<P nl = 2n su h that(�nvj�1)k := Xjlj<P+n nl (�nvj)2k�l; (40)whenever vj := Svj�1: 17

Proof:Consider the ase n = 1. Assuming (39), we an write(�vj�1)k = Pjlj<P l(vj2k+2�l � vj2k�l)= Pjlj<P l((�vj)2k+1�l + (�vj)2k�l)= Pjlj�P+1 1l (�vj)2k�l;with 1l := l + l+1 whi h proves the result. The ase n > 1 follows byindu tion.We use the restri tion operators for the �nite di�eren es through the follow-ing lemma.Lemma 1 Let S be a data dependent subdivision rule whi h reprodu es poly-nomials of degree N . Then there exists a onstant D > 0; depending only onn, su h thatk�nvjk`1 � 2�nk�nvj�1k`1 +Dk�n+1vjk`1; 0 � n � N: (41)for all j � 0 and v0 2 `1.Proof:Sin e (�nvj�1)k = Pjlj�P+n nl (�nvj)2k�l with Pl nl = 2n, we also have(�nvj�1)k = 2n(�nvj)2k + Xjlj<P+n nl ((�nvj)2k�l � (�nvj)2k) (42)It follows that(�nvj)2j := 2�n[(�nvj�1)k + Xjlj<P+n l(�n+1vj)2k�l℄; (43)with l := Pl�1k=0 nk . In a similar way, we obtain(�nvj)2k+1 := 2�n[(�nvj�1)k + Xjlj<P+ndl(�n+1vj)2k�l℄: (44)The laim follows with D := 2�nmaxfPjlj<P+n j lj;Pjlj<P+n jdljg.Remark 1 Note that, sin e the restri tion operator is linear, we also have,k�nvj��nevjk`1 � 2�nk�nvj�1��nevj�1k`1+Dk�n+1vj��n+1evjk`1; (45)for vj = S(vj�1)vj�1 and evj = S(evj�1)evj�1.18

The main ingredient for our analysis of the stability of quasilinear subdivisions heme is the following result.Lemma 2 Let S be a quasilinear subdivision rule, whi h reprodu es polyno-mials up to the order N . Assume that S is ontinuously dependent on thedata. Then for n = 0; : : : ; N , and � > �1(Sn+1); we havek�n+1vj ��n+1evjk`1 � C�j�j�1Xl=0 k�nvl ��nevlk`1�; (46)where C depends in a ontinuous non-de reasing way on �maxfkvlk`1; kevlk`1; l =0 : : : j � 1g�.Proof:It is enough to give the proof for n = 0; sin e it is similar for larger values ofn. If � > �1(S1), there exists a onstant K su h that for all initial data v0,k�vjk`1 � K�jk�v0k`1: (47)Moreover there exists an integer L su h thatk�vjk`1 � �Lk�vj�Lk`1; j � L: (48)Assuming that j � L, we havek�vj ��evjk`1 = kS1(vj�1) � � �S1(vj�L)�vj�L � S1(evj�1) � � �S1(evj�L)�evj�Lk`1� Aj +Bj:where Aj = kS1(vj�1) � � �S1(vj�L)(�vj�L ��evj�L)k`1;and Bj = kS1(vj�1) � � �S1(vj�L)�evj�L � S1(evj�1) � � �S1(evj�L)�evj�Lk`1:By (48), we obtain Aj � �Lk�vj�L ��evj�Lk`1: (49)In order to estimate Bj, we de�ne for i > j � LGi := S1(vi�1) � � �S1(vj�L)�evj�L � S1(evi�1) � � �S1(evj�L)�evj�L;19

andKi := S1(vi�1)S1(vi�2) � � �S1(vj�L)�evj�L�S1(evi�1)S1(vi�2) � � �S1(vj�L)�evj�L;Li := S1(evi�1)S1(vi�2) � � �S1(vj�L)�evj�L�S1(evi�1)S1(evi�2) � � �S1(evj�L)�evj�L:We thus have Bj = kGjk`1 � kKjk`1 + kLjk`1 (50)Re alling the boundedness and ontinuous dependen y on the data of thes heme S1, i.e. kS1(v)k`1 � B1; (51)and kS1(v)� S1(ev)k`1 � C1kv � evk`1; (52)where C1 depends in a ontinuous non-de reasing way on maxfkvk`1; kevk`1g,we an estimate the �rst term a ording tokKjk`1 � C1BL�11 kvj�1 � evj�1k`1k�evj�Lk`1; (53)where C1 depends in a ontinuous non-de reasing way on maxfkvj�1k`1; kevj�1k`1g,and the se ond term by kLjk`1 � B1kGj�1k`1: (54)Therefore, we obtainkGjk`1 � C1BL�11 kvj�1 � evj�1k`1k�evj�Lk`1 +B1kGj�1k`1where C1 depends in a ontinuous non-de reasing way on maxfkvj�1k`1; kevj�1k`1g,Similarly, we havekGj�1k`1 � C1BL�21 kvj�2 � evj�2k`1k�evj�Lk`1 +B1kGj�2k`1;where C1 depends in a ontinuous non-de reasing way on maxfkvj�2k`1; kevj�2k`1g,and thereforekGjk`1 � C1BL�11 (kvj�1�evj�1k`1+kvj�2�evj�2k`1)k�evj�Lk`1+B21kGj�2k`1;where C1 depends in a ontinuous non-de reasing way onmaxfkvj�1k`1; kevj�1k`1; kvj�2k`1; kevj�2k`1g:20

