Wavelet decompositions of non-refinable shift invariant spaces

Transcription

1 Wavelet ecopositions of non-refinable shift invariant spaces S. Deel an D. Leviatan Abstract: The otivation for this wor is a recently constructe faily of generators of shift-invariant spaces with certain optial approxiation properties, but which are not refinable in the classical sense. We try to see whether, once the classical refinability requireent is reove, it is still possible to construct eaningful wavelet ecopositions of ilates of the shift invariant space that are well suite for applications. AMS Subect classifications: 4C4, 65T6, 4A5, 4A44 Key wors: wavelets, non-stationary wavelets, shift-invariant spaces, approxiation orer. Authors affiliation: E-ail: School of Matheatical sciences Tel-Aviv University Tel-Aviv 69978, Israel S. Deel: shai.eel@turboiage.co D. Leviatan: leviatan@ath.tau.ac.il

2 Introuction In classical refinable wavelet theory ([Ch], [Da], [M] one begins with a finitely generate shift invariant (FSI space { } S Φ : = span Φ, Z, where Φ is a finite set an the closure is taen in soe Banach space X. Typically, S ( Φ is selecte to have approxiation orer N. This eans that for any h > an f X where h ( X g S( Φ E f, S Φ : = inf h f g Ch f, (. h { } S Φ : = span h Φ, Z, an is a sei-nor, easuring the soothness of the eleents of X. X To allow the construction of wavelets associate with S ( Φ, one assues that the shift invariant space is two-scale refinable, naely S S / X X Φ Φ. (. One then selects a copleentary set of generators, so calle wavelets, Ψ so that / It is easy to see that (.3 can be ilate to any given scale J Z that is, Assue J f S J Φ so that J J J Φ fφ fφ fψ plays the role of a low resolution approxiation to etail. Typically, if f S Φ = S Φ + S Ψ. (.3 J J+ J+ S Φ = S Φ + S Ψ. J + J J = +, where fφ S( Φ, f S J f Φ, while f J + Ψ. Then, J Ψ fφ is the ifference between the two, the J Ψ J J f Φ is a sufficiently sooth function or J is sufficiently large, then fφ J Ψ. Uner certain conitions (.3 leas to a wavelet ecoposition i.e., any J f S J Φ Φ possesses a ecoposition J J + J + J + 3 f an S Φ = S Ψ + S Ψ + S Ψ +, (.4 J Φ f = f + f + f +. (.5 J J J J3 Φ Ψ Ψ Ψ

3 In applications, FSI spaces are use as follows. Let f be soe signal that one wishes to approxiate. Using property (., one chooses a fine enough scale J Z an coputes an approxiation J f fφ S J Φ. (.6 J In soe applications there is no nee to further ecopose the approxiation f Φ into the wavelet su (.5. Typical exaples are curve an surface (linear approxiations in CAGD or re-sapling in iage processing. However, the wavelet ecoposition (.4 is effective in applications that require a copact representation of the signal such as copression, enoising, segentation, etc. S Φ be a non-refinable FSI space. Naely, S S / Φ Φ. There are any exaples of non-refinable FSI spaces that perfor well in approxiations of type (.6. In fact, there is an interesting recent construction [BTU] of shift invariant spaces that are optial in soe approxiation theoretical sense an are not two-scale refinable. Nevertheless, we woul still lie to ecopose the space S Let J Φ into a su of ifference (wavelet spaces in the sense of (.4 (see [CSW] for a ifferent approach. Since our FSI space is not refinable we nee to replace S ( Φ by a ifferent space play the role of a low resolution space an a (wavelet space S ( Ψ to serve as a ifference space in a ecoposition siilar to (.3, naely, / S( Φ = S( Φ + S( Ψ. In this wor we show that such eaningful ecoposition techniques exist. They allow us, to further ecopose S / S S Φ = Φ + Ψ an so on an to obtain a non-stationary wavelet ecoposition siilar to (.4, i.e., J J+ J+ J+ 3 3 S Φ = S Ψ + S Ψ + S Ψ + Thus, the (non-stationary sequence { Φ } is a eans to obtain the non-stationary wavelet sequence { Ψ }. The sequence { Φ } is also use to eterine the (linear approxiation properties of the S Φ to wavelets. It is interesting to note that our techniques enable us to recover the stationary choice Φ =Φ, Ψ =Ψ, whenever S ( Φ is two-scale refinable an S( Φ / = S( Φ + S( Ψ. Another interesting question aresse in this wor is the following. Let S ( Φ be an optial non-refinable FSI space uner soe approxiation theoretical gauge. Obviously, if S ( Φ has an optial approxiation property, no constructe S / S then ass how close are the approxiation properties of Φ Φ can inherit this exact property. One S Φ to those of S ( Φ? Another question is the following. In what way (if any are wavelets that ecopose ilations of optial non-refinable FSI spaces better than nown existing wavelets? In Section we present the basic theory on the structure of shift invariant spaces which serves as fraewor throughout the wor. We also present soe new regularity results that are require for the 3

4 wavelet constructions in Section 3.. In section 3 we construct non-stationary wavelet ecopositions of shift invariant spaces which are not require to be two-scale refinable. There are two such constructions. The Superfunction wavelet construction escribe in Section 3. is inspire by the superfunction theory of [BDR], [BDR], [BDR3]. In Section 3. we introuce Cascae wavelets. Their construction exploits properties of the Cascae operator (see for exaple [Da]. In Section 4 we first present results on approxiation fro shift invariant spaces. We then procee to ustify the constructions of Section 3, by showing that our non-stationary sequence { Φ } inherits the approxiation properties of the ecopose non-refinable shift invariant space. Consequently, the non-stationary wavelet sequence { Ψ } span etail spaces an are therefore suitable for signal processing applications. Shift invariant spaces Shift invariant spaces are a special case of invariant subspaces in Banach spaces. Here we use the fraewor of [BDR] an present results that are require for the constructions in Section 3. Definition. For any Z we enote the linear shift operator Definition. Let V be a close subspace of p (SI space if it is invariant uner the operators { } V S : span{ (, } S by S ( f : f ( =. L R, p. We say that V is a shift invariant S Z. We say that a set Φ generates V if = Φ = Φ Z. We say that V is a finite shift invariant (FSI space, if there exists a finite generating set Φ, Φ= n, such that V = S( Φ. In such a case we say that V is of length n. We enote len( V : = in Φ V = S( Φ. An SI space V is calle a principal shift invariant (PSI space if len( V =. { } To approxiate functions with arbitrary precision one uses ilates of shift invariant spaces. For a h given subspace V an h R + we enote by V the ilate space { } h V : = / h V. We note that is if S ( ϕ is a PSI space, then for, S ( ϕ is a FSI space of length. We now restrict our iscussion to L R. It is well nown that Fourier techniques appear naturally in the analysis of SI spaces. The following is siple characterization of SI spaces in the Fourier oain. Lea.3 [BDR] Let are equivalent:. f S Φ. S Φ be an FSI subspace of L ( R an let f L R. Then the following 4

