Biorthogonal Spline-Wavelets on the Interval Stability and. RWTH Aachen Aachen. Germany. e{mail:

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1 Biorthogonal Spline-Wavelets on the Interval Stability and Moment Conditions Wolfgang Dahmen Institut fur Geometrie und Praktische Mathematik RWTH Aachen 556 Aachen Germany e{mail: WWW: Angela Kunoth Weierstrass{Institut fur Angewandte Analysis und Stochastik (WIAS) Mohrenstr Berlin Germany e{mail: kunoth@wias-berlin.de WWW: Karsten Urban Institut fur Geometrie und Praktische Mathematik RWTH Aachen 556 Aachen Germany e{mail: urban@igpm.rwth-aachen.de WWW: August 5, 996 AMS subect classication. 5A, 35Q3, 65F35, 65N3, A7, A63. Keywords. Multiresolution analysis on the interval, biorthogonal wavelets, moment conditions, Riesz bases, discrete Sobolev norms.

2 Abstract This paper is concerned with the construction of biorthogonal multiresolution analyses on [ ] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal B-splines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a rst step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly stable bases on each level. As a second step these initial complements are then proected into the desired complements spanned by compactly supported biorthogonal wavelets. Since all generators and wavelets on the primal as well as on the dual side have nitely supported masks the corresponding decomposition and reconstruction algorithms are simple and ecient. The desired number of vanishing moments is implied by the polynomial exactness of the dual multiresolution. Again due to the polynomial exactness the primal and dual spaces satisfy corresponding Jackson estimates. In addition, Bernstein inequalities can be shown to hold for a range of Sobolev norms depending on the regularity of the primal and dual wavelets. Then it follows from general principles that the wavelets form Riesz bases for L ([ ]) and that weighted sequence norms for the coecients of such wavelet expansions characterize Sobolev spaces and their duals on [ ] within a range depending on the parameters in the Jackson and Bernstein estimates.

3 Contents Introduction 3. Background and Motivation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3. Biorthogonal Multiresolution in L (IR) : : : : : : : : : : : : : : : : : : : : : : : : : : : : :.3 The Layout ofthepaper : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 Some General Concepts 6. Two Scale Relations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6. Stability and Approximation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7.3 Stable Completions and Biorthogonal Bases : : : : : : : : : : : : : : : : : : : : : : : : : : 9. Changing Bases Continued : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 Biorthogonal Multiresolution in L ([ ]) 3. Boundary Functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3. Spline Multiresolution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Biorthogonalization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3. Direct and Inverse Estimates, Norm Equivalences : : : : : : : : : : : : : : : : : : : : : : : 3.5 Renement Matrices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Biorthogonal Wavelets on [ ]. An Initial Stable Completion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :. Biorthogonal Wavelet Bases : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 5 Computational Issues and Example 3 5. Some Ingredients of the Construction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 5. Basis Transformations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Example d =3 d ~ =5: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 References 6