By iteration, and sin e Gj�L := �evj�L ��evj�L = 0, we obtainBj � C1BL�11 k�evj�Lk`1� LXl=1 kvj�l � evj�lk`1�; (55)where C1 depends in a ontinuous non-de reasing way on maxfkvlk`1; kevlk`1; l =0 : : : j � 1g: Adding (49) and (55), we thus obtaink�vj��evjk`1 � �Lk�vj�L��evj�Lk`1+C1BL�11 k�evj�Lk`1� LXl=1 kvj�l � evj�lk`1�:Combining this estimate with (47) givesk�vj��evjk`1 � �Lk�vj�L��evj�Lk`1+C2�j�L� LXl=1 kvj�l � evj�lk`1�; (56)with C2 = 2C1Kkev0k`1. If j � L � L, we also havek�vj�L��evj�Lk`1 � �Lk�vj�2L��evj�2Lk`1+C2�j�2L� 2LXl=L+1 kvj�l � evj�lk`1�;and thereforek�vj ��evjk`1 � �2Lk�vj�2L ��evj�2Lk`1 + C2�j�L� 2LXl=1 kvj�l � evj�lk`1�:After [ jL ℄ iterations, we thus obtaink�vj��evjk`1 � �L[ jL ℄ max0�l�L�1 k�vl ��evlk`1+C2�j�L� jXl=1 kvj�l � evj�lk`1�:For the values l = 0; � � � ; L � 1, as well as in the ase 0 � j < L, we simplyuse k�vl ��evlk`1 � 2kvl � evlk`1 it follows thatk�vj ��evjk`1 � C�j� jXl=1 kvj�l � evj�lk`1�; (57)where C = 2maxf1; ��Lg(1 +KC1)kev0k`1; (58)21

depends in a ontinuous non-de reasing way on maxfkvlk`1; kevlk`1; l = 0 : : : j�1g.We are now ready to give ondition for the stability of the quasilinear subdi-vision s hemes for various norms measuring S1v� S1ev. We begin with theuniform norm.Theorem 3 Let S be a quasilinear subdivision rule whi h reprodu es on-stants. Assume that S is ontinuously dependent on the data and that �1(S1) <1. Then for all data v0 and ev0, we havekS1v0 � S1ev0kL1 < Ckv0 � ev0k`1; (59)where C depends in a ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g:Also for s < � log(�1(S1))= log 2 we have thatk�vj ��evjk`1 � C2�sjkv0 � ev0k`1: (60)Proof:It suÆ es to prove that for all j > 0kvj � evjk`1 < Ckv0 � ev0k`1; (61)with C independent of j, sin e we then havekf j � ef jkL1 < Ckv0 � ev0k`1; (62)and therefore (59) by letting j go to +1.Let � be su h that �1(S1) < � < 1. Let us denote �j := kvj � evjk`1 and�j := k�vj ��evjk`1. By Remark 1 and Lemma 2, these sequen es satisfythe following inequalities( �j � �j�1 +D�j;�j � C�j(�j�1 + � � �+ �0);where C depends in a ontinuous non-de reasing way on maxfkvlk`1; kevlk`1; l =0 : : : j � 1g. However, we remark that sin e �1(S1) < 1, we have kSjvk`1 �Kkvk`1 with K a onstant independent of j and v, and therefore we havethat C depends in a ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g.22

