WAVELET-GALERKIN-METHODS: AN ADAPTEDBIORTHOGONAL WAVELET BASISSTEPHAN DAHLKEInstitut f�ur Geometrie und Praktische MathematikRWTH AachenTemplergraben 555100 AachenFederal Republic of GermanyILONA WEINREICHInstitut f�ur Geometrie und Praktische MathematikRWTH AachenTemplergraben 555100 AachenFederal Republic of GermanyABSTRACT. In this paper we construct a compactly supported biorthogonal waveletbasis adapted to some simple di�erential operators. Moreover, we estimate the con-dition numbers of the corresponding sti�ness matrices.Key Words: compactly supported wavelets, biorthogonal bases, Galerkin methods.AMS Subject classi�cation: 34K10, 34A50, 42C05, 46N05, 65N30.1 IntroductionIn recent studies, wavelet decompositions were used for the numerical solution ofintegral and (ordinary and partial) di�erential equations. The objective of this paperis the construction of a biorthogonal wavelet basis adapted to a given di�erentialoperator. We only consider some one-dimensional model problems (leading to simpleordinary di�erential equations) to explore the principle possibilities of the adaptionof wavelets to a given problem. It seems that our approach can be generalized topartial di�erential equations. This will be studied in a forthcoming paper.In general, a function is called a wavelet if the scaled (by 2j; j 2 Z) and integertranslated versions of form an orthonormal basis of L2(R). We want to usethese functions as a basis for a Galerkin-approach. The structure of the resultingsti�ness matrix depends on the wavelets and the given di�erential operator. There-fore, it is natural to look for a family of wavelets such that, for a given di�erential1

operator, the sti�ness matrix has a simple structure and a bounded condition numberto facilitate an e�cient and accurate numerical treatment of such problems.The most e�cient way to construct wavelets is to use the concept of multiresolu-tion analysis. In this paper, we construct a multiresolution analysis approximationsuch that the resulting wavelets are adapted to simple model problems of the formu(2m)(x) = g(x), this e�ects that the sti�ness matrix splits into blocks. Moreoverit is shown that even for more general elliptic di�erential operators the conditionnumber of the sti�ness matrix is independent of the re�nement level. (To focus hereon principal estimates which are most conveniently formulated for settings based onshift invariant spaces we con�ne the discussion to model problems involving periodicboundary conditions.)This paper is organized as follows: In section 2 we review the concept of stationarysubdivision. This is the basic tool for the construction of the adapted multiresolutionanalysis. In section 3 we brie y recall the construction of (orthonormal and biortho-gonal) wavelet bases with the use of multiresolution analysis. In section 4 and 5 wedevelop the adapted biorthogonal wavelet basis. In section 6 we study the relation ofour construction to other approaches and give some generalizations to more compli-cated di�erential operators. Finally, we check in section 7 that the condition numberof the sti�ness matrix is independent of the re�nement level.2 Stationary SubdivisionThe general setting for stationary subdivision schemes (SSS) can be described asfollows. Let lp(Z) denote the space of sequences � = f�kgk2Z such thatk�klp = (Xk j �k jp)1=p <1; 1 � p <1;k�kl1 = supk2Z j �k j <1; p =1:For a given sequence a = fakgk2Z of real numbers let S = Sa be de�ned by(Sa�)k =Xl ak�2l�l; k 2 Z: (1)The sequence a will be referred to as themask of the operator Sa. We will henceforthassume that a is �nitely supported. Associating the value (Sma �)k with the point2�mk we say that Sa is (uniformly) convergent if for each � 2 l1(Z) there existsa continuous function f� on R such thatlimm!1 kf�(2�m�)� Sma �kl1 = 02

Convergence and structural properties of the limits of subdivision schemes for a gen-eral multivariate setting are extensively studied in [1], where the multivariate versionof the following basic theorem is proved.Theorem 2.1 If Sa converges and there exists a � 2 l1(Z) such that f� 6= 0 thenthe mask satis�es Xl2Z ak�2l = 1; k 2 Z: (2)Moreover, there exists a unique continuous function � with compact support satisfyingthe functional equation: �(x) =Xk ak�(2x� k); x 2 R; (3)the translates of � form a partition of unityXk �(x� k) = 1; x 2 R; (4)and the limiting function f� has the representationf�(x) =Xk �k�(x� k); x 2 R: (5)The function � is called the re�nable function associated with a. It can be shownthat supp� � conv(supp a): (6)Now it is natural to ask how convergence of a SSS can be checked. To formulatea convergence criterion it is convenient to introduce the symbol of a mask a as theLaurent polynomial a(z) =Xk akzk:Then one has the following theorem, proved in [6].Theorem 2.2 Suppose that for some N � 1a(z) = (1 + z)NQ(z); (7)where Q is a Laurent polynomial such that for some k 2 N, r 2 Z+Qk(z) = Q(z)Q(z2) � ::: �Q(z2k�1);satis�es kQkk1 := maxjzj=1 j Qk(z) j< 2�kr; Q(1) = 2�N+1: (8)Then Sa converges and the corresponding re�nable function � belongs to Cr0(R).3