4 Introduction The obective of this paper is the construction of biorthogonal multiresolution analyses and corresponding wavelets on the interval [ ] with the following properties: (i) The primal multiresolution consists of spline spaces for any desired degree d ;. (ii) For a given degree d ; of the splines and any ~ d IN, ~ d d such that d + ~ d is even the dual multiresolution has degree ~ d ; of polynomial exactness. (iii) As a consequence of (ii) the biorthogonal spline wavelets have the corresponding number ~ d of vanishing moments. (iv) All generators and wavelets on the primal as well as on the dual side have nitely supported masks so that decomposition and reconstruction algorithms are simple and fast. (v) The wavelets form Riesz bases for L ([ ]). Moreover, discrete norms based on these wavelet expansions characterize Sobolev spaces and their duals on [ ] within a range depending on the regularity and degree of exactness of the involved multiresolution analyses.. Background and Motivation The issue of constructing wavelets on the interval has been recently addressed in several papers (see e.g. [AHJP, CQ, CDJV, CDV, Me]). However, as far as we know none of these approaches meets the above complete list of requirements. While [CDV, CQ] focus on orthogonal decompositions [AHJP] does address biorthogonal multiresolution but fails to build in any polynomial exactness of the dual spaces. Furthermore, neither is it proved there that the central biorthogonalization of properly adusted spanning sets is actually possible nor are the Riesz basis property and related Sobolev norm equivalences established which are of fundamental importance for many applications. It is perhaps instructive to point out why the above requirements are important and why we found it worthwhile investing some further technical eort into their realization. First a few general comments. One important property of wavelets on the line is that the wavelet representation of many operators is (nearly) sparse which is crucial for fast numerical processing. The near sparseness is an immediate consequence of a suciently high number of vanishing moments, see (iii). This, in turn, is implied by the corresponding polynomial exactness of the dual multiresolution throughout the respective domain as required in (ii) above. As for (v), the stability of the multiscale transformations forming the reconstruction and decomposition procedures is known to be equivalent to the Riesz basis property of the wavelet bases. The Sobolev norm equivalences, in turn, are equivalent to the fact that for elliptic problems diagonal scalings of stiness matrices relative to wavelet bases yield uniformly bounded condition numbers and thus facilitate fast iterative solvers [DK, DPS3]. Wavelet schemes for the approximate solution of saddle point problems stemming, for instance, from a weak formulation of the Stokes problem or mixed formulations of second order scalar elliptic equations lead to further examples where the requirements (ii) and (v) are essential. Here it is important to construct pairs of trial spaces for pressure and velocity, say, which are compatible in the sense that the so called Ladysenskaa-Babuska-Brezzi condition is satised. In [DKU] the construction of families of such spaces for any spatial dimension and any degree of exactness was based on suitable biorthogonal multiresolution. Since the pressure is discretized there by the primal multiresolution while velocities are represented in terms of the dual multiresolution it is important that both spaces have sucient polynomial exactness to guarantee accurate solutions. In particular, realizing a higher degree of exactness for the velocities requires the ability of raising the exactness of the dual multiresolution independently of the degree of the primal one as in (ii) above. Again preconditioning of the resulting matrices is based on (v). Another context where the above conditions are relevant is the numerical solution of boundary integral equations. While conventional boundary element methods usually give rise to densely populated matrices wavelet based discretizations often lead to nearly sparse matrices [BCR, DPS, DPS, PS]. The analysis in [DPS3, Schn] yields precise conditions on the wavelets that guarantee asymptotically optimal eciency. By this we mean that the compressed stiness matrices contain only an amountofnonvanishing entries of 3

5 order N, N being the number of unknowns, that diagonal scalings produce uniformly bounded condition numbers, and that the solutions to the compressed systems still exhibit the same asymptotic accuracy as those to the unperturbed problems. When the boundary surfaces are represented by parametric mappings it is convenient to construct the wavelets on the surface by means of parametric mappings of wavelets dened on the unit square [DS]. Thus, again tensor products of wavelets on the unit interval form the core of the construction. Specically, when the integral operators have nonpositive order asymptotic optimality requires for the primal system a higher number of moment conditions than the degree of exactness which rules out orthogonal decompositions and stresses the importance of (ii) and (iii). Also when the operators have negative order preconditioning the compressed matrices requires the validity of Sobolev norm equivalences also for Sobolev spaces of negative order. Again (ii) is needed for this purpose. These are some instances where the results available in the literature were not sucient and thus lead us to the present investigation. One could add other examples such as appending boundary conditions by Lagrange multipliers [K]. This leads to similar requirements on the wavelets dened on the boundary curve or surface. Furthermore, biorthogonal wavelet bases are used for the pressure computation when employing divergence free wavelets for the Stokes problem [U]. Our starting point is a family of biorthogonal multiresolution analyses on IR developed in [CDF]. Specifically, we conne the discussion to the case where the primal multiresolution is generated by cardinal B-splines. Polynomial splines have numerous practical advantages over other types of scaling functions, among them explicit analytic representations and minimal support relative to their smoothness.. Biorthogonal Multiresolution in L (IR) L (IR) iscalledrenable with mask a = fa k g k Z, a k IR, if (x) = k Z a k (x ; k) x IR a:e: (..) We say that two renable functions, ~ form a dual pair if (;k) ~ = k k Z (..) IR where in the sequel for any domain IR (f g) := It is well-known that and ~ can be normalized so that Z (x)dx = Z Z f(x)g(x)dx: ~(x)dx =: (..3) IR IR Let us abbreviate for any collection C L () S(C) = clos L (span C) the L closure of the linear span of C. Itwillbeconvenient to write for g L (IR) g [ k] := = g( ;k) k Z: Thus, dening S = S(f [ k] : k Zg) S ~ = S(f ~ [ k] : k Zg) (..) renability isknown to imply [ ::: S S + ::: clos L ( S )=L (IR) Z [ (..5) ::: S ~ S+ ~ ::: clos L ( ~S )=L (IR) Z