If we now onsider the positives non-de reasing sequen es ��j and ��j de-�ned by ��0 = �0, ��0 = �0 and satisfying( ��j = ��j�1 +D ��j;��j = C�j(��j�1 + � � �+ ��0); (63)we learly have �j � ��j and �j � ��j. Using the last equality from (63) andthe fa t that ��j is in reasing, we get��j � Cj�j ��j�1: (64)Combining this with the �rst equality in (63) , we obtain��j � (1 + CDj�j)��j�1; (65)and therefore ��j = jYl=0 (1 + CDl�l)�0: (66)Clearly the produ t Q1l=0 (1 + CDl�l) is onvergent, and by taking its loga-rithm, one easily he k that its limit is bounded by CD �(1��)2 . Therefore, weobtainkvj � evjk`1 = �j � CD �(1� �)2�0 = CD �(1� �)2kv0 � ev0k`1; (67)whi h proves our �rst laim sin e the onstant C of Lemma 2 depends ina ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g. For the se ond laim we note that�j � ��j � Cj�j ��j�1 � C2D �(1� �)2 j�jkv0 � ev0k`1 � C2�sjkv0 � ev0k`1;with the last onstant C depends in a ontinuous non-de reasing way onmaxfkv0k`1; kev0k`1g:We next address the stability in H�older norm Cs for 0 < s < 1.Theorem 4 Under the assumptions of Theorem 3, we havekS1v0 � S1ev0kCs < Ckv0 � ev0k`1; (68)for all s > 0 su h that s < � log (�1(S1))log 2 , where C depends in a ontinuousnon-de reasing way on maxfkv0k`1; kev0k`1g.23

Proof:Let � be su h that s < � log �log 2 < � log (�1(S1))log 2 , i.e. �1(S1) < � < 2�s < 1.Let us de�ne f = S1v0, ef = S1ev0, and F = f � ef . We also re all f j andef j de�ned by the interpolation of vj and evj a ording to (11), and we de�neF j = f j � ef j. As in the proof of Theorem 1, we an writekF j+1 � F jkL1 � Ck�vj ��evjk`1: (69)From Theorem 3, we thus obtainkF j+1 � F jkL1 � C2�sjkv0 � ev0k`1; (70)where C depends in a ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g:It follows that kF � F jkL1 � C2�sjkv0 � ev0k`1: (71)For jx� yj � 1 and j su h that 2�j�1 < jx� yj � 2�j,jF (x)� F (y)j � jF (x)� F j(x)j+ jF (y)� F j(y)j+ jF j(x)� F j(y)j� 2kF � F jkL1 + jF j(x)� F j(y)j� C2�sjkv0 � ev0k`1 + 2�jk(F j)0kL1� C2�sjkv0 � ev0k`1 + k�vj ��evjk`1� C2�sjkv0 � ev0k`1� Cjx� yjskv0 � ev0k`1;up to a multipli ative hange in C. This on ludes the proof.Finally, we address stability in the H�older norm Cs for s > 1.Theorem 5 Let S be a quasilinear subdivision rule whi h reprodu es polyno-mials up to degree N . Assume that S is ontinuously dependent of the data,and that �1(Sn+1) < 2�n for some n 2 f0; � � � ; Ng. We then havekS1v0 � S1ev0kCs < Ckv0 � ev0k`1; (72)for all s > 0 su h that s < � log (�1(Sn+1))log 2 , where C depends in a ontinuousnon-de reasing way on maxfkv0k`1; kev0k`1g.Proof:We shall use indu tion on n in a similar way as in the proof of Theorem 2.24

For n = 0, the result is proved by Theorem 4. For n = 1, we assume that�(S2) < 1=2. We de�ne f , ef , F , f j, ef j and F j as in the proof of Theorem4. We re all the sequen es wj := 2j�vj and ewj := 2j�evj, and the fun tionsgj := Pk2ZZwjk e'(2j � �k) and egj := Pk2ZZ ewjk e'(2j � �k).We already know from the proof of Theorem 2 that gj and egj uniformly onverge to g = f 0 and eg = ef 0. Therefore Gj := gj � egj onverges to G = F 0.Sin e s < � log (�1(eS2))log 2 , we obtain by similar arguments as in the proof ofTheorem 3 that k�wj �� ewjk`1 � C2(1�s)jkw0 � ew0k`1: (73)Note that, we use the fa t that, a ording to Remark 1, we also have theinequality kwj � ewjk`1 � kwj�1 � ewj�1k`1 +Dk�wj �� ewjk`1; (74)with onstant 1 for the �rst term. We then use the same type of argumentsas in the proof of Theorem 4 to derive thatjG(x)�G(y)j � Cjx� yjs�1kw0 � ew0k`1 � 2Cjx� yjs�1kv0 � ev0k`1;whi h gives the desired result. Iterating this argument for n > 1, we obtainthe general result.Appli ationIn this se tion we apply the results of the previous se tion to quasilinearsubdivision s hemes based on ENO and WENO interpolation te hniques inthe point values setting as des ribed in Example 1 of Se tion 2. Remark thatthe smoothness of the limit fun tions based on ENO interpolation te hniquesis inherently limited in the following sense: if the data v0k are su h that thesten il sele tion always avoids a singularity point on the oarse grid, then thelimit fun tion will not be di�erentiable at this point. Similarly, we annotexpe t ontinuity in the ENO ell-average setting.We treat here the parti ular ase of 4 point interpolation, i.e. M = 4.The asso iated s heme S1 is de�ned by a rule of the type(S1(v)�w)k := Xjk�2lj�4 bk;l(v)�wl; (75)25