Concerning the smoothness of the solution of the functional equation (3), we introducethe di�erence operator rf(�) := f(�) � f(� � 1): The following theorem is proved in[1].Theorem 2.3 Let Sa be a convergent subdivision scheme with re�nable function �.Suppose there exists a convergent subdivision scheme Sq with re�nable function � suchthat for some r 2 N rrSa = 12rSqrr: (9)Then � 2 Cr(R) and �(r) = rr�. Moreover, (9) is equivalent toa(z) = (1 + z)r2r q(z): (10)3 BiorthogonalityIn this part we summarize brie y the basic concepts used in the wavelet setting. Themost important tool for the construction of wavelets is the multiresolution analysisapproximation of functions introduced by Mallat [14]. It is de�ned as follows:De�nition 3.1 A multiresolution approximation of L2(R) is a sequence fVjgj2Z ofclosed subspaces of L2(R) such that8j 2 Z; Vj � Vj+1; (11)1[�1 Vj = L2(R); 1\�1 Vj = f0g; (12)f(�) 2 Vj () f(2�) 2 Vj+1; (13)f(�) 2 V0 () f(� � k) 2 V0: (14)There exists a function � whose integer translates form a Riesz basis of V0, (15)this means that V0 is the closed linear span of �(� � k) and that there exist constantsc1; c2 2 R+ such thatc1k�kl2 � kXk �k�(� � k)k2 � c2k�kl2; 8� 2 l2(Z): (16)4

A natural way to construct a wavelet orthonormal basis is to �nd a function inthe orthogonal complementW0 of V0 in V1 whose translates are orthonormal, i.e.,< (� � k); (� � l) >= ZR (x� k) (x� l)dx = �kl; k; l 2 Z; (17)< �(� � l); (�) > = 0; l 2 Z; (18)and whose translates span W0,W0 = spanf (� � n); n 2 Zg: (19)Hence, de�ning Wj = ff 2 L2(R) j f(2�j �) 2 W0g;one has Vj+1 = Vj �Wj; L2(R) = �j2ZWjand the functions jk := 2j=2 (2j � �k)form a complete orthonormal system for L2(R). is called the orthonormalwavelet associated with �. Compactly supported orthonormal wavelets were �rstconstructed by I.Daubechies in [8]. It is possible to construct the correspondingmultiresolution analysis with the aid of SSS. The interrelation between SSS andorthonormal wavelets is clari�ed in [6], where among other things the following resultis proved.Theorem 3.2 Let Sa be a convergent SSS with re�nable function �. If the maskcoe�cients aj satisfy: Xk akak+2r = 2�0r; (20)then the function � satis�es the orthogonality condition< �(�); �(� � k) >= �0k (21)and generates a multiresolution analysis. Moreover, the function (x) =Xk (�1)ka1�k�(2x� k) (22)is the unique compactly supported orthonormal wavelet associated with �.Remark 3.3 The function � can be constructed with the aid of Theorem 2.2, leadingto a family N of wavelets depending on the parameter N (see [6], [8]). It is shownthat the assumptions of 2.2 are satis�ed for r = 0; k = 1. Therefore, the re�nablefunction �N and the associated wavelets N are continuous. Actually, it can be shown[8] that �N has smoothness cN , for some constant c, as N tends to in�nity, by using aconvergence result covered by Theorem 2.2. Precise results concerning the smoothnessof solutions of functional equations of type (3) are shown in [9].5

Orthonormal wavelet bases can be generalized a little bit further. In many applica-tions, it is convenient to deal with biorthogonal wavelet bases. In this case onehas two hierarchical sequences of approximation spaces� � � � V�2 � V�1 � V0 � V1 � V2 � � � (23)� � � � ~V�2 � ~V�1 � ~V0 � ~V1 � ~V2 � � � :Now the space W0 will be a complement to V0 in V1, but not necessarily the orthog-onal complement. The orthogonality condition from above is now replaced by theconditions ~W0 ? V0; W0 ? ~V0: (24)As above, one looks for functions ; ~ whose translates span W0; ~W0 and moreoversatisfy the biorthogonality condition< jk; ~ j0k0 >= �kk0�jj0 ; jk := 2j=2 (2j � �k): (25)This concept was introduced by Cohen, Daubechies and Feauveau in [4]. It is possibleto construct compactly supported biothogonal wavelet bases with SSS. One has:Theorem 3.4 Let Sa und Sb be convergent SSS with re�nable functions �; ~�. If themask coe�cients aj and bj satisfyXk akbk+2r = 2�0r; (26)then the functions � and ~� form a dual pair< �(�); ~�(� � k) >= �0k; (27)and �; ~� generate a multiresolution analysis. Moreover, the functions (x) =Xk (�1)kb1�k�(2x� k); (28)~ (x) =Xk (�1)ka1�k ~�(2x� k)form a biorthogonal wavelet basis, i.e.,< jk; ~ j0k0 >= �jj0�kk0 (29)and V1 = V0 �W0; ~V1 = ~V0 � ~W0; V0 ? ~W0; ~V0 ?W0; (30)with W0 = spanf (� � k) j k 2 Zg; ~W0 = spanf ~ (� � k) j k 2 Zg: (31)6

Furthermore, the sets f jkgk2Z and f ~ j0k0gk02Z form a Riesz basis, i.e.,c1k�kl2 � kXk �k jkk2 � c2k�kl2; � 2 l2(Z); (32)~c1k�kl2 � kXk0 �k0 ~ j0k0k2 � ~c2k�kl2; � 2 l2(Z);with constants c1; c2; ~c1; ~c2 independent of j and j 0 .Proof:(27) can be derived from (26) by following the lines of the proof of Proposition 10in [6]. (With one sequence fakgk2Z replaced by fbkgk2Z.) To show that � (and, inthe same way ~�) generates a multiresolution analysis, it is su�cient to check that �satis�es Z 10 (Xk j �(x� k) j)2dx < 1; (33)kXk �k�(� � k)k2 � ck�kl2; (34)to apply then Theorem 2.2 in [12].(33) is obvious since � is compactly supported and continuous. From (27) it is clearthat the integer translates of � are linearly independent, i.e.,Xk �k�(x� k) = 0 =) �k = 0; 8 k 2 Z;for an arbitrary sequence f�kgk2Z: Furthermore it is known that the linear indepen-dence of the translates implies the stability condition (34) (see [12], x5 for details).(29) follows immediatly from (26) - (28) ( see [4], Lemma 3.7).(30), (31) follow from (27), (28) and the following formulas proved in [7]:�(2x)=Xk<�(2�); ~�(�� k)>�(x� k) +Xk<�(2�); ~ (�� k)> (x� k) (35)�(2x� 1) =Xk<�(2 ��1); ~�(�� k)>�(x� k) +Xk<�(2 ��1); ~ (�� k)> (x� k):(36)Finally, (32) is a consequence of (29), since (29) implies the linear independence ofthe translates of and ~ . 2 7