6 and \ \ S = ~S = fg: Z Z Moreover, if ~ have compact support it is easy to see that for c = fc k g k Z `( Z), c k (;k) k Z L(IR) kck`( Z) k Z c k ~ (;k) L(IR) (..6) which due to [ k] L(IR) = kk L(IR) (..7) implies the uniform stability of the scaled dilates, k Z c k [ k] L(IR) kck`( Z) (..8) and likewise for ~. Here a < b means that a can be bounded by some constant multiple of b uniformly in any parameters on which a and b may depend. Similarly, we write a b if a < b and b < a. ~ are called the generators of the multiresolution sequences S = fs g Z, ~ S = f ~ S g Z. Moreover, it will be convenient to refer to S and ~ S as primal and dual multiresolution. Recall that under the given circumstances the polynomial exactness of the spaces S determines their approximation power. We say is exact of order d if all polynomials of degree at most d ; can be written as a linear combination of the integer translates (;k). In fact, dening ~ r (y) := () r ~ (;y) one has then, in view of (..), the explicit representation which will be used frequently later on. IR (..9) x r = r ~ (k)(x ; k) x IR r = ::: d; (..) k Z The concept of biorthogonal wavelets consists now of nding complement spacesw W ~ of S S ~ in S + S ~ +, respectively, satisfying ~W?S W? S ~ (..) so that, by (..5), It is known [CDF] that dening new masks W? ~ W r 6= r: (..) b k := (;) k ~a ;k ~ bk := (;) k a ;k k Z (..3) such spaces W W ~ are generated by the dilates and translates of the functions (x) := b k (x ; k) ~ (x) := ~ bk (x ~ ; k) (..) k Z which satisfy ~ (;k)ir = ~ (;k) IR = k Z ~ (;k) = k k Z: (..5) IR Note that if ~ is exact of order ~ d, one immediately infers from (..) and (..) that has ~ d vanishing moments, i.e., Z IR x r (x)dx = r = ::: ~ d ; : (..6) Finally, a fact of primary importance is that the collections f [ k] : k Zg,f ~ [ k] : k Zg form (biorthogonal) Riesz bases of L (IR) which means that kvk L (IR) (v ~ [ k]) IR (v [ k] ) IR v L (IR): (..7) k Z k Z In fact, the latter relations can be extended to norm equivalences for a certain range of Sobolev spaces. The obective of the subsequent investigation is to construct biorthogonal wavelet bases for L ([ ]) retaining as many properties of the above setting as possible. 5