where bk;l , are the oeÆ ients asso iated to the interval �jl := [(k�l)2�j; (k�l+1)2�j℄. In the parti ular ase of four point ENO interpolation, the di�er-en es are al ulated with one of the following rules:8>><>>: �vj+12k;0 := 1116�vjk � 14�vjk+1 + 116�vjk+2;�vj+12k;1 := 116�vjk�1 + 12�vjk � 116�vjk+1;�vj+12k;2 := � 116�vjk�2 + 14�vjk�1 + 516�vjk; (76)obtained respe tively from ea h ase of (2) . By symmetry, we an also writethe rule for the odd di�eren es8>><>>: �vj+12k+1;0 := 516�vjk + 14�vjk+1 � 116�vjk+2;�vj+12k+1;1 := � 116�vjk�1 + 12�vjk + 116�vjk+1;�vj+12k+1;2 := 116�vjk�2 � 14�vjk�1 + 1116�vjk; (77)These rules allow us to estimate the joint spe tral radius of S1, a ording tothe following result.Lemma 3 In the ase of ENO four point subdivision s heme, one hassupu;w2`1 kS1(u)S1(w)k`1 < 1 (78)and therefore �1(S1) < 1.Proof:Noti e �rst that the `1 norm of the operator de�ned in (76) and (77) satis�eskS1(v)k`1 = supk Xl jbk;l(v)j = 1116 + 14 + 116 = 1: (79)For �xed u; w 2 `1(ZZ), we have that�S1(u)S1(w)�k;l := Xk02ZZ (S1(u))k;k0(S1(w))k0;l: (80)and therefore kS1(u)S1(w)k`1 is estimated bysupkPl j(S1(u)S1(w))k;lj � supkPlPk0 j(S1(u))k;k0jj(S1(w))k0;lj� supkPk02S(k) �jbk;k0(u)jPl jbk0;l(w)j�;26

where S(k) is the sele ted sten ils for k. Sin e S(k) in ludes three onse utiveintegers, it always in lude a pair (2m; 2m + 1). From (76) and (77) , wenoti e that either Pl jb2m;l(w)j = 5=8 or Pl jb2m+1;l(w)j = 5=8. Therefore,there exist k0 2 S(k) su h thatXl jbk0;l(w)j = 5=8: (81)Sin e k0 2 S(k), we also have jbk;k0(u)j � 116 . It follows thatPk0 �jbk;k0(u)Pl jbk0;l(w)j� = 58 jbk;k0(u)j+Pk0 6=k0 jbk;k0(u)jPl jbk0;l(w)j� 58 jbk;k0(u)j+Pk0 6=k0 jbk;k0(u)j� 1 + (58 � 1)jbk;k0(u)j� 1� 38 116 = 125128 < 1:A more pre ise estimation of kS1(u)S1(w)k`1 an be obtained by an ex-pli it omputation for ea h di�erent sten il ombinations. This leads to thesharper bound �1(S1) � supu;w2`1 kS1(u)S1(w)k1=2`1 = 916p2: (82)As a onsequen e of Theorem 1 and (82) , we obtain the following smooth-ness result of the limit fun tion, in the parti ular ase of four point ENOinterpolation. subdivision.Theorem 6 In the ase of ENO four point interpolation, the limit fun -tion of the subdivision s heme is bounded and belongs to Cs for all s <� log( 916p2)log 2 � 0:6601499:We �nally turn to WENO interpolation de�ned in Se tion 1. The s hemeS1 is de�ned by a rule of the type(S1(v)�w)k := Xjk�2lj�6 bk;l(v)�wl: (83)The rule for the di�eren es has the form of a onvex ombination of the rules(76) , namely�vj2k := ��216 �vj�1k�2 + 4�2+�116 �vj�1k�1 + 11�0+8�1+5�216 �vj�1k+�4�0��116 �vj�1k+1 + �016�vj�1k+2; (84)27