Remark 3.5 a) Using the symbols a(z); b(z) it is easy to see that (26) is equivalentto a(z)b(z) + a(�z)b(�z) = 4 (37)(See [6] for a similar calculation; the following fact is proved there in Proposition 15:Xk akak+2r = 2�0r ()j a(z) j2 + j a(�z) j2= 4 ) (38)Moreover, using Fourier transformation, it is easy to see that (28) is equivalent to ̂(�) = �z2 b(�z)�̂(�=2) =: c(z)2 �̂(�=2); z = e�i �2 ; (39)~̂ (�) = �z2 a(�z) ~̂�(�=2) =: d(z)2 ~̂�(�=2); z = e�i �2 :b) Biorthogonal bases can be studied in a more general setting (see [4]). Let fakgk2Z,fbkgk2Z be two sequences related by (26), not necessarily �nitely supported, but withsu�cient decay. Under certain conditions on the mask the in�nite products�̂(�) = 1Yj=1 a(e�i �2j )2 ; (40)~̂�(�) = 1Yj=1 b(e�i �2j )2 ;converge and give rise to (in general not compactly supported) solutions of the func-tional equation (3) associated with fakgk2Z and fbkgk2Z. Under additional condi-tions, � and ~� de�ned by (40) generate a multiresolution analysis and form a dualpair, leading to a (not necessarily compactly supported) biorthogonal wavelet basis.4 A general formula for b(z)In this chapter we consider the simple one-dimensional model problemu(2m)(x) = g(x) on [0;M ]; m;M 2 N; (41)either with Dirichlet boundary conditionsu(j)(0) = u(j)(M) = 0; j = 0; :::;m� 1;or with periodic boundary conditionsu(j)(0) = u(j)(M); j = 0; :::; 2m� 1;8

together with the compatibility conditionZ M0 g(x)dx = 0:We treat (41) by means of a Galerkin approach. To do this, we use the spaces Vj ofa multiresolution analysis as approximation spaces. (It is shown in [10] that, undercertain conditions, this approach converges.) Therefore, the weak formulation of (41)is to �nd u 2 Vj such that<< u; uk >>:=< u(m); u(m)k >=< g; uk >= g�(uk);where fukgk2Z is a basis of Vj . In practice, one does of course not use the wholespace Vj . For the Dirichlet problem, one deals with the �nite-dimensional subspacespanned by all integer translates of the generator � having support in [0;M ]. In theperiodic case, one identi�es the points 0 and M . This means that, in this case, oneessentially deals with approximation spaces consisting of periodic functions. They canbe extracted from the original multiresolution analysis by using periodic coe�cientvectors.The structure and the condition of the resulting sti�ness matrix depend on the dif-ferential operator and on the functions uk. The optimal form would be a diagonalmatrix. To get this form, one could try to use the functions f�jkgk2Z as a basis andto look for a re�nable function � whose integer translates are orthogonal with respectto the energy norm, i.e.,<< �(� � k); �(� � l) >>=< �(m)(� � k); �(m)(� � l) >= �kl: (42)But it turns out that such a function does not exist. This can be seen as follows. Leta(z) be the symbol of �. Then �(m) has the symbol 2ma(z). Condition (42) impliesthat j a(z) j2 + j a(�z) j2= 2�2(m�1): (43)(This can be proved in the same way as (37).) But (43) is a contradiction to condition(2), which implies a(1) = 2.The next we could expect is orthogonality with respect to di�erent re�nement levels.As we will see, this is possible. To carry out such a construction, we use the functionsf�(� � k); 2j=2 (2j � �l)gk;l2Z;j�0( according to Theorem 3.4) as a basis. (We will always assume that the interval[0;M ] is big enough. Otherwise, one has to start with a higher re�nement level.)Similar things are done in [11] by using "classical" orthonormal wavelets. Biorthogo-nal wavelets have the advantage that, because one deals with two sequences fakgk2Z;fbkgk2Z one has more exibility. 9

A natural candidate for a biorthogonal wavelet basis adapted to the model problem(41) would be a system having the property<< �(� � k); (� � l) >>= 0; k; l 2 Z: (44)The condition (44) implies orthogonality of di�erent re�nement levels with respectto the energy norm, and consequently the sti�ness matrix splits into blocks (For theDirichlet case, the structure is partially disturbed, see Chapter 7). In fact, since there�nement equation (3) is valid for �, (44) has the consequence<< jk; j0l >>= 0; j 6= j 0 ; << �k; jl >>= 0: (45)This leads to the following problem: Let Sa be a given convergent SSS. Is it possibleto �nd a second SSS Sb such that Sa and Sb give rise to a biorthogonal system whichsaties�es (44) ?To answer this question, we derive a general formula for the symbol b(z).We haveLemma 4.1 Let Sa be a convergent SSS and ~� a biorthogonal generator with symbolb(z). A necessary and su�cient condition for (44) isb(z) = 4a(z)g(�)j a(z) j2 g(�)+ j a(�z) j2 g(� + 2�) (46)where g(�) is de�ned byg(�) :=Xn (� + 4�n)2m j �̂(�=2 + 2�n) j2 : (47)Proof: We have<< �(� � k); (� � l) >> = < �(m)(� � k); (m)(� � l) >= ZR �(m)(x� k) (m)(x� l)dx= 12� ZR d�(m)(� � k)(�) d (m)(� � l)(�)d�= 12� ZR e�i(k�l)� d�(m)(�) d (m)(�)d�= 12� ZR e�i(k�l)��2m�̂(�) ̂(�)d�= 12� ZR e�i(k�l)��2m[12a(z)�̂(�2)][12c(z)�̂(�2)] d�= 18� Xn2Z Z[0;2�)+2�n e�i(k�l)��2ma(z)c(z) j �̂(�2) j2 d�:10