5 . There exist T perioic functions { } τ such that f ˆ = τ ˆ. We see that we can regar the generators of an FSI space as vectors spanning a finite iensional vector space, with perioic functions playing the role of coefficients in the representations. Thus, we turn to Fourier base techniques. For each f L Φ R we enote ( ˆ( π fˆ : = f w+ w, The bracet operator [ ]: L L L Z R R T is efine by w T. fˆ, gˆ ( w : = fˆ, ˆ w g w l ( Z, w T. It is easy to see that the Fourier expansion of f ˆ, gˆ is fˆ, gˆ w f, g + e L( R Z Observe that if f, g are copactly supporte, then the bracet so we have an equality in (.. For f L ( R the function [ ] iw, (. fˆ, gˆ is a trigonoetric polynoial an f, f L T is calle the autocorrelation of f. Auto-correlations play a aor role in our analysis. They are use in the efinitions of stability constants, error ernels an fine error estiation constants. Our analysis requires the following siple result on the convergence of auto-correlations. Lea.4 Assue that L ( R Then for any we have the convergence Proof It is easy to see that we also have such that supp, supp ( L Ω where Ω is a boune oain. ˆ ˆ( ˆ, ˆ,. (. C( T R for any. By virtue of (. we have that ˆ ˆ(,, ( ˆ, ˆ are trigonoetric polynoials of uniforly boune egree. Therefore, the convergence of the Fourier coefficients ( ˆ, ˆ,, ˆ, ˆ = =, iplies the convergence (.. 5

6 We now procee to present regularity results for shift invariant spaces in L R. The otivation for woring with regular shift invariant spaces coes fro applications where it is require to have a stable representation or approxiation of signals. Stability iplies that sall changes in the input function o not change uch the representation an sall changes in the representation change the reconstructe function only a little. We begin with efinitions an notions fro [BDR]. Let S ( Φ be an SI space. The range function associate with The spectru of S ( Φ is efine by or equivalently S : ˆ { w } S Φ is J w = span Φ (.3 { } S σ S Φ : = w T ij w >, (.4 { } ˆ ˆ σs Φ : = w T, w, for soe Φ. It can be shown ([BDR] that the range an spectru of an SI space are invariants of the space. In particular they o not epen on the generating set. If i J S ( w const a.e. we say that S is regular. Observe that regularity iplies a full spectru. In the other irection, a full spectru iplies regularity only in the PSI case. We say that Φ is a basis for S if for each f S τ where f ˆ τ ˆ τ are uniquely eterine. Observe that if T \ σ S ( Φ is of positive easure = an Φ Φ there are perioic functions then S ( Φ oes not have a basis. The set Φ is calle a stable generating set or a stable basis (for its span if there exist constants A B < < such that for every c = { c, } l ( Φ Z, Z Φ, Z Ac c Bc l Φ, Z L( R Φ l Φ ( Z. (.5 It can be shown that a stable basis is inee a basis. Since stable bases are necessary for applications, the next result leas towars the construction of regular spaces. Theore.5 [BDR] Let S ( Φ be an FSI space. Then generating set. Furtherore, an FSI space is regular if an only if it is the orthogonal su of len( S( Φ regular PSI spaces. S Φ is regular if an only if it contains a stable We recall the connection between the efinition of stability (.5 an the notion of the range function (.3 for the siple case of PSI spaces (see [RS] Theore.3.6 for the general case of FSI spaces. Theore.6 [Me] A function L A ˆ, ˆ B, a.e. R is stable iff there exist < A B< such that 6

7 Assue that we have constructe a non regular FSI subspace S ( Φ of a regular FSI space S ( Φ n so that len( S( Φ = < n= len( S( Φ n. We can certainly efine S ( Ψ as the orthogonal copleent of S ( Φ in S Φ S Ψ = S Φ. n S Φ such that But the ecoposition will have two unesirable features. First, there is no choice of generators Φ, Ψ so that S Φ = S Φ, S Ψ = S Ψ an Φ Ψ is stable. Seconly, the ecoposition ay be {, } soewhat reunant, naely, len( S n S ( Φ such that S( Φ S( Φ S( Φ n, len( S( Φ = an S ( Ψ >. We will show that this can be fixe by constructing Φ is regular. In oing so we ensure that the orthogonal copleent is also regular an of length n. Hence, such a correction can prouce a stable an efficient ecoposition of S ( Φ n. Lea.7 Let S ( Φ be a regular FSI space an let S ( Φ. Then there exists ϕ S S( S( ϕ an S ( ϕ is a regular PSI subspace of S ( Φ. n Φ, such that Proof If S ( is regular, we are one. Otherwise, by Corollary 3.3 in [BDR], we ay assue the n ecoposition S( Φ = S( so that each i= i orthonoral basis for S ( i. Therefore there exists a unique representation ˆ, ˆ w = δ functions. Since, supp n = i. Define ϕ S i= σs τ S is a (regular PSI subspace an the shifts of i are an i n ˆ = ˆ τ with i i τ i perioic i = for, n we have that [ ˆ, ˆ] = τ an so i= Φ by ϕˆ τ ˆ τˆ n = +, τ ( w i i = τ ( w n i= i ( w T \ σs, else. Then [ ϕϕ ˆ, ˆ] = τ + τ an we can conclue the following. The space n i= i = supp ([ ˆ, ˆ] = supp( supp( i σs ϕ ϕϕ τ τ = n i= S ϕ is regular since n ( T \ σs ( supp ( τ supp ( τi i= 7