By symmetry, we an also write the rule for the odd di�eren es�vj2k+1 := �216�vj�1k�2 + �4�2��116 �vj�1k�1 + 11�2+8�1+5�016 �vj�1k+4�0+�116 �vj�1k+1 � �016�vj�1k+2: (85)Note that in both formulas, �0, �1 and �2 vary with k. We then have thefollowing result for the joint spe tral radius of S1.Lemma 4 In the ase of WENO interpolation, one hassupu;w2`1 kS1(u)S1(w)k`1 < 1 (86)and therefore �1(S1) < 1.Proof:From (84) and (85) we have thatXl jb2k;l(v)j � �0 + 58(�1 + �2) � 1; (87)and Xl jb2k+1;l(v)j � �2 + 58(�1 + �0) � 1; (88)and therefore kS1(v)kl1 � 1. For �xed u; w 2 `1(ZZ), we have that�S1(u)S1(w)�k;l := Xk02ZZ (S1(u))k;k0(S1(w))k0;l: (89)We re all that kS1(u)S1(w)k`1 is estimated bysupkPl j(S1(u)S1(w))k;lj � supkPlPk0 j(S1(u))k;k0jj(S1(w))k0;lj� supkPk0 s:t: jk�2k0j�6 �jbk;k0(u)jPl jbk0;l(w)j�;We remark that the set fk0 s:t jk�2k0j � 6g in ludes �ve onse utive integers,and then it always in lude a quadruplet (2m; 2m + 1; 2m + 2; 2m + 3). Wethen again remark that one of the rules (84) or (85) for the di�eren es is ontra tive, sin e we haveXl jb2k;l(v)j+Xl jb2k+1;l(v)j = 54�1 + 138 (�0 + �2) � 138 < 2: (90)28

Consequently, there exists p and q in f0; 1g su h thatPl jb2m+p;l(w)j � 1316 < 1and Pl jb2m+2+q;l(w)j � 1316 < 1. We also derive from the rules (84) or (85)that we always haveminfjbk;2m+p(u)j; jbk;2m+2+q(u)jg � 1=16: (91)Therefore, there exist k0 su h that Pl jbk0;l(w)j � 1316 and jbk;k0(u)j � 1=16.It follows thatPk0 �jbk;k0(u)Pl jbk0;l(w)j� = 1316 jbk;k0(u)j+Pk0 6=k0 jbk;k0(u)jPl jbk0;l(w)j� 1316 jbk;k0(u)j+Pk0 6=k0 jbk;k0(u)j� 1 + (1316 � 1)jbk;k0(u)j� 1� 316 116 = 253256 < 1:A more pre ise estimation of kS1(u)S1(w)k`1 an be obtained by an ex-pli it omputation for ea h di�erent sten il ombinations. This leads to thesame sharper bound as in ENO ase�1(S1) � supu;w2`1 kS1(u)S1(w)k1=2`1 = 916p2: (92)As a onsequen e of Theorem 1 and (92) , we obtain the following smooth-ness result of the limit fun tion of the subdivision pro ess, based on WENOinterpolation:Theorem 7 In the ase of WENO interpolation, the limit fun tion of thesubdivision s heme is bounded and belongs to Cs for all s < � log( 916p2)log 2 �0:6601499:Although they are bounded, the nonlinear operators based on ENO te h-niques are unstable. The ENO te hniques use a numeri al riterion in thesele tion pro ess of the sten il. If the two terms in the numeri al riterionare lose to zero, then a small hange at the round o� level would hange thedire tion in the numeri al riterion and hen e the sten il. In this situation,there is no hope to have stability. In ontrast, WENO interpolation basedon the weights introdu ed in [7℄ is stable.29

Proposition 4 In the ase of WENO interpolation, the subdivision operatorgiven in (2) with the weights de�ned in (7) is ontinuous with respe t to thedata.Proof:Let u; eu 2 `1(ZZ). From the de�nition of the subdivision operator operatorwe have kS(u)� S(eu)k`1 = supk Xl jak;l(u)� ak;l(eu)j: (93)In the parti ular ase of WENO interpolation we obtainkS(u)� S(eu)k`1 � j�0 � e�0j+ j�1 � e�1j+ j�2 � e�2j:where �2; �1; �0; e�2; e�1; e�0 represent the weights of the left and of the rightsten il for u and eu. From the de�nition of the weights in x 2, we havej�i � e�ij = j aia0+a1+a2 � eaiea0+ea1+ea2 j� j ai�eaia0+a1+a2 j+ jeai( 1a0+a1+a2 � 1ea0+ea1+ea2 )j� 1a0+a1+a2 [2jai � eaij+Pj 6=i jaj � eajj℄;and therefore kS(u)� S(eu)k`1 � 4a0 + a1 + a2 Xi jai � eaij:From (7) we have that jbij � C0kuk2̀1 where C0 > 0; onstant independenton u and w. It follows thata0 + a1 + a2 = d0(�+b0)2 + d1(�+b1)2 + d2(�+b2)2� P2i=0 di�+C0kuk2̀1 = 3�+C0kuk2̀1 (94)Using straightforward omputations we also obtainjai � eaij = di 2�+bi+ebi(�+bi)2(�+ebi)2 jbi � ebij� 2di�3 jbi � ebij: (95)From (7) , we obtainjbi � ebij � C1ku+ euk`1ku� euk`1; (96)30