where we used Plancherel's theorem and the functional equations (3), (39). Splittingup the sum into even and odd terms we conclude that:<< �(� � k); (� � l) >>= 18� Z 2�0 e�i(k�l)� Xn2Z a(e�i 12 (�+2�n))c(e�i 12 (�+2�n))(� + 2�n)2m j �̂(12(� + 2�n)) j2 d�= 18� Z 2�0 e�i(k�l)�[a(e�i 12 �)c(e�i 12 �)Xn2Z(� + 4�n)2m j �̂(12� + 2�n) j2+a(�e�i 12 �)c(�e�i 12 �)Xn2Z(� + 2� + 4�n)2m j �̂(12� + � + 2�n) j2]d�:Therefore, condition (44) is equivalent to the following formula:18� Z 2�0 e�i(k�l)� h(a(z)c(z)g(�) + a(�z)c(�z)g(� + 2�)i d� = 0 8k; l 2 Z: (48)A neccessary and su�cient condition for (48) isa(z)c(z)g(�) + a(�z)c(�z)g(� + 2�) = 0: (49)Taking into account the formula (39) and the biorthogonality condition (37), we seethat we have to solve the following linear system:0@ a(z) a(�z)z�1a(�z)g(� + 2�) �z�1a(z)g(�) 1A0@ b(z)b(�z) 1A = 0@ 40 1A : (50)This yields b(z) = 4a(z)g(�)j a(z) j2 g(�)+ j a(�z) j2 g(� + 2�) ;proving the Lemma. 2For a given SSS Sa every part of (46) is computable. But, in general, b(z) will not bea �nite Laurent polynomial and therefore it will not give rise to a SSS with �nitelysupported mask. For example, take the (centralized) B-Spline~N4(x) = 1[0;1) � ::: � 1[0;1)| {z }4-times (x+ 2): (51)Then one gets b(z) = 2 cos4( �4) + 415 cos2( �4) sin4( �4)cos8( �4) + sin8( �4) + 215 sin4( �4) cos4( �4) : (52)11

In this case, the coe�cients of b(z) have exponential decay, but they are not �nitelysupported. Nevertheless, the general formula (46) can make sense in the more generalsetting of Remark 3.5 b).To get a �nite Laurent series from (46), we have to construct a SSS Sa such that thedeterminant of the matrix in (50) is a constant. (Or, more generally, such that itdivides the numerator in (46)) . This will be done in the following section.5 An adapted biorthogonal wavelet basisIn this part we construct two convergent SSS Sa and Sb which give rise to a biorthog-onal wavelet basis such that (44) is satis�ed. As stated above, the crucial point is to�nd a SSS such that b(z) de�ned by (46) is a Laurent polynomial. To do this, weuse Theorem 2.3, which tells us how to work with derivatives of re�nable functions.The basic idea is to choose Sq to be the SSS appearing in Theorem 3.2. Then, ac-cording to Theorem 2.3, the derivatives of � would be di�erences of translates of theDaubechies-generator. Because of (21) it would be easy to calculate scalar productsof the form (44). The following theorem shows that this choice is the right one.Theorem 5.1 Let SqN be a convergent SSS according to Theorem 3.2 with symbolqN(z) = (1 + z)NQ(zjN) (53)and re�nable function �N . (We use the notation Q(�jN) to indicate the dependencyon the parameter N , see Remark 3.3.) For a �xed m 2 N and for N su�cientlylarge, the SSS Sa and Sb associated with the symbolsa(z) = (1 + z)m2m qN(z) = (1 + z)2m N+mQ(zjN); (54)b(z) = (2z)m(1 + z)N�mQ(zjN)converge and give rise to a dual pair and a biorthogonal wavelet basis in the sense ofTheorem 3.4, with arbitrarily high regularity as N tends to in�nity. Moreover, thefollowing relation holds:< �(m)N (� � k); (m)N (� � l) >= 0; k; l 2 Z; (55)where �N is the re�nable function of Sa and N is de�ned by (28).Proof: We have to show that Sa and Sb converge and that a(z) and b(z) are relatedby (46).The convergence of Sa is no problem. In fact, since �N is constructed by usingTheorem 2.2, we know that kQ(�jN)k1 < 1:12