7 .3 The Layout of the Paper The standard derivation of the above facts makes heavy use of Fourier techniques (see e.g. [CDF]). When working on the interval such techniques are not directly applicable which forces us to resort to alternative tools. Therefore we briey collect in Section a few general concepts which will serve that purpose and will guide later constructions. Some of these results are known, some are implicit in various studies and some are simply folklore. Nevertheless, we hope that the reader will benet from putting them briey together since we feel that they help making several somewhat technical developments more transparent. The main tools are stable completions [CDP] and associated stability criteria as well as a mechanism for generating from some initial multiscale decomposition of a multiresolution space other complements which correspond to biorthogonal wavelets. Moreover, we recall a general criterion for establishing the Riesz-basis property and Sobolev norm equivalences based on direct and inverse estimates [D]. The results presented in this section provide the guideline for the whole subsequent development. In Section 3 we construct spline multiresolution spaces of arbitrary degree on [ ] along with a dual multiresolution satisfying requirements (i) and (ii) above. The basic idea is quite familiar. To construct two collections of spanning sets of multiresolution sequences that are candidates for biorthogonal bases on [ ] one has to form special boundary near basis functions by forming xed linear combinations of translates of scaling functions in such away that the resulting linear spans are still nested and contain all polynomials up to the original degree of exactness (see also [AHJP, CQ, CDV]). Since (see (3..5) below) the support of the dual generator ~ is at least as large as that of the degree of exactness of the dual multiresolution determines the number of summands appearing in the boundary near basis functions. Due to these modications the resulting collections of functions have equal cardinality but are, of course, no longer biorthogonal. Apparently, the fact that a biorthogonalization is actually possible has never been established in previous investigations, not even in the case where no exactness is enforced on the dual side. After completion of this paper we became aware of a similar approach [Ma] where, however, again the biorthogonalization is not rigorously ustied. Therefore, we invest some eort in proving that the involved linear systems are always nonsingular which is the main result in Section 3. Moreover, applying the results from Section yields discrete norms which are equivalent tosobolevnormson[ ] for a range depending on the regularity and the exactness of the generators. Section is devoted to the construction of biorthogonal wavelets. Here our approach diers again in an essential way from previous studies. It is divided into two steps. By adapting a factorization result for biinnite B-spline renement matrices from [DM] to the case at hand, we rst construct in a systematic way a family of initial stable completions. Once this has been accomplished we can again make use of the results in Section to derive next biorthogonal wavelet bases. The primal and dual wavelets all have compact support. Moreover, combining the stability of the completions with the norm equivalences from Section 3 readily conrms that these wavelet bases are Riesz bases for L ([ ]). Furthermore, weighted coecient norms are shown to be equivalent to Sobolev norms within certain ranges of Sobolev exponents. Finally, in Section 5 we briey comment on the actual computation of the various ingredients of the construction and display a list of lter coecients as well as plots of the generators and wavelets for the example d =3 ~ d = 5. Data for several other cases can be obtained from the authors. Some General Concepts. Two Scale Relations When trying to carry over the results from Section. to the interval [ ] we have togiveupon translation invariance. The arguments which we will employ actually hold in greater generality and since they will be used in dierent settings we will formulate the main facts in sucient generality to permit their exible application. For later use we record a few facts from [CDP]. Let H be some Hilbert space with inner product h i and norm kk H.We are interested in spaces of the form S( ),, IN xed, where = f k : k ghare uniformly stable in the sense that uniformly in.herewehave used the shorthand notation T c := c k k : kck`() k T ck H c `( ) (..) k 6

8 It is known that nestedness S( ) S( + ) and stability imply the existence of matrices M = (m l k ) l+ k such that k = l+ m l k + l k : (..) Of course, in the case (..) treated in Section. the renement coecients m l k = a l;k are independent of. However, in particular, when working on the interval it will be much more convenient to regard the above renement relation as a matrix relation. In fact, viewing as above as a vector (..) takes the form T = T +M : (..3) Moreover, denoting by [ Y ] the space of bounded linear operators from a normed linear space into the normed linear space Y one has [CDP] M [`( ) `( + )] km k = O() (..) where km k := sup u`( ) kuk`( )= km uk`(+) : Although in all our applications the index sets will be nite we remark that the results remain valid for innite sets as well where the corresponding matrix-vector operations are to be understood in the sense of absolute convergence. Thus, the situation in Section. is covered as well.. Stability and Approximation Exploiting our shorthand notation a little further we dene for any two collections Hthe matrix h i := (h i) : In particular, for v H, hv i denotes the (row) vectors with entries hv k i, k. We will make use also of the following observation whose proof will be included for the convenience of the reader. Lemma. Let H = L () where isadomaininir n or a manifold and let ~ Hhave the following properties: (a) and ~ are biorthogonal, i.e., h ~ i = I: (..) (b) k k k H k ~ k k H <. (c) and ~ are locally nite, i.e., setting k := supp k ~ k := supp ~ k k there exists a constant C< such that #fk : k \ k 6= g #fk :~ k \ ~ k 6= g C: (..) Then (i) f g := f g f~ g := f~ g are uniformly stable. (ii) Let be a domain with Lipschitz boundary and let l () denote the space ofpolynomials of total degree (at most) l ; on. If l () S( ) then inf kv ; v k < L() v S() hl kvk H l () v H l () where h := sup k fdiam ~ k diam k g. Here H l () is the usual Sobolev space with norm kk H l (). 7