8 = ( T \ σs( σs( = T. Finally, ˆ χ ˆ σs( ϕ = iplies that S( S( ϕ. Lea.8 Let VU, be FSI spaces where V U. Then len( V len( U. Proof This is a irect consequence of the fact that the shift an orthogonal proection into an SI space P generate V. coute. This iplies that if Φ= { } generate U, then { } i Theore.9 Let U be a regular FSI. Then for any FSI subspace S regular subspace S ( Φ of length such that ( V S Φ S Φ U. i Φ U of length there exists a Proof The proof is essentially a Gra-Schit type construction, where we construct the correction ( S Φ as an orthogonal su of regular PSI spaces. We use inuction on the length Φ =. The case = follows by virtue of Lea.7. Assue the clai is true for <. Denote Φ = {,, }, where Φ {,, = }. Then by the inuction hypothesis there exists a regular FSI subspace S ( Φ such that S( Φ S( Φ U, an len S( Φ =Φ =. By [BDR] the orthogonal copleent in U of S ( Φ by W, is a regular FSI space. Let S( ψ : = PW S(. Observe that ( S Φ S Φ which by Lea.8 contraicts len( S( woul iply.7, we can fin a regular PSI space S ( such that, enote S ψ is not trivial since this Φ =. Using again Lea S S W. ( ψ ( Since by Theore.5 the orthogonal su of two regular FSI spaces is regular, we have that S ( Φ =Φ is a regular FSI subspace of U. To conclue, observe that S ( require properties of inial length, len( S( Φ =Φ = an that S( Φ S ( Φ. Next we iscuss the special structure of the orthogonal proection into SI spaces. Φ, Φ also possesses the 8

9 Lea. [BDR] Let Φ be a basis for an FSI space proection PS ( Φ f is given by where G ( Φ ˆ : = ˆ ( ψ, ˆ an Gˆ ψ, ( fˆ Φ ( f ˆ, ψ ˆ. ψ Φ S Φ an let f L et Gˆ ( fˆ PS f = Φ et G ( ˆ Φ Φ is obtaine fro ( ˆ R. Then the orthogonal ˆ, (.6 G Φ by replacing the -th row with In the PSI case the forula for the orthogonal proection (.6 leas to the efinition of the natural ual. For any L R, the natural ual is efine by its Fourier transfor where we interpret /=. ˆ ˆ : =, (.7 ˆ, ˆ Equation (.6 iplies that in the PSI case ˆ P f = fˆ, ˆ S(. Transforing this bac to the tie oain we obtain the well nown quasi-interpolation representation for the orthogonal proection, naely, P f = f, S. (.8 Z An FSI space V is calle local if there exist a finite set of copactly supporte functions, Φ, such that V = S( Φ. In applications copactly supporte generators are frequently use to iniize the tie an space coplexities of the algoriths. An exaple is Daubechies [Da] construction of copactly supporte orthonoral wavelets. Observe that a local FSI is always regular ([BDR]. We require the following result on the special case of orthogonal proections of local SI spaces into local SI spaces. Theore. Let VU, be local FSI spaces. Then the orthogonal proection of V into U is a local FSI subspace. In particular it is a regular FSI space. Proof Let U = S( Φ, V S = Ψ be so that ΦΨ, are copactly supporte generating sets for UV, respectively. Using the coutativity of the orthogonal proection into an SI space an the shift operator, PV = PS Ψ = S P Ψ. Thus, it suffices to prove that for each ψ Ψ, there exists a we have that U U ( U copactly supporte function ψ U, such that ( ψ ( ψ S = S P U. By virtue of (.6 we have et G ˆ ( ψˆ P ψ U = ˆ. (.9 Φ et G ( Φˆ 9

10 Since the set Φ is copose of copactly supporte functions, it follows fro (. that the eleents of ˆ et G Φ ˆ is also a trigonoetric polynoial so the Graian G( Φ are trigonoetric polynoials. Thus, that ( ˆ etg Φ a.e. on T. Let ψ S( P U ψ be efine by its Fourier transfor, ψ : et G( ˆ P Uψ = Φ. Then the constructe generator ψ has the require copact support property. Inee, fro (.9 we have the representation ψ = ˆ ( ψˆ ˆ where each ( ˆ Φ et G etg ˆ ψ is a trigonoetric polynoial. This eans that ψ is a finite su of copactly supporte functions hence it is copactly supporte. To conclue we observe that since ( ˆ G Φ a.e., we have that P ( et Uψ = G Φ ψ, thus S( ψ S( P U ψ et =. The following theore is the ain result of this section. It provies eaningful ecopositions of FSI spaces with goo approxiation properties to an orthogonal su of two FSI subspace. Naturally, there are any ways to represent FSI spaces as a su of two FSI subspaces. But our construction is such that the first subspace inherits the goo approxiation properties of the ecopose space, so that the secon subspace is a ifference (wavelet space. The ey to the construction is the use of an auxiliary reference space. The unerlying principal which ustifies this approach is superfunction theory [BDR] an is elaborate upon in Section 4. Theore. Let U be a (local regular FSI space of length lu. Let V be a (local FSI space of length lv < lu. Then U can be ecopose U = U W such that:. U is a (local regular FSI space of length lu = l. V. W is a (local regular FSI space of length lw = l U l V. 3. W Proof V.. Let U = PV. Note that U U is an FSI subspace of U with ( in (, len U l l = l. Without loss U V V of generality, U is regular, otherwise, by virtue of Theore.9, we can replace it by a regular subspace of U, containing U an of the sae length, which we will continue to call U. Observe that in the local case, Theore. iplies that U is local.. Since U is (local regular, by (Theore 3.38 Theore 3.3 in [BDR] its orthogonal copleent in U enote by W is (local regular an of length l l l ( ψ S ψ,, i is (local regular for i lw. Let W = S( ψ ψ l W U V generating set with that property. Define W: S( ψ,, ψ lw W V.,, where W. By Theore.5 it is always possible to fin a = where lw = l U l V. Then clearly

11 3. We conclue the construction by setting U to be the orthogonal copleent of W in U. By (Theore 3.38 Theore 3.3 in [BDR], U is a (local regular subspace of U of length l = l l = l. U U W V Exaple.3. Let ψ, be any nown pair of univariate sei-orthogonal scaling function an wavelet, e.g., B- splines an B-wavelets [Ch]. Define U = S( / an V = S(. Then, since S( S( / above construction recovers the (refinable ecoposition, the ( ( ψ ( / S S = S. (.. Let S ( be a univariate regular PSI space that is not refinable. Assue that provies L approxiation orer. Select U = S( /, ecoposition ( ( ψ ( / S S S V = S. Then the above construction fins a =, S( ψ S(, which in soe sense iics the refinable ecoposition (.. Furtherore, we show in Section 4. that inherits the approxiation orer fro while the wavelet ψ has vanishing oents. 3 Non-stationary wavelets Our first results are siple oifications of the classical sybol approach to wavelet construction for the non-refinable setting. Assue S ( ϕ / where ϕ L ( R is stable. Define the sybol iw P( w : = pe, where = pϕ(. (3. Z To ustify the pointwise valiity of (3. an resolve technical ifficulties concerning convergence, we f L T is in the Wiener Z require that these sybols be taen fro the Wiener algebra. Naely, Algebra ( f W if its Fourier coefficients are in l ( Z. The following partitioning of the lattice Z, nown to be useful in the analysis of refinable functions, is also useful in our non-refinable setting ( e Z = + Z, { } e E E : =,. (3. We begin with a stability lea (see [Ch] Theore 5.6 for the univariate case.