Therefore, it follows from (54) that the conditions of Theorem 2.2 are satis�ed fora(z) for k = 1; r = m.Now we want to prove (46). The crucial part is the computation of the function g(�).To do this, we introduce the Laurent seriesI(z) = Xk << �N (�); �N(� � k) >> zk (56)= Xk < �(m)N (�); �(m)N (� � k) > zk:Using Poisson summation formula, we getI(z) =Xn j d�(m)N (�2+2�n) j2=Xn (�2+2�n)2m j b�N(�2+2�n) j2= 2�2mg(�); z = e�i�=2:(57)Therefore, we have to compute the coe�cients of I(z). This can be done by usingTheorem 2.3. From (54) it follows that:Xk < �(m)N (�); �(m)N (� � k) > zk =Xk < rm�N (�);rm�N (� � k) > zk: (58)To compute the right-hand side, we want to use the identity< rm�N (�);rm�N(� � k) >= mXj=�m 2mj +m!(�1)j�jk: (59)(59) can be proved by induction. For m = 1 we get< r�N(�);r�N(� � k) >= < �N (�)� �N (� � 1); �N (� � k)� �N (� � k � 1) >= < �N (�); �N(� � k) > � < �N(�); �N (� � k � 1) >� < �N (� � 1); �N(� � k) > + < �N (� � 1); �N (� � k � 1) >= ���1k + 2�0k � �1k:Assume that (59) is true for m� 1. Then we get< rm�N(�);rm�N(� � k) >=< rm�1�N(�)�rm�1�N (� � 1);rm�1�N (� � k)�rm�1�N (� � k � 1) >= Pm�1j=�m+1 � 2m�2j+m�1�(�1)j�jk �Pm�1j=�m+1 � 2m�2j+m�1�(�1)j�j k�1�Pm�1j=�m+1 � 2m�2j+m�1�(�1)j�j+1k+Pm�1j=�m+1 � 2m�2j+m�1�(�1)j�j+1 k+1= Pmj=�m �2m�2j+m �(�1)j�jk + 2Pm�1j=�m+1 � 2m�2j+m�1�(�1)j�jk +Pm�2j=�m+2 � 2m�2j+m�2�(�1)j�jk= Pmj=�m � 2mj+m�(�1)j�jk: 13

Again by induction one can show the identitymXj=�m 2mj +m!(�z)j = mXk=0 mk ! mXl=0 ml !(�z)k�l: (60)Using (59) and (60) we get from (57) and (58)g(�) = 22mI(z) = 22mXk < rm�N (�);rm�N (� � k >)(�z)k= 22mXk ( mX�m 2mj +m!(�1)j�jk)zk= 22m mX�m 2mj +m!(�z)j= 22m mXk=0 mk ! mXl=0 ml !(�z)k�l = 22m j 1 � z j2m :Inserting this into (46) leads to:4a(z)g(�)j a(z) j2 g(�)+ j a(�z) j2 g(� + 2�)= 4 � 2�m(1 + z)mqN(z)22m j 1 � z j2m2�2m j 1 + z j2mj qN(z) j2 22m j 1� z j2m +2�2m j 1� z j2mj qN(�z) j2 22m j 1 + z j2m= 4 � 2m(1 + z)mqN (z)j 1 + z j2m (j qN(z) j2 + j qN(�z) j2)= 2m(1 + z)mqN(z)j 1 + z j2m= 2m(1 + z)m(1 + z)NQ(zjN)(1 + z)m(1 + z�1)m= (2z)m(1 + z)N�mQ(zjN)where we have used (38).It remains to study the convergence of Sb. We want to use the Theorem 2.2 again.In [8], the following formula is proved:kQ2(zjN)k1 = supjzj=1 j Q(zjN)Q(z2jN) j� N(1627)N=2: (61)Using (61) we getsupjzj=1 j (2z)mQ(zjN)(2z2)mQ(z2jN) j� 22mN(1627)N=2: (62)14

As N tends to in�nity, the right-hand side of (62) gets arbitrary small. Therefore, byTheorem 2.2, we get a family of biorthogonal wavelet basis of arbitrary high smooth-ness. 2Now let us take a look at the special case m = 1.From (62), we can conclude that ~� is C0 for N � 16 and that ~� is C1 for N � 23. Butthe estimate in (61) is very rough. In fact, it can be seen that Sb converges for muchsmaller values of N . For example, in �gure [1]-[4], we plotted �; ~� and the resultingwavelets ; ~ for N = 10. ~� seems to be C1. This is not surprising. According toTheorem 2.3, the de�nition of Sa (a multiplication of the symbol qN(z) with (1+z)2 )can be interpreted as a kind of integration. From (54) we see that Sb is constructedby almost the opposite operation (by dividing qN (z) by (1+z)2 ): This is some kind ofdi�erentiation.('Partial integration' of a compactly supported wavelet basis yields anadapted biorthogonal wavelet basis !) From [8] we know that �10 has smoothnessC2:902. Therefore, ~�10, the 'derivative' of �10, should be C1.On the other hand, for the applications we have in mind, the smoothness of ~� is notthe most important thing. All we have to know is that ~� exists, at least as a functionin L2(R). This is true for much smaller values of N .Corollary 5.2 For m = 1; N � 4; ~� de�ned by (40) exists as a function in L2(R).Moreover, � and ~� form a dual pair and give rise to a biorthogonal wavelet basis.Proof: We want to use the following theorem, proved by Cohen, Daubechies andFeauveau (see [4], Prop. 4.9).Theorem 5.3 Assume that a(z) and b(z) satisfy (37) and can be factored as:a(z) = (1 + z)NP (z);b(z) = (1 + z) ~N ~P (z):Furthermore suppose that, for some k; ~k > 0;kPk(z)k1 = kP (z) � P (z2) � ::: � P (z2k�1)k1 < 2k=2;k ~P~k(z)k1 = k ~P (z) � ~P (z2) � ::: � ~P (z2~k�1)k1 < 2~k=2:Then �; ~� 2 L2(R) and < �; ~�(� � n) >= �0n.15

We want to use this theorem for k = 1 and P (z) = 2zQ(zjN). Therefore we have toshow that kQ(zjN)k1 < 2�1=2:The following formula is proved in [6] (by using the Bernstein representation forQ(zjN) and estimating the leading coe�cient)kQ(zjN)k1 � 2�2(N�1) 2N � 1N � 1 !:The right-hand side tends to zero as N tends to in�nity.For N = 4 we get kQ(zjN)k1 � 3564 < 2�1=2: 2Similar results can be shown for higher values of m.