where C1 > 0; onstant independent on u and eu, and thereforeXi jai � eaij � 6C1�3 (kuk`1 + keuk`1)ku� euk`1: (97)Combining (94) and (97) , we therefore obtainkS(u)� S(eu)k`1 � [2C1�3 (kuk`1 + keuk`1)(�+ C0kuk2̀1)℄ku� euk`1; (98)whi h on ludes the proof.We an thus apply the results of x 4 to derive the following result.Theorem 8 In the ase of WENO four point interpolatory te hniques, de-�ned in (2) , with the weights satisfying (7) , the subdivision s heme is L1stable and Cs stable for all s < � log(�1(S1))log 2 � 0:6601499:AppendixWe shall brie y sket h some smoothness and stability results in the spa esLp and Bsp;q whi h generalize those obtained in x 3 and x 4. The Besov spa esBsp;q roughly represent the fun tions with s derivatives in Lp. They an bede�ned through the n-th order Lp modulus of of f ,!n(f; t)Lp = supjhj�t k�nhfkLp; (99)where �nhf is the usual n�th order �nite di�eren e operator�nhf = nXm=0 (�1)m�nm�f(�+ hm):For p; q � 1; s > 0; the Besov spa es Bsp;q onsists of the fun tions f 2 Lpsu h that �2sj!n(f; 2�j)Lp�j�0 2 `q: (100)Here n is an integer stri tly larger than s. A natural norm for su h a spa eis then given bykfkBsp;q := kfkLp + k�2sj!n(f; 2�j)Lp�j�0k`q :31

Remark 2 For q = 1, (100) simply means that k�nhfkLp � Chs. Inparti ular, one has Cs = Bs1;1 when s is not an integer. More generally,one has W s;p = Bsp;p if s is not an integer and Hs = W s;2 = Bs2;2 for all s.We an study the onvergen e of quasilinear subdivision s hemes in Lpa ording to the following natural de�nition.De�nition 7 A subdivision s heme is alled Lp onvergent if, for every �niteset of initial ontrol points v0 2 `p(ZZ), there exists a fun tion f 2 Lp, alledthe limit fun tion, su h that limj!1 kf j � fkLp = 0; (101)where f j is the fun tion de�ned in (11) .One easily he k that we havekf jkLp � 2�j=pkvjk`p: (102)Therefore, similar onvergen e and smoothness results an be obtained, basedon the `p study of the Sn. We assume boundedness of S in the `p sense whi hmeans that for all v 2 `p(ZZ), kS(v)k`p � B; (103)where kAk`p := supfkAwk`p ; kwk`p = 1g, and we de�ne the `p joint spe tralradius �p(S) := lim supj!1 sup(u0;���;uj�1)2(`p(ZZ))j kS(uj�1); : : : ; S(u0)k 1j̀p: (104)It an easily be he ked that Proposition 2 extends to the `p joint spe tralradius, i.e. �p(Sn+1) � �p(Sn)=2. Note that onvergen e of the subdivisions heme implies �p(S) � 21=p sin e otherwise S1v0 = 0 for all initial data v0in view of (102) . Therefore, the above result shows that we always have�p(Sn) � 21=p�n: (105)With su h de�nitions, we have the following results, similar to Theorem1 and Theorem 2 32