12 Lea 3. Let S ( ϕ / have a sybol P W such that P( w+ π e >, e E w T, an assue that L S. ϕ R is stable. Then is a stable generator for Proof The proof for the univariate case can be foun in [Ch] Theore 5.6. To obtain the proof for the ultivariate case one uses the lattice (3.. We observe that the following result, which is well nown for the refinable case = ϕ, is still vali for the ore general case. Theore 3. Let ϕ L ( R be a basis for S ( ϕ an let ψ, S ( ϕ / are the sybols of ψ, respectively. A necessary an sufficient conition for {, } S ( ϕ / is ( π ( π. Assue PQ W, where PQ, ψ to be a basis for w PQ, : = P w Q w+ P w+ Qw, w T. (3.3 Furtherore, if ϕ is stable, then an ψ are stable bases of S ( an S ( ψ, respectively. Proof The proof basically follows the etho of [Ch] Theore 5.6 with the observation that refinability ( = ϕ is not require. Next we iscuss the special case of a ecoposition S( ϕ / S( S( ψ orthogonality constraint S( S( ψ. Definition 3.3 Let ϕ L ( R an ψ, S ( ϕ /. In case S( S( ψ S( ϕ / ecoposition sei-orthogonal an ψ, a sei-orthogonal pair. = +, with the aitional =, we call the, but the shifts of, respectively ψ, are not necessarily orthogonal to each other. Assue has a two-scale sybol P W so that w w ˆ ( w = P ϕˆ. Note that the ter sei-orthogonality coes fro the fact that S( S( ψ Recall that the natural ual (see (.7 can be use to copute the orthogonal proection into S (. For the ual we also have the following ual two-scale relation ˆ [ ] [ ] [ ϕϕ ˆ, ˆ]( ˆ ˆ ˆ, ˆ ˆ, ˆ ˆ, ˆ [ ] ˆ = = P ϕ = P ϕ.

13 Hence ˆ = G ( ϕˆ ( Denoting where G : = [ ϕϕ ˆ, ˆ] [ ˆ, ˆ]( P. (3.4 G: = G, (3.5 it is easy to see that we have the uality relation ( π ( π P w G w + P w+ G w+. (3.6 Equippe with the notion of the ual sybol, we now characterize the univariate sei-orthogonal (wavelet copleent of a given generator in a space of type / S ϕ. Theore 3.4 Let S ( ϕ / with a two-scale sybol P W, where ϕ an are stable. Assue further that G W, where G is efine by (3.5. Then, ψ S ( ϕ / copleent such that S( ϕ / S( S( ψ is a stable sei-orthogonal = with a two-scale sybol Q W if an only if where K W oes not vanish on T. iw ( π ( Q w = e G w+ K w, (3.7 Proof The proof is siilar to [Ch] Theore 5.9. Using the above we can always copleent any generator by a sei-orthogonal counterpart. In particular, in the case of local spaces, this gives us a etho to construct a (inial copactly supporte generator, as one in [Ch], by a proper selection of the perioic function K. Naely, assue ϕ, are stable an copactly supporte an that the sybol P of is a trigonoetric polynoial. By (3.4, the choice K [ ˆ, ˆ] = in (3.7 leas to the following two-scale sybol iw [ ϕϕ ˆ, ˆ]( π ( π Q w = e w+ P w+. (3.8 It is easy to see that for copactly supporte ϕ,, the above sybol prouces a copleentary copactly supporte wavelet. We conclue this section with the following observation. Let ϕ be stable an two-scale refinable such that S( ϕ / S( ϕ S( ψ = + is a ecoposition where PQ, are the corresponing sybols of ϕψ., In iage coing applications perfect reconstruction subban filters bans erive fro the sybols, PQ are use in iscrete settings (see Section 7.3. in [M]. In any applications, one is not require to unerstan wavelet theory but siply to ipleent an efficient iscrete filtering process. Furtherore, coputational steps, that see necessary accoring to sapling theory, are orinarily neglecte (see the 3

14 iscussion in [M] pp , but still goo coing results are obtaine. How can one explain this phenoenon? A plausible explanation can be given using the results of this section. As is well nown in the signal processing counity, the perfect reconstruction ecoposition conition (3.3 is a property of the sybols PQ, an oes not epen on the generator ϕ. Assue that conition (3.3 hols for the two-scale sybols PQ, an replace the generator ϕ by soe other stable generator which nee not be refinable. Then, by Theore 3., the functions, ψ S ( / that have PQ, as their two-scale sybols are a basis for S ( /. This eans that (3.3 is a universal property of the two-scale sybols PQ, an the subban filters erive fro the, regarless of the unerlying functions. Furtherore, we will see in Section 4.3 that if in aition, the sybols PQ, have certain approxiation properties, then the corresponing basis {, } theory, whenever S ( has goo approxiation properties. ψ provies a ecoposition which is eaningful in the context of wavelet 3. Non-stationary Superfunction wavelets In this section we present the construction of non-stationary wavelets inspire by the superfunction techniques of [BDR]. In our case the proection is one fro a stationary reference space, but the superfunction an wavelet spaces are non-stationary. The abstract ecoposition of Theore. alreay tells us that, given a reasonable FSI space U, we can ecopose it into U = U W using a reference space V, with len( V < len( U, such that W V an U, V are of the sae length. The heuristics of the superfunction ecopositions presente in this section is ustifie in Section 4. where the approxiation properties of the ecoposition subspaces are iscusse in etail. Theore 3.5 Let U L R be a (local regular FSI space. Let V be a (local FSI space with len( V = len( U. Then there exists a sequence of subspaces U,. U an /. U W = U. 3. W V. W are (local regular FSI spaces with len( U len( U W, such that =, len( W ( len( U =. Proof Since ilation by,, preserves the property of (localness regularity, regular FSI of length len( U. By Theore., len( U len( V len( U W V U is a (local / / U can be ecopose into U = U W where = =, an such that U, W are (local regular. We now continue an / ecopose U in the sae anner. By repeate ecoposition we obtain an half-ultiresolution with the require properties. Corollary 3.6 Let U L R be a (local regular FSI space. Let V be a (local FSI space with len( V = len( U. Then for any scale J Z we have the following foral wavelet ecoposition / 4