Figure 1: Generator � for N = 10 16

Figure 2: Dual generator ~� for N = 10Figure 3: Wavelet for N = 10 17

Figure 4: Dual wavelet ~ for N = 106 Special SolutionsIn this chapter we study some special solutions of the formulas (49) and (54). Thereare two things we have in mind: First, we show that some wavelet bases constructedindependently by other authors can be interpreted as a special case of our construc-tion. Secondly, we give an outlook how our approach can be generalized to morecomplicated di�erential operators.First let us take a look at (54) for the casem = 1. If we use the "simplest" Daubechies-generator (leading to the Haar basis)�1 = 1[0;1) = N1; (63)whose symbol is q1(z) = 1 + z; (64)we get a(z) = (1 + z)2 q1(z) = (1 + z)22 ; b(z) = 2z: (65)18

a(z) is exactly the symbol of the B-spline N2. Using (28) we get 1(x) =Xk (�1)kb1�k�1(2x� k) = 2N2(2x): (66)Up to multiplication by a constant, the above expression is the univariate version ofYserentant's hierarchical basis [16]. This example is in a certain sense pathologicalsince the SSS associated with b(z) does obviously not converge. But, since the biortho-gonality condition (37) is still satis�ed, it is clear that the translates of 1 span thewhole space W0 ( see formula (72) below). Because of (66) we can (for m = 1)interpret our biorthogonal wavelets as a generalized hierarchical basis. They havethe same properties, but higher smoothness, connected with a larger support.Now, let us take a look at (49) for the case m = 2. If we use the centralized B-spline~N4, we get a(z) = 2 cos4(�4); g(�) = 283 sin4(�4)[2 cos2(�4) + 1]: (67)Inserting this into (49) we obtain the condition[2 cos2(�4) + 1]c(z) + [2 sin2(�4) + 1]c(�z) = 0; z = e�i �2 : (68)One solution of (68) isc(z) = �z6 (2 sin2(�4) + 1) = 12(16 � 23z + 16z2): (69)The wavelet associated with c(z) is (x) = Xk ck ~N4(2x� k) (70)= 12(16 ~N4(2x)� 23 ~N4(2x� 1) + 16 ~N4(2x� 2)):This is a shifted version of the spline wavelet adapted to the problem (41) for m = 2which was constructed by Lorentz and Madych ([13], formula (5.2)). They derive thisresult by "guessing" the right solution from the following formula concerning splinepre-wavelets given by Chui and Wang in [2]: m(x) = 2�m+1 2m�2Xk=0 (�1)kN2m(k + 1)N (m)2m (2x� k): (71)Solution (69) is in a certain sense "noncanonical" since it does not give rise to abiorthogonal wavelet basis. (But it has the advantage that it leads to a compactlysupported wavelet. This is not true for the biorthogonal basis in the spline case, see19

for example (52).) Nevertheless, it is possible that even for noncanonical solutions of(49) the translates of the resulting wavelet span the whole complement W0 of V0in V1 and form a Riesz basis so that they can be used for a Galerkin approach. ( Butnevertheless, there might be solutions for which this is not true.) Recently Jia andMicchelli derived certain conditions how this can be checked (see [12], x 4.) In ourcase, it turns out that the claim stated above is ful�lled i�det a(z) a(�z)c(z) c(�z) ! 6= 0; j z j= 1: (72)For c(z) according to (69) we get:det a(z) a(�z)c(z) c(�z) ! = z3((2 cos2 �4 + 1) cos4 �4 + (2 sin2 �4 + 1) sin4 �4)6= 0; j z j= 1:When dealing with biorthogonal wavelet bases, condition (72) is always satis�ed.Furthermore, biorthogonal systems have the advantage that one only needs �nitelymany coe�cients for the decomposition. This is not always true for the non-canonicalsolutions of (49).The condition (72) can be used to treat more general di�erential operators. Forexample, consider the problem�u00 + �u = g on [0;M ]: (73)The weak formulation is<< u; v >> +� < u; v >=< g; v >= g�(v): (74)Therefore, to construct an adapted biorthogonal wavelet basis, we look for a function such that< �0(� � k); 0(� � l) > +� < �(� � k); (� � l) >= 0 8k; l 2 Z: (75)A similar calculation as in the proof of Lemma (4.1) shows that (75) is equivalent toa(z)(g(�) + �f(�))c(z) + a(�z)(g(� + 2�) + �f(� + 2�))c(�z) = 0; (76)where g(�) = Xn (� + 4�n)2 j �̂(�=2 + 2�n) j2f(�) = Xn j �̂(�=2 + 2�n) j2 :20

One solution of (76) isc(z) = za(�z)(g(� + 2�) + �f(� + 2�)): (77)For this special solution we getdet0@ a(z) a(�z)c(z) c(�z) 1A = det0@ a(z) a(�z)za(�z)(g(� + 2�) + �f(� + 2�)) �za(z)(g(�) + �f(�)) 1A= �z[j a(z) j2 (g(�) + �f(�))+ j a(�z) j2 (g(� + 2�) + �f(� + 2�))]:It is a well-known fact that a(z) and a(�z) have no common zero on the unit circle(see [3], Theorem 2.1). On the other hand, since the translates of � form a Rieszbasis, there exist constants c1; c2 such thatc1 �Xn j �̂(� + 2�n) j2� c2;(see [15], Chapter II, Theorem 1).Therefore, since g(�) � 0, we see that (72) holds for � > 0. De�ning by � jl thewavelets associated with (77), the whole adapted wavelet basis would consist of thesystem f�(� � k); �22j jl gk;l2Z; j � 0: (78)(The factor 2�2j is caused by di�erentiation in the scalar product << �; � >>).For example, take the multiresolution analysis generated by the centralized B-spline~N2(x) = 1[0;1) � 1[0;1)(x+ 1):Then we get a(z) = 2 cos2(�4)f(�) = 1� 23 sin2(�4)g(�) = 16 sin2(�4):The resulting symbol c(z) isc(z) = 2z sin2 �4(16 cos2 �4 + �(1 � 23 cos2 �4))= ( �12 � 2)z3 � �2 z2 + (4 + 56�)z � �2 + ( �12 � 2)z�1:In �gure [5]-[8] we plotted the wavelets � for some special values of �. This techniquecan now be extended to higher-order problems in a similar way.21