Theorem 9 Let S be a quasilinear subdivision rule whi h reprodu es on-stants. If �p(S1) < 2 1p then S is Lp- onvergent. Moreover, the limit fun tionf belong to Bsp;q for all s < � log(�p(S1))log 2 + 1=p.Proof:By similar arguments as in the proof of Theorem 1, we establish thatkf j+1 � f jkLp � C2�j=pk�vjk`p � C�j2�j=pk�v0k`p; (106)for � su h that �p(S1) < � < 2 1p , from whi h we obtain the Lp onvergen e off j to some f 2 Lp. If jhj � 1 and j is su h that 2�j�1 < jhj � 2�j, we havekf � f(�+ h)kLp � 2kf � f jkLp + kf j � f j(�+ h)kLp� C�j2�j=pk�v0k`p + 2�jk(f j)0kLp� C(�j2�j=pk�v0k`p + 2�j=pk�vjk`p)� C�j2�j=pk�v0k`p� Cjhjsk�v0k`p;with s = � log �log 2 + 1=p. Therefore f 2 Bsp;1 for all s < � log(�p(S1))log 2 + 1=p.Sin e Btp;1 � Bsp;q when t > s, it follows that we also have f 2 Bsp;q for alls < � log(�p(S1))log 2 + 1=p.Theorem 10 Let S be a quasilinear subdivision rule whi h reprodu es poly-nomials up to the order N . If �p(Sn+1) < 2 1p�n for some n � N , the limitfun tion f is in Bsp;q for all s < � log(�p(Sn+1))log 2 + 1=p.Proof:We use exa tly the same arguments as those used in the proof of Theorem2. For n = 0, the result is proved by Theorem 9. For n = 1, we re all thesequen e wj := 2j�vj and the fun tion gj := Pk2ZZ wjk e'(2j ��k):We get thatg := eS1�v0 belongs to Bsp;q for s < � log(�p(eS1))log 2 + 1=p and satis�es f 0 = g.Therefore f 2 Bsp;q for all s < � log �1(S2)log 2 . Iterating this argument for n > 1,we obtain the general result.33

We �nally want to generalize the stability results given in Theorem 3 andTheorem 4 to the Lp norm and Bsp;1 norm. A �rst possibility is to pro eedin a similar way as in the proof of these results, repla ing the assumptionson the spe tral radius of S1 or Sn in `1 by assumptions of their spe tralradius in `p similar to those in theorems 9 and 10, and to assume ontinuousdependen y with respe t to the data in the sense wherekS(vj)� S(evj)k`p � Ckf j � ef jkLp = C2�j=pkvj � evjk`p:However this last assumption is too restri tive in view of the fa tor 2�j=p. Inparti ular, it is not ful�lled by the WENO point value subdivision s heme.In the following, we show that Lp (resp. Bsp;1) stability an be obtained by ombining the L1 (resp. Cs) stability with the fa t that the subdivisions heme is lo al.Theorem 11 Let S be a quasilinear subdivision s heme whi h reprodu es onstants and whi h is ontinuously dependent on the data in the sense of(14) . Assume that �1(S1) < 1. Then we havekS1v0 � S1ev0kLp < Ckv0 � ev0k`p; (107)where C depends in a ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g.Proof:For all j > 0, we havekf j � ef jkpLp � 2�jkvj � evjkp̀p = 2�j Xk2ZZ kvj � evjkp̀p(ZZ\[2jk;2j(k+1))): (108)We also have2�jkvj � evjkp̀p(ZZ\[2jk;2j(k+1))) � kvj � evjkp̀1(ZZ\[2jk;2j(k+1))): (109)Using the L1 stability result established in Theorem 3, together with thefa t that our s heme is lo al, we obtain thatkvj � evjk`1(ZZ\[2jk;2j(k+1))) � Ckv0 � ev0k`1(ZZ\[k�2M;k+2M℄) (110)where C depends in a ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g.Elevating this last estimate to the power p and summing on k, we thus obtainfrom (108) that kf j � ef jkLp � Ckv0 � ev0k`p; (111)34

where C depends in a ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g:The laim follows by letting j tend to +1 in the above inequality.We �nally give a stability result in Besov norms.Theorem 12 Let S be a quasilinear subdivision rule whi h reprodu es poly-nomials up to the order N , whi h is ontinuously dependent of the data inthe sense of (14). Assume that �1(S1) < 1 and that for some n � N ,�p(Sn+1) < 21=p�n. Then we havekS1v0 � S1ev0kBsp;q < Ckv0 � ev0k`p; (112)for all s < � log(�p(Sn+1))log 2 + 1=p, where C depends in a ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g.Proof:For n = 0, we pro eed as in the proof of Theorem 4. Let � be su h thats < � log �log 2 + 1=p < � log (�p(S1))log 2 + 1=p, i.e. �p(S1) < � < 21=p�s < 21=p.Re alling F j := f j � ef j and its Lp limit F = f � ef , we �rst establishkF j+1 � F jkpLp � C2�jk�vj ��evjkp̀p; (113)where C depends in a ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g,by the same te hnique as in the proof of Theorem 1, In order to estimatethe right hand side, we use the same lo alization te hnique as in the proofof Theorem 10, i.e.2�jk�vj ��evjkp̀p = 2�jPk2ZZ k�vj ��evjkp̀p(ZZ\[2jk;2j(k+1)))� Pk2ZZ k�vj ��evjkp̀1(ZZ\[2jk;2j(k+1)))� Pk2ZZ k�vj ��evjkp̀1(ZZ\[2jk;2j(k+1)))� CPk2ZZ 2�spjkv0 � ev0kp̀1(ZZ\[k�2M;k+2M℄)� C2�spjkv0 � ev0kp̀p:In the third inequality, we have used the lo al version of the estimate k�vj��evjk`1 � 2�sjkv0 � ev0k`1 used in the proof of Theorem 4. It follows thatkF � F jkpLp � C2�sjkv0 � ev0k`p; (114)35