15 where : U J J WJ = W = S Ψ V, are non-stationary (local regular wavelet spaces. =, (3.9 Clearly, the fact that we construct only half-ultiresolutions is not a real restriction. By ilating the construction to any given (fine scale, it can be use to approxiate any function in L R at any require level of accuracy. Also, since we have ensure that each wavelet space W is regular, by [BDR] Corollary 3.3, one ay select for each an orthonoral wavelet basis for W. Fro the orthogonality W W for, any selection of orthonoral bases Ψ for W (with the appropriate noralization provies an orthonoral basis for U, J Z. Next we iscuss actual constructions that realize the ecoposition of Theore 3.5. There are two strategies we can eploy. First, we can follow the etho of Theore. by constructing the superfunction spaces U using proection an then copleenting the by the wavelet spaces W. The secon approach is to construct the wavelet space first using ethos ostly applie for wavelet constructions over (ultivariate non-unifor gris (see [LM], [LMQ]. Let ϕ, L ( R such that supp ( ϕ, supp, ϕ ϕ J, with, N. We wish to fin copactly supporte generators ψ, so that S( ϕ / = S( S( ψ an S( ψ S( wavelet ψ. Assue supp( ψ [, y], y N. Since ψ S ( ϕ / yϕ unnowns { q } = where The assuption that supp ψ yϕ. We begin with the construction of the, we nee to copute y + qϕ(. = =, iplies the following y+ constraints ψ, =, =,, y. In orer to have a non-trivial solution, the nuber of constraints ust be strictly saller than the nuber of unnowns. Thus, y+ + y ϕ +. nuber of orthgonality constraints nuber of unnowns The sallest possible value y = + leas to the following efinition for ψ (up to a ultiplicative constant ϕ ϕ 5

16 ψ ( x = et, ϕ, ϕ + ϕ, ϕ, ϕ + ϕ, ϕ, ϕ + ϕ + ϕ + ϕ ϕ ( x ϕ ( x + ϕ, where we have enote : = (, ϕ : = ϕ(. We see that q efine by the Gra atrix Thus, we obtain the following result. Theore 3.7 Let ϕ, L ( R where, with supp Assue that the sequence { } ϕ+ is ϕ = where the inor + ϕ : = et Gra. (3. ϕ ϕ ϕ+ ϕ + ϕ ϕ+ ϕ, ϕ, supp = efine by (3. is not ientically zero. Then for qϕ(, ϕ q = we have that S( ψ S( an ψ : = =,, ϕ, N. supp ψ +. ϕ Exaple 3.8. Let ϕ = = N, where N is the univariate B-spline of orer. Then (see [LM] the B-splines fulfill the conitions of Theore 3.7. Since supp ( N =, we recover the result of Chui that the support of the B-wavelet (inially supporte sei-orthogonal wavelet is of size.. Let ϕ = = OM 4 where OM4: = N4 + N 4 /4. This generator, constructe in [BTU], has optial approxiation properties, but is not two-scale refinable (see Exaple 4.6. Then, ψ S( OM / efine by ψ = qom 4( with { } = 4 q given (up to a ultiplicative constant by the table below, is stable an fulfills the orthogonality conition S S( OM ψ. 4 q, , , , ,

17 Even before the analysis of approxiation properties is presente, it is easy to see that ψ has all the require properties of a wavelet: q oscillate in sign. The coefficients { } The coefficients { q } as high pass filters have four vanishing oents. The function ψ has four vanishing oents. q or ψ is closer to zero than the corresponing one of the cubic B-spline wavelet with the sae support size. In fact, with the right noralizations, the fifth (non vanishing oent of { } Still, accoring to our theory, the wavelet ψ constructe in Exaple 3.8 is only the first wavelet in a series of non-stationary wavelets that ust be constructe if one wishes to ecopose spaces of the type J S OM 4. The next wavelets in the sequence ψ, ψ 3, still have four vanishing oents an as we shall see, their fifth oent reains closer to zero than the fifth oent of the cubic wavelet. In such exaples, the price pai for reoving the refinability property is that the support of the constructe wavelets ight grow. Once the wavelet ψ is constructe, one ay construct a copleentary superfunction as follows. Assue supp( ψ + such that S( ψ S( ϕ. Now we assue the conitions of Theore 3.7 again, this tie allowing ψ to play the role of the reference generator. This leas to the construction of a generator S ( ϕ / with S( S( ψ S( ϕ / = an supp = +. ϕ ψ ϕ ϕ ϕ Observe that S( ψ S( iplies P /S( S( S( ϕ PSI space, Corollary.6 in [BDR] iplies that S( P /S( 3. Non-stationary Cascae wavelets. Since by Theore. =. S ϕ P S /S ( ϕ is a local It is well nown that the cascae operator can be use to obtain a refinable function corresponing to a subivision schee, or equivalently, a solution of a two-scale functional equation. Given a as P = { p}, we efine the cascae operator C by Z C f : = p f Z. R one iterates = C +. For our construction we require the general results of [R] on the cascae operator. We have an initial generator, possibly not refinable, but with goo approxiation properties. We woul lie to Starting with an initial function L p ecopose the space S ( J, corresponing to a certain scale J, into a su of eaningful wavelet 7

18 subspaces. By carefully choosing an appropriate cascae operator an applying it to, we obtain a sequence of generators =C such that:. The sequence { } converges in soe (or all p -etrics to a refinable function which is a fixe point of the operator C.. The spaces S ( satisfy a nesting property, i.e., S( / S. { } Such a cascae sequence can be use to construct a wavelet type ecoposition of the space S J length in the following way. First we construct for each level so that / foun we can (forally ecopose The orthogonality S S S Ψ of a copleent FSI space ( S S Ψ = S. Once such a non-stationary sequence of spaces is S Ψ Ψ for Inee, we will construct wavelet generators J S = J+ = Ψ. siplifies the construction of a stable basis for S ( Ψ that are a stable basis for ( J S Ψ with stability constants A, B which are uniforly boune fro below an above, i.e., < A A B B. Then, fro the orthogonality S S Ψ Ψ, we can ieiately erive that their union is a stable basis for S ( with stability constants boune fro below an above, respectively, by AB., The following is a siple for of Theore 3..8 in [R]. Theore 3.9 [R] Let W p R be a two-scale refinable an stable generator for. J S. Denote by C : = C the corresponing cascae operator. Let g be a boune stable copactly supporte function for which ˆ ˆ n g O = near the origin. If the shifts of g provie approxiation orer, then the cascae algorith converges at the rate { } in, n C g A L g. p( R We see that by a careful selection of the unerlying refinable function we not only ensure convergence of the cascae process, but we can also estiate the convergence rate. For exaple, a typical application = I+ D N be a stable of Theore 3.9 in our setting for the univariate case is as follows. Let generator where N is the B-spline of orer an D is soe hoogeneous ifferential operator of. Select the cascae operator C egree n. Then, near the origin we have ( N ( w Cw. As we shall see, provies the sae approxiation orer as the conitions of Theore 3.9 are satisfie. N N an therefore, 8