Figure 5: Wavelet � for � = 1Figure 6: Wavelet � for � = 6 22

Figure 7: Wavelet � for � = 12Figure 8: Wavelet � for � = 48 23

7 Estimation of the condition numberIn this chapter we check that, when using the biorthogonal wavelet basis contructedin Chapter 5, the condition numbers of the resulting sti�ness matrices are uniformlybounded. This result is a natural by-product of our construction since we can makeintensive use of the orthogonality condition ful�lled by our biorthogonal wavelet ba-sis. A short time after we �nished the preprint version of this paper, W. Dahmen andA. Kunoth (independently from us) derived general estimates for condition numbersarising from Galerkin methods for elliptic boundary value problems [5]. Their resultsare based on profound approximation theoretic considerations. These technical di�-culties can be avoided in our case by using the special orthogonality.Estimations of condition numbers were also performed by Ja�ard [11], but with re-spect to orthonormal wavelets of non-compact support whose construction is highlydependent on the domain.Let L be a given (one-dimensional) elliptic di�erential operator of second order andlet us consider the problem Lu = g on [0;M ] (79)with periodic boundary conditionsu(0) = u(M); u0(0) = u0(M):(In general, one has a di�erential equation of the form� ddx(a(x)dudx) + b(x)u(x) = g(x)with a(x); b(x) continuous and periodic, 0 < �1 � a(x) � �2 <1; 0 � b(x) � �3 <1.) The weak formulation of this problem isB(u; v) = g�(v) =< g; v > : (80)Furthermore we assume the resulting energy norm B(u; u)1=2 being equivalent eitherto the Sobolev norm, i.e., kuk21;2;[0;M ] � B(u; u) � 0kuk21;2;[0;M ] ; 0 2 R+; (81)or to the Sobolev semi-norm� j u j21;2;[0;M ]� B(u; u) � �0 j u j21;2;[0;M ] �; �0 2 R+:To estimate the condition number it is enough to estimate the Sobolev-norm fromboth sides by the l2-norm of the wavelet coe�cients. Unless stated otherwise, we willhenceforth use the biorthogonal wavelet basis constructed in Chapter 5.24

Let f 2 Vj be a given function (as stated above fVjgj2Z denotes the multiresolutionanalysis according to Theorem 5.1). Then f has the decompositionf = Xk2Z dk�(� � k) + JXj=0Xl2Z�jl jl (82)= Xk2Z dk�(� � k) + JXj=0Xl2Z < f; ~ jl > jlWe de�ne PV0f :=Xk dk�(� � k); (83)which is the projection onto the space V0:Then we haveTheorem 7.1 Let fVjgj2Z be the multiresolution analysis according to Theorem 5.1.Then there exist constants K;K 0 2 R+ such thatK(j PV0f j21;2 + j f j2) �j f j21;2� K 0(j PV0f j21;2 + j f j2) 8f 2 Vj; j � 0; (84)where j � j is de�ned by j f j2= JXj=0 22jXk j �jk j2 : (85)Furthermore, there exist constants ~K; ~K 0 such that~K(kPV0fk21;2+ jkfkj2) � kfk21;2 � ~K 0(kPV0fk21;2+ jkfkj2) 8f 2 Vj ; j � 0; (86)where jk � kj is de�ned by jkfkj2= JXj=0(22j + 1)Xk j �jk j2 : (87)Proof: To prove (84) it is enough to show that there exist constants K1;K2 2 R+such that K1kfk22 �j f j21;2� K2kfk22; f 2 W0: (88)(88) has the consequence22jK1kfk22 �j f j21;2� 22jK2kfk22; f 2 Wj: (89)From (55) we getj f j21;2=j PV0f j21;2 + JXj=0 jXk �jk jk j21;2; f 2 Vj: (90)25

Finally, let us recall (32):K3Xk j �jk j2� kXk �jk jkk22 � K4Xk j �jk j2 : (91)Putting these formulas together we getj PV0f j21;2 + j f j2 = j PV0f j21;2 + JXj=0 22jXk j �jk j2� j PV0f j21;2 + 1K3 JXj=0 22jkXk �jk jkk22� j PV0f j21;2 + 1K3K1 JXj=0 jXk �jk jk j21;2� max(1; 1K3K1 )(j PV0f j21;2 + JXj=0 jXk �jk jk j21;2)� max(1; 1K3K1 ) j f j21;2 :The right-hand side of (84) follows in the same way.It remains to prove (88): The right-hand side is the well-known Bernstein inequality.It can be proved as follows Let f 2 V0 ) f(x) = Pk �k�(x� k))j f j21;2 = kf 0k22 = kXk �k�0(� � k)k22= ZR jXk �k�0(x� k) j2 dx� ZR(Xk j �k jj �0(x� k) j1=2j �0(x� k) j1=2)2dx� ZRXk j �k j2j �0(x� k) jXk j �0(x� k) j dx� d1Xk j �k j2 ZR j �0(x� k) j dx� d2Xk j �k j2� d2c22 kXk �k�(� � k)k22= d2c22 kfk22;where we used the Cauchy-Schwarz inequality and the stability condition (16).26