where C depends in a ontinuous non-de reasing way on maxfkv0k`1; kev0k`1g.For jhj � 1 and j su h that 2�j�1 < jhj � 2�j, we then writek�hFkLp � 4kF � F jkLp + k�hF jkLp� C2�sjkv0 � ev0k`p + 2�jk(F j)0kLp� C2�sjkv0 � ev0k`1 + 2�j=pk�vj ��evjk`p� C2�sjkv0 � ev0k`p� Cjhjskv0 � ev0k`p;whi h proves the result for q = 1 and therefore for all q sin e Btp;1 � Bsp;qwhen t > s. For n > 0 we use exa tly the same argument as in Theorem 5.The results of this Appendix an be applied to the Lp analysis of ENOand WENO subdivision s hemes in a similar way as in Se tion 5. We end thisAppendix with a smoothness result in the ell averages setting. We onsiderthe predi tion operator de�ned in Example 2 of Se tion 1. An estimationof kS1(u)S1(v)S1(w)k`1 an be obtained by an expli it omputation for ea hdi�erent sten il ombinations. This leads to the same bound for ENO andWENO interpolation:�1(S1) � supu;v;w2`1 kS1(u)S1(v)S1(w)k1=3`1 = 1:2365: (115)As a onsequen e of Theorem 9, the following result holds:Theorem 13 In the ase of three ell averages ENO interpolation and inthe ase of three ell averages WENO interpolation the quasilinear subdivisionoperator S is L1- onvergent. Moreover, in both situations, the limit fun tion,belong to Bs1;q for all s < � log(1:2365)log 2 + 1 � 0:69371.Referen es[1℄ Arandiga, F. and R. Donat (1999) A lass of nonlinear multis ale de- omposition, preprint, University of Valen ia, submitted to Numeri alAlgorithms. 36

[2℄ Amat, S., F. Arandiga, A. Cohen and R. Donat (1999) Tensor prod-u t multiresolution analysis with error ontrol, preprint, University ofValen ia, to appear in Signal Pro essing.[3℄ Amat, S., F. Arandiga, A. Cohen, R. Donat, G. Gar ia, and M. VonOehsen (1999) Data ompression with ENO S hemes: a ase study ,preprint, University of Valen ia, to appear in Appl. Comp. Harm. Anal.[4℄ Candes, E. and D. Donoho (1999), Ridgelets: a key to higher-dimensional intermitten y ?, Phil. Trans. Roy. So . to appear.[5℄ Candes, E. and D. Donoho (1999), A Surprisingly E�e tive Nonadap-tive Representation For Obje ts with Edges, To appear in Curves andSurfa es, L. L. S humaker et al. (eds), Vanderbilt University Press,Nashville, TN.[6℄ Cavaretta, A.S., W. Dahmen and C.A. Mi helli (1991), Stationary Sub-division, Memoirs of Amer. Math. So .,Volume 93.[7℄ Chan T., X.-D Liu and S. Osher (1994) Weighted essentially non-os illatory s hemes, Journal of Comput. Phys., 115:200-212.[8℄ Baraniuk, R., R.L. Claypoole, G. Davis, G and W. Sweldens (1997),Nonlinear Wavelet Transforms for Image Coding, Pro . 31st AsilomarConferen e.[9℄ Baraniuk, R., R.L. Claypoole, G. Davis, G and W. Sweldens (1999),Nonlinear Wavelet Transforms for Image Coding via Lifting s heme,submitted to IEEE Trans. on Image Pro essing.[10℄ Cohen, A. (1999) Wavelets in numeri al analysis, Handbook of Nu-meri al Analysis, vol. VII, P.G. Ciarlet and J.L. Lions, eds., Elsevier,Amsterdam.[11℄ Cohen, A. and R. Ryan(1995)Wavelets and multis ale signal pro essing,Chapman and Hall, London.[12℄ Cohen, A., W. Dahmen, I. Daube hies and R. DeVore (1999) Tree ap-proximation and optimal en oding, IGPM report 174, RWTH-Aa hen.37

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[25℄ Jiang G. and C.-W. Shu (1996) EÆ ient implementation of weightedENO s hemes, Journal of Comput. Phys., 126:202-228.[26℄ Mallat, S. (1998). A wavelet tour of signal pro essing, A ademi Press,New York.

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