19 In contrast to the convergence acceleration sought in [R] using a sart choice of initial see, in our settings there are cases where slow convergence is preferable. As we shall see in Section 4.3, this is the case whenever the initial function has better properties then the liit function. In such a case the first few levels of the cascae process have properties that are close to the properties of the initial function. This is useful in applications, since in practice only the first levels of the cascae are use. Definition 3. Let be an initial function for the cascae process C efine by a refinable. Let =C an assue li L ( R basis for S ( / a Cascae Wavelet sequence.. We call any sequence { } Ψ such that { +, + } Ψ is a For the rest of the section we assue that the ass of the cascae operators are finitely supporte, hence also the corresponing refinable function. We now show that the cascae process interpolates the stability of the enpoints,. Theore 3. Let L operator associate with a stable L R be a stable copactly supporte initial function an let C be a cascae unifor stability constants < A B< R. If li = where : L( R =C, then there exist such that A ˆ, ˆ B for all. Proof For, let A, B be in/ax values of ˆ, ˆ. Since the Cascae as is finitely supporte, by Lea.4 we have the convergence A A, B B where AB, are the in/ax values of ˆ, ˆ. Thus, we nee only prove that each A >. iw To this en, let P( w = pe be the trigonoetric polynoial corresponing to the finite Z as of the cascae operator C. Since is stable, we have P( w+ πe >, e E w T, (3. where we have use the lattice ecoposition (3.. Inee, otherwise P( w πe soe w T. Then, by the refinability of ( w P( w e ˆ ˆ ( w e e E + =, e E, for ˆ, ˆ = + π, + π =. Since is copactly supporte, ˆ, ˆ is a trigonoetric polynoial an by Theore.6, this contraicts the stability of. We can now apply Lea 3. inuctively to obtain that each A >. An ieiate consequence of the bouns obtaine in Lea 3. an Theore 3. is the following. 9

20 Corollary 3. Assue that = an let an be as in Theore 3.. Assue further that S ( = { } an let { } = ( / + ψ+ ψ be a univariate Cascae wavelet sequence such that S S = S for all. If the two-scale sybols of the wavelets satisfy. Q W with an Q B <, C ( T Q w + Q w+ π A >, w T,. then for any J Z the ilate non-stationary wavelet set S J / { J ( } J ψ, Z. Next we use the general tools presente at the beginning of this section to construct, for a univariate cascae sequence { }, a sequence of sei-orthogonal wavelets { ψ } = = is a stable basis for for which the conitions of Corollary 3.3 hol. Assue an are as in Theore 3.. Following (3.4 an (3.5 we efine for Since ˆ, ˆ > ˆ ˆ, G = P. ˆ, ˆ ( is a trigonoetric polynoial for, by Wiener s lea [K], we have that G W for each. By Theore 3.4, any wavelet sybol where Q of the for ψ such that / iw ( π ( S S ψ = S has a + + (3. Q w = e G w+ K w, (3.3 K W never vanishes. Recall that in this local setting we can use (3.8 to choose { } { Q } are trigonoetric polynoials an thus construct { } select ψˆ Q ( / ˆ ( / = where This is equivalent to the selection K = ˆ, ˆ in (3.3. We alreay now that ψ is a seiorthogonal copleent to K so that ψ with copact support. For each we iw ˆ ˆ Q w : = e, w P w. so that / ψ S S = S. Also, observe that since the autocorrelation ˆ ˆ, an P are trigonoetric polynoials, so is Q. Thus, the { ψ } s have copact

21 support. Furtherore, we can uniforly boun their support ue to the convergence an the fact that we are using a finitely supporte cascae as. It reains to show that the conitions specifie in Corollary 3.3 on the wavelet sybols are et. To this en, by Theore 3. there exist < A B < such that for each we have A ˆ, ˆ B. (3.4 Hence Q ˆ ˆ w, P BP = : B<. (3.5 Also, (3.4 together with (3. iply ( ˆ ˆ ( ˆ ˆ,, ( ( π Q w + Q w+ π = w P w + w+ π P w+ π By virtue of Corollary 3.3 we can conclue that S J. A P w + P w+ A>. / { J ( } J ψ, Z is a stable basis for (3.6 4 Approxiation properties We recall that in classical refinable setting, it is a stanar practice to construct wavelets fro a given ultiresolution analysis of scaling function(s. Any reasonable wavelet construction ensures that the (linear approxiation properties of wavelets are erive irectly fro the approxiation properties of the scaling function(s. Let us briefly review this point. Throughout this chapter we use the stanar notation for the error of approxiation E f, V : = inf f g, where V X is a close subspace of a Banach space X. First recall that a close subspace V L p for any function f in the Sobolev space Wp ( R X g V X R is sai to provie L p approxiation orer if (, h (, E f V C V f h. (4. p Most results on approxiation fro shift invariant spaces use the Sobolev sei-nor of the approxiate function for the constant in (4., naely, a Jacson-type estiate, h E f V Ch f. (4., p V W p