9 Proof: By (b) and (c), one has for k k T ck L ( k) k k c k A < k k c k where k := fk : k \ k 6= g. Summing over k and taking (..) into account provides k T ck L() kck`( ). Furthermore, let v := T c so that by (a) and (b), c k = hv ~ k i < kv k L (~ k). Again summing over k and using (..) yields kck`( ) < kv k L() which proves (i). As for (ii), one has for v H l () and any P l () kv ;hv ~ i k L( k) kv ; P k L( k) + khv ; P ~ i k L( k) < kv ; P k L ( k) + kv ; P k L(^ k) where ^ k := [f~ k : k k g. Since P was arbitrary (ii) follows from a classical Whitney-type estimate, squaring and summing over k, and taking again (..) into account. Remark. The results of Lemma. are readily extended to the case L p () ~ L q () where p + q =, <p q<, replacing Hl () by W l p(). The proof of Lemma. already indicates the usefulness of the proectors which are obviously adoints of each other. Q v := hv ~ i Q v = hv i ~ (..3) Now suppose again that H = L () where is some suciently smooth manifold so that Sobolev spaces H s are well-dened for the range of indices under consideration. First recall the following fact from [CDP]. Remark.3 Let f g be uniformly stable. The Q dened by (..3) are uniformly bounded if and only if f~ g is uniformly stable as well. Moreover, the Q satisfy Q l Q = Q l l (..) if and only if the ~ are also renable, i.e., there exist matrices ~ M = (~m l k ) l+ k dening uniformly bounded mappings from `( )! `( + ) such that ~ T = ~ T + ~ M : (..5) Note that then (..) implies M T ~ M = ~ M T M = I: (..6) The relevance of Lemma. and Remark.3 is explained by the following criterion from [D] for the validity of Sobolev norm equivalences and the Riesz basis property. Theorem. Let f g f ~ g be uniformly stable, renable, biorthogonal collections and let the Q be dened by (..3). If the Jackson-type estimate and the Bernstein inequality inf kv ; v k < L() v V ;s kvk H s () v H s () s n (..7) kv k H s () < s kv k L() v V s (..8) hold for V = S( ) and V = S( ~ ) with n = d ~ d and = t d = ~ t ~ d,respectively, then kvk H s = s k(q ; Q ; )vk L () A = s (;~ t t): (..9) Here we have used the convention that Q ; := and that for s< H s () means the dual (H ;s ()) of H ;s () relative to the dual form induced byh i. 8