To prove the left hand side, we use the following inequality of the Jackson-type:kf � P ~V0fk2 � C j f j1;2; (92)where P ~V0 denotes the orthogonal projection onto the space ~V0. (Clearly, P ~V0 is thebest approximation of f in the space ~V0. (92) is a simple consequence of the fact that,according to formula (4), the constant functions can be built from the translates of~�.) By using the fact that W0 ? ~V0, we get for a function f 2 W0:kfk22 =< f; f >=< f � P ~V0f; f > � kf � P ~V0fk2kfk2� C j f j1;2 kfk2:To prove (86), it is su�cient to show that there exist constants K5;K6 2 R+ suchthat K5(kPV0fk22 + JXj=0Xk2Z j �jk j2) � kfk22 � K6(kPV0fk22 + JXj=0Xk2Z j �jk j2): (93)Then (86) follows by adding (93) and (84). To show (93) we �rst remark that thesystem f jkgj;k2Z forms a Riesz basis in the following global sense: 9K7;K8 such thatK7Xj Xk j �jk j2� kXj;k �jk jkk22 � K8Xj Xk j �jk j2 : (94)(94) was proved by Cohen, Daubechies and Feauveau (see [4], Theorem 3.8. Similarthings are also done in [7]). From (94) we getkPV0fk22 + JXj=0 Xk2Z j �jk j2 = kXj<0 Xk2Z�jk jkk22 + JXj=0 Xk2Z j �jk j2� K8Xj<0Xk j �jk j2 + JXj=0Xk2Z j �jk j2� max(K8; 1)(Xj Xk j �jk j2)� max(K8; 1)K7 kXj;k �jk jkk22= max(K8; 1)K7 kfk22:The right-hand side of (93) follows in the same way. 227

Theorem 7.1 can now be used to show that the condition number of the sti�nessmatrix is independent of the re�nement level. Although 7.1 is formulated as a globaltheorem concerning properties of the multiresolution analysis according to Theorem5.1, it can be checked that, because of the periodic boundary conditions, all argumentsremain unchanged in the local situation. Now, let Aj denote the sti�ness matrix atlevel j, then (86) and (84), respectively, have the consequence~K(�; ~A0�) � (�;Aj�) � ~K 0(�; ~A0�) (95)for any coe�cient vector �. ( The brackets denote the Euclidean inner product.)~A0 is a diagonal matrix perturbated by the sti�ness matrix A0 in the upper leftcorner. Therefore, after computing the Cholesky decomposition L of ~A0 (which ischeap because of A0 is in general very small) (95) implies that the spectral conditionnumber of L�1AjL�T is bounded by ~K0~K .For the Dirichlet problem one has the di�culty that the boundedness of the domainrequires the introduction of some special elements. ( The reason is that for thereconstruction of the translates of the generator having support at the boundary oneneeds wavelets partially located outside the domain, see [13] for details.) In this caseone has no longer full orthogonality of levels with respect to the Sobolev-semi-norm.Similar problems occur for the wavelet basis according to (78). Because of the factor2�2j it is not clear that the constants in (91) are independent of j. Nevertheless,because of the nice structure of the sti�ness matrix, such a wavelet basis might beuseful in practice.Higher-order problems can be treated in a similar way.Note The arguments in this chapter were partially inspired by Yserentant's investi-gations in [16].Acknowledgement The authors want to thank Wolfgang Dahmen for many stimu-lating discussions and Ehrhard Behrends for helpful remarks.References[1] Cavaretta, A.S., Dahmen, W., and C.A.Micchelli, Stationary Subdivision, Mem-oirs of Amer. Math. Soc., Vol.93, # 453,1991.[2] Chui, C. and Wang, I., On compactly supported spline wavelets and a dualityprinciple, CAT Report #219, Texas A&M University, 1990.[3] Chui, C. and Wang, I., A general framework of compactly supported splines andwavelets, CAT Report, Texas A&M University, 1990.28

[4] Cohen, A., Daubechies, I., and J.C.Feauveau, Biorthogonal bases of compactlysupported wavelets, Preprint 1990.[5] Dahmen, W., and A. Kunoth, Multilevel Preconditioning, 1992, to appear inNumerische Mathematik.[6] Dahmen, W., and C.A.Micchelli, On stationary subdivision and the constructionof compactly supported wavelets, in Multivariate Approximation and Interpo-lation, Edited by W.Haussmann, K.Jetter, ISNM 94, Birkhaeuser Verlag, 1990,69-89.[7] Dahmen, W., and C.A.Micchelli, Dual wavelet expansions for general scalings,in preparation.[8] Daubechies, I., Orthonormal bases of wavelets with compact support, Comm.Pure and Appl. Math. 41 (1987), 909-996.[9] Eirola, T., Sobolev characterization of solutions of dilation equations, Preprint1991.[10] Glowinski, R., Lawton, W., Ravachol, M., and E. Tenenbaum, Wavelet solutionsof linear and nonlinear elliptic, parabolic and hyperbolic problems in one spacedimension, Preprint, 1989, Aware, Inc., Cambridge, Mass.[11] Ja�ard, S., Wavelet methods for fast resolution of elliptic problems, in: M�ethodesd'ondelettes pour la r�esolution d'�equations aux d�eriv�ees partielles, S. Ja�ard,LAMM-Report, 1990.[12] Jia, R.Q., and C.A. Micchelli, Using the re�nement equations for the construc-tion of Pre-wavelets II: Powers of two, in: Curves and Surfaces, P.J. Laurent, A.Le M�ehaut�e and L.L. Schumaker (eds.), Academic Press, New York, 1991.[13] Lorentz, R.A., and W. Madych, Spline wavelets for ordinary di�erential equa-tions, GmD-Report 1990, to appear.[14] Mallat, S., Multiresolution approximation and wavelet orthonormal bases of L2,Trans. Amer. Math. Soc., 315 (1989), 69-88.[15] Meyer, Y., Ondelettes et Op�erateurs I, Hermann Editeurs des Sciences et desArts, Paris 1990.[16] Yserentant, H., On the multilevel splitting of �nite element spaces, Numer.Math.49 (1986), 379-412. 29