22 If V = S( Φ is an FSI space we write C Φ for C V. In wavelet theory it is a coon practice to ensure that the so calle scaling functions provie approxiation orer. Also recall that a generator ϕ of a PSI space S ( ϕ satisfies the Strang-Fix (SF conitions of orer if ϕˆ an D ( α ϕˆ π = for all Z \ an α <. (4.3 It is well nown that, uner certain il restrictions, if ϕ satisfies the SF conitions of orer then the polynoials of egree can be represente using a superposition of the integer shifts of ϕ, an S ( ϕ provies approxiation orer. On the other han, wavelets shoul have the coplientary feature of vanishing oents. That is, ψ is a wavelet if for all polynoials of egree, p Π pψ =. R The connection between approxiation orer of the scaling functions an the wavelets is siple.,, L ψ S ϕ = S S Ψ. It can be shown that if ϕ, ϕ Ψ R, where Ψ= { } an Assue provie approxiation orer then all ψ Ψ have vanishing oents. In such a case the space S ( Ψ will be orthogonal to all polynoials of egree. In this section we show that the neste sequence of non-stationary ( scaling function spaces we have constructe using the Superfunction or Cascae ethos, beginning with soe given non-refinable shift invariant space, inherits the approxiation properties of the initial space. Also, the neste spaces share unifor approxiation properties. Specifically, we provie siultaneous estiates using unifor constants for the approxiation of functions fro these spaces. Consequently, our non-stationary wavelet spaces will have the esire vanishing oents property. This is what aes the suitable for signal processing applications. We now state a Strang-Fix type result that will becoe useful in Section 4.3. It is quite basic, but hanles the case of approxiation fro a sequence of PSI spaces. First we nee the following efinitions. Let ( R E enote the space of boune easurable functions that ecay faster than an inverse of a polynoial of egree +, i.e., Definition 4. Let f E ( R { ε } ( + + ε R E : = f f x C + x, for soe >. n if the Poisson Suation Forula hols for all f ( x suation forula for g L R is. We say that f satisfies the Poisson suation conition of orer Z, n <, x R. Recall that the Poisson ( = ˆ ( π g x g e π Z ix.

23 The above requireent hols for exaple if f is copactly supporte, continuous an of boune variation. Theore 4. Let { } following conitions hol for each. be a sequence of easurable univariate functions an. Assue the. (uniforly boune support supp [ LL, ]. (unifor boun M.. 3. (Poisson Suation The Poisson suation conition of orer hols for. ˆ =, 4. (Strang-Fix ( l ( ˆ π =, l =,,,. Then, there exist constants C, C which epen on L, M, (but o not epen on p such that: (i For any ( R f W p (ii For any ( R f L p h (, ( W p ( R E f S Ch f,. (4.4 p h (, ( ω (, E f S C f h,. (4.5 p p Proof The proof essentially follows the approach of [DL] Chapter 3 Section 7, with the observation that the constants can be estiate using values of the erivatives of the Fourier transfor at the origin. Conitions an ensure that this can be achieve. Naely, there are constants C,, C such that ( n ˆ, n,. Cn 4. L approxiation fro shift invariant spaces For the case of p =, two tools allow the analysis to be both elegant an powerful, the Hilbert space geoetry an the Plancharel-Parseval equality. The latter allows us to carry out the analysis in the frequency oain. An excellent survey of L approxiation fro shift invariant spaces is [JP]. Henceforth we enote H ( : = W ( R R. Definition 4.3 [BDR] For L R, efine the error ernel L [ ππ, ] Λ by ˆ Λ : = ˆ, ˆ, (4.6 3

24 where / is interprete as. Applying Fourier ethos one can use the error ernel (4.6 to obtain L estiates. The following theore characterizes the approxiation orer of an SI space, by the existence of a superfunction. The superfunction is require to have an error ernel (4.6 with fast ecay to zero about the origin. Theore 4.4 [BDR3] Let V be an SI space. Then V provies approxiation orer such that h E f V Ch f., V H if an only if there exists V for which Λ L ( B, for soe neighborhoo B of the origin. As prove in [BU] the ernel (4.6 can also be use to prouce very accurate error estiates. Theore 4.5 [BU] Assue that E ( R is stable with orer. Then for any function f H + ( R ˆ = an provies L approxiation (, h ( + = + H E f S Ch f O h R, ˆ( C ( = π. (4.7! One of the results in [U] is that the leaing constants of type C in (4.7 are uch saller for the B-Spline generators than for the Daubechies orthonoral scaling functions [Da]. Since the wavelets inherit in soe sense this constant fro the scaling functions, it ight explain the epirical evience in iage coing that spline wavelets outperfor the Daubechies wavelets with the sae nuber of vanishing oents. Exaple 4.6 OM, O-Mos (Optial Maxiu Orer an Minial Support The generator OM ([BTU], [TBU] iniize for a given support size (an approxiation orer, the constant C in (4.7. For each orer, OM can be efine as the outcoe of applying a ifferential operator I + D to the B-spline N, where D is hoogeneous of egree. It is easy to see that for any ifferential operator of the type I D I+ D N is piecewise polynoial with +, the resulting egree an support size. Also, since the SF conitions reain vali, OM provies approxiation orer. The O-Mos functions are continuous for the even orers. For exaple, OM 4 = N4 + N4, 4 ( 4 OM 6 = N6 + N6 + N The (noralize gains in sapling ensity brought by using O-Mos instea of the b-splines are C C N 4 OM 4 /4.463 C N6, COM6 /

25 We augent the L superfunction theory with a ore careful treatent of constants. We cobine the finer error estiates of [BU] relate to optial constants with the superfunction theory of [BDR]. We show that the superfunction provies asyptotically exactly the sae approxiation as the full space, with the sae (sharp leaing constant. First we require the following Lea 4.7 [BDR] Let V be an SI space. Then for any f, g L ( R (, (, (, (, E f V E f S Pg E f V + E f S g. V Theore 4.8 Let V be an FSI space which provies approxiation orer, such that for any r function f H R, r, (, h ( r V E f V Ch f + O h. (4.8 r Then there exists a superfunction V such that for any f H ( R, r one has r Proof Let f H (, h ( r V H E f S Ch f + O h. R. We use a ilate version of (4.8 / / (, (, V (,, H E f h V = h E fv h Ch f + C V r f h. h r H Select = Pg, where g is the ultivariate sinc-function V g sinπ x i : = i= π x, i g = χ. [ ππ, ] It is well nown (see for exaple [JP] that h ( E f S g h f. (4.9 r, r H By virtue of Lea 4.7, (4.8 an (4.9 we obtain h / (, ( = ( (, ( E f S h E f h S (,, + / h E f h V E f h S g r r Ch f + C V, r, f h + h f V r r H H 5