10 The norm equivalence (..9) suggests identifying stable bases (and ~ ) of the particular spaces W := (Q ; Q ; )H and W ~ := (Q ; Q ; )H which due to (..) agree with (Q ; Q ; )S( ) and (Q ; Q ;)S( ~ ), respectively. It is well-known that such collections ~ are actually biorthogonal when properly normalized. In fact, the relation S( ~ ; )? W can easily be conrmed as follows. For any v Hone has by (..) and (..5) h(q ; Q ; )v ~ ; i = E E Dhv ~ i M ~ T ~ ; ; Dhv ~ ; i ; ~ ; = hv ~ M T ; ~ i;hv ~ ; i =:.3 Stable Completions and Biorthogonal Bases Given two collections, ~ of biorthogonal functions our goal is to determine next the corresponding collections, ~ of biorthogonal wavelets. Our strategy is to accomplish this in two steps. In many cases it is possible to identify some initial complement ofs( )ins( + ). Then one can proect this complement onto the desired complement (Q + ; Q )S( + ) while preserving stability and compact support of the basis functions. First we need a stability criterion for complement bases. We can formulate this again for the above general Hilbert space setting. Proposition.5 ([CDP]) Suppose that f g is uniformly stable and (..3) holds. Then f [ g for S( + ) is uniformly stable if and only if there exists M [`(r ) `( + )] r := + n such that T = T +M (.3.) and M =(M M ) [`( [r ) `( + )] is invertible and satises = O() : (.3.) In fact, one has c M ; km k M ; ; u u `([r r T u + T u r H c km k ) where c, c are theconstants from the stability relation (..). Writing M ; = G = ; G G, one obtains the reconstruction formula u u r `([r ) (.3.3) T + = T G + T G : (.3.) Given and M,any M [`(r ) `( + )] such that (.3.) holds is called a stable completion of M [CDP]. Clearly the pair of two scale relations (..3), (.3.) together with (.3.) gives rise to cascadic decomposition and reconstruction algorithms whose structure is analogous to the classical wavelet schemes on the real line. Their description in terms of the matrices M e G e, e f g, can be found e.g. in [CDP]. Recall that uniform stability off [ g does by no means imply the Riesz basis property (..7) of the union of the complement bases for all. This is where biorthogonality comes into play (see [D, D]). Thus, given some stable completion M of M we need next a mechanism to generate the stable completion corresponding to the particular complements (Q ; Q ; )S( ). It can be based on the following observation from [CDP]. Suppose in the sequel that the biorthogonal collections f g, f~ g are both uniformly stable and renable with renement matrices M, ~M, i.e., T =T +M ~ T = ~ T + ~ M : (.3.5) Proposition.6 ([CDP]) Let f g, f~ g, M and ~M berelated asabove. Suppose that M is some stable completion of M and that G = M ;.Then M := (I ; M ~M T ) M (.3.6) 9

11 is also a stable completion and G = M ; has the form Moreover, the collections form biorthogonal systems, so that G = ~ M T G : (.3.7) := M T + ~ := G ~ + (.3.8) h ~ i = I h ~ i = h ~ i = (.3.9) (Q + ; Q )S( + )=S( ) (Q + ; Q )S( ~ + )=S( ~ ): (.3.) In view of (.3.5), (.3.9) implies that the collections = [ [ ~ :=~ [ [ are biorthogonal, ~ h ~ i = I ; (.3.) where with r ; := ; k := k ~ ; k := ~ k. Recall that h ~ i = h k ~ k i J = f( k) :kr ; g: ( k) ( k )J Note that with L = ; ~M T M (.3.) the new complement functions k are obtained by updating the initial complement functions k by a linear combination of the coarse generators k. In fact, by (.3.6) and (.3.5), i.e., T =T +M = T + M + T +M L = T + T L k = k + (L ) l k l k r : (.3.3) l The following result is an immediate consequence of Theorem. and Proposition.6. Corollary.7 Under the assumptions of Theorem. one has kvk H s = ; kr s hv ~ ki A = s (;~ t t): (.3.) Note that, in particular, for s = the Riesz basis property relative tol () of the, ~ iscovered.. Changing Bases Continued Clearly relation (.3.) describes a change of bases involving two successive scales. In addition we will also have to change bases within a given scale. It is therefore convenient to use the following simple mechanism for identifying the new renement relations as well as corresponding stable completions. To this end, suppose that is renable with renement matrix M and some stable completion M. Suppose we performachange of bases in S( ), i.e., we will need the corresponding new renement relation. = C (..)

12 Remark.8 For, as above one has where Moreover, is the corresponding stable completion where C ;T M ; G = C T + G C T + conrming (..) and (..3). Further- Proof: T =(C )T =( + )T M C T =T +C ;T more, one has ; C ;T +M C T C;T +M C ;T T = T +M (..) M = C ;T +M C T : (..3) G C T + G C T + M = C ;T +M (..) +M C T = C ;T +M =: G : (..5) C T I C ;T I G C T + which proves the claim. = C ;T +M G C T + = I 3 Biorthogonal Multiresolution in L ([ ]) 3. Boundary Functions Suppose now that ~ form a dual pair of renable functions as in Section. with supp =[` `] (3..) and that is exact of order d. Let a = fa g` k denote the mask of where a k=` k := for k<` or k>`. The essence of the following observation is well-known (see e.g. [AHJP, CDV]). However, since the precise role of the various parameters chosen here will matter and since the renement relation below diers somewhat from the ndings in [AHJP] we will include a proof of the following facts. Lemma 3. Suppose that and dene Then one has where L `;d+r := `; m=;`+ `+`; `;d+r L = ;(r+=) + `;d+r L + r ~ (m) [+ m] + ` ;` (3..) r ~ (m) [ m] IR+ r = ::: d; : (3..3) `+`; m=`+` m=` r ~ (m) `; :=;= ~ r (m) [+ m] q=d m;` e and bxc (dxe) is the largest (smallest) integer less (greater) than or equal to x. C! r = ::: d; (3..) r ~ (q) a m;qa (3..5)

13 Proof: In view of (..), one has m Z ~ r (m) [ m](x) = = ( x) r r = ::: d; : (3..6) Inserting the renement equation on IR (..) written in the form into (3..6) yields or, equivalently, [ m] = ;= = ( x) r = t Z (+)= ( + x) r = t Z m+` t=m+` b t;` c ;= a t;m [+ t] m Z (3..7) m=d t;` e b t;` c r m=d t;` e ~ r (m) a t;m ~ r (m) a t;m Comparing this with (3..6) for exchanged by + results in the identity Now by denition (3..3), (3..6) yields L `;d+r(x) = = ( x) r IR+ ; = ;(r+=) b t;` c ;r r ~ (t) = m=;`+ Splitting the rst sum and using (3..3) gives m=d t;` e [+ t] (x) [+ t] (x): ~ r (m) a t;m: (3..8) r ~ (m) [ m](x) IR+ m=` L `;d+r = ;(r+=) L + `;d+r + = ;(r+=) L + `;d+r + ; m=` ~ r (m) [+ m](x) IR+ ; m=` m=` m+` r ~ (m) ;= r ~ (m) [ m](x) IR+ : m=` r ~ (m) [+ m] IR+ ; r ~ (m) [+ m] IR+ t=m+` m=` a t;m [+ t] IR+ upon inserting (3..7). Exchanging the order of summation in the last term yields L `;d+r = ;(r+=) L + `;d+r + ; t=`+` m=` m=` `+`; = ;(r+=) + `;d+r L + + m=`+` r ~ (m) [+ m] IR+ ;= ~ r (m)a t;m [+ t] IR+ m=` r ~ (m) [+ m] IR+ ;(r+=) ~ r (m) ; ;= s=` ~ r (s) a m;s r ~ (m) [ m] IR+ [+ m] IR+ :

14 Substituting (3..8) in the last sum, one obtains `+`; `;d+r L = ;(r+=) + `;d+r L + + m=`+` m=` b m;` c ;= q=d m;` e `+`; = ;(r+=) + `;d+r L + + `+`; m=`+` m=` ;= `; r ~ (m) [+ m] IR+ ~ r (q) a m;q ; b m;` c q=` r ~ (m) [+ m] IR+ q=d m;` e which, in view of (3..5), is the asserted relation (3..). ~ r (q) a m;q ~ r (q) a m;q [+ m] IR+ [+ m] IR+ Note that ~ r (y) = Z IR (x + y) r ~ (x)dx = r is a polynomial of degree r whose coecients ; r i a recursion (see e.g. [DM] as well as Section 5). 3. Spline Multiresolution R IR i= r Z y i i IR x r;i ~ (x)dx (3..9) x r;i ~ (x)dx can be computed exactly with the aid of We will specify now the primal multiresolution as follows. Let us denote for a sequence of knots t i ::: t i+d by[t i ::: t i+d ]f the d-th order divided dierence of f C d (IR) at t i ::: t i+d. Setting x d