26 r CV h f r + O( h. Next we present a siilar result for local shift invariant spaces. We require the following superfunction result for the local case. Theore 4.9 [BDR] Let V be a local FSI space. Let g be any copactly supporte function (not necessarily in V. Then there exists a copactly supporte function V, such that for every f L R H (, (, (, E f S E fv + E f S g. (4. Theore 4. If in aition to the assuptions of Theore 4.8 we further assue that V is local, then r for each r > there exists a copactly supporte function r V f H R one has (, h ( r r V H E f S Ch f + O h. such that for every Proof The etho of proof is very siilar to Theore 4.8, only this tie we apply Theore 4.9 with the selection g = Nr, where N r is the tensor-prouct B-spline of orer r 4. Approxiation properties of the non-stationary Superfunction wavelets We now go bac to the superfunction ecopositions of Section 3. an verify that the nonstationary half-ultiresolution inherits the approxiation properties of the initial space an the reference space. First, we nee the following result. R have approxiation orer an assue S( / = S( S( Ψ where S S( has approxiation orer. Furtherore: Theore 4. Let, L Ψ. Then. If for all functions f H then for all functions f H R an h > the following two estiates hol (, h H ( R, E f S C h f R an h >, r. If for all functions f H (, h H (, h H E f S C h f, C C + C. R E f S Ch f, (4. R R, r > an h > the following two estiates hol (, h ( r +, ( H R E f S C h f O h (, h ( r + H E f S C h f O h, (4. R 6

27 R an h >, r then for all functions f H Proof. Let f H (, h ( r + H E f S C h f O h, C C C R +. R an h >. Since, have approxiation orer, we can obtain a ilate version of (4. for both generators / / ( (, ( ( / H ( R, ( / E f h, S h Ch f H E f h S h C h f R. Since S( Ψ S(, it follows that space an the shift coute we get We now apply Lea 4.7 to erive P S S S / ( ( ( S ( S(. Since the orthogonal proection onto an SI ( P S = S P. / / r. Let f H h / (, ( = ( (, ( / h E( f ( h, S( P / S( / / h E( f ( h, S( + E( f ( h, S( E f S h E f h S R. Then the sae arguents yiel ( C + C h f. H h / / (, ( ( (, ( + ( (, ( ( + + ( ( + ( E f S h E f h S E f h S r r C C h f C, f C, f h. We are now reay to ustify the superfunction construction of Theore 3.5. Corollary 4. Let, L( R have approxiation orer an let { } H be so that for all : (i S( S S / (ii S S(. If, Ψ =, Ψ. satisfy (4., then we have the unifor estiate for any f H ( R 7

28 . If, + h (, ( ax (, H E f S C C h f R,. (4.3 r satisfy (4., then we have the unifor estiate for any f H R, r >, + h r (, ( ax (, + H E f S C C h f O h,. (4.4 R Proof The proof is by inuction. We only prove (4.3, the proof for (4.4 being siilar. The estiate (4.3 is certainly true for the initial function. Assue that has approxiation power. By Theore 4. we see that the generator, constructe using the proection etho of Theore 3.5, inherits the approxiation power with a constant C C C +. The relation leas to the unifor boun n C C C + n= + ( C C n ax, n= + ax ( C,. C Exaple 4.3 Select, in Corollary 4. to be OM4: = N4 + N 4 /4 (see Exaple 4.6. Then for any f r H R, r > 4 Therefore for all, 5 h 4 r (, ( 4 OM H E f S C h f O h,. R /4 N C 4 N4 OM C 4 C C 4 Assue { } ψ are any non-stationary (copactly supporte wavelets, copleenting the halfultiresolution generate by { } where OM 4 generates both the initial space an the reference space. Then, these wavelets have a sharp constant saller then the B-wavelets of [Ch] with a gain of about %. This result is not very surprising. We have shown (Exaple 3.8 that we can choose the first wavelet ψ such that supp ψ = 7, which is exactly the support size of the cubic B-wavelet. But as explaine in Section 3., for any such non-stationary wavelet sequence, the support of the wavelets in general grows. 8

29 4.3 Approxiation properties of the non-stationary Cascae wavelets The first results of this section verify that the application of a cascae operator with goo properties to a given function with goo approxiation properties yiels a function that inherits these properties. These results are connecte to the nown so calle zero conitions on the as sybol (see Section 3. in [JP]. The ain ifference with previous wor is that we use zero conitions on the cascae as when applie to non-refinable functions. Lea 4.4 Assue ( R trigonoetric polynoial efine by E satisfies the SF conitions of orer an let P Π N be a iw r r + e iw P( w = R( w = pe, (4.5 r= N with r, r,, = an R. Then efine by is in ( R E an satisfies the SF conitions of orer. p (, (4.6 = N Proof The Fourier equivalent of (4.6, is the two-scale relation Since ˆ ( w w w = P ˆ. (4.7 P Π N, the su in (4.6 is finite so that E SF, it follows that ˆ R ˆ R an ˆ, ˆ C R. Since satisfies =. It is quite easy to show that satisfies the other SF conitions (4.3. This is one using the two-scale relation (4.7 an the ultivariate for of Leibniz rule. It is nown [R] that any univariate generator that provies approxiation orer is a convolution of a B-spline of orer an a tepere istribution. Thus, the sallest support possible for a given approxiation orer is. Next we see that the B-spline cascae operator can help preserve this optial feature. R satisfies the SF an Poisson suation conitions of orer Corollary 4.5 Assue that L an has (inial support size. Then there exists S ( / that provies approxiation orer an has (inial support size. Proof Observe that by Theore 4. provies approxiation orer. We ay assue that [ ] supp, (we can always shift the construction below to this interval an then bac. Select P N, the (inially supporte two-scale sybol of the B-Spline of orer, efine by 9

30 Clearly, for iw + e iw PN ( w = pe =, = P N conition (4.5 hols. Thus by Lea 4.4, efine by ˆ ( w w w P ˆ =, N p +. (4.8 = is in L ( R, has copact support an satisfies the SF an Poisson suation conitions of orer. Using Theore 4. this iplies that provies approxiation orer. Also, since p = for all,,, has the require (inial support property. Thus, we see that a goo cascae operator is actually an algorith to extract a superfunction fro the FSI space S ( /. We nee to verify that the cascae process preserves approxiation properties in a unifor sense. The next result overcoes this technical point. Corollary 4.6 Let be a univariate function with copact support that satisfies the SF an Poisson suation conitions of orer. Let P be a finite as of type (4.5 associate with a cascae operator C an a refinable function L ( R an let : =C be so that,. supp [ LL, ] for all <,. M for all <. Then the following hol,. There exists a constant. There exists a constant f W p R, p C such that for any h (, ( E f S Ch f. C such that for any for any p Lp ( R f L p R, p h (, ( ω (, E f S C f h. p p Proof It is easy to see that uner our assuptions, conitions -3 of Theore 4. hol. Also by Lea 4.4 it follows that the SF conitions of orer hol for all functions in the sequence an so conition 4 of Theore 4. is also fulfille. We now apply Theore 4. to obtain the require estiates. Rear It is interesting to observe that for the last result we i not require that the cascae sequence converges to a refinable function, ust that it reaine boune in soe box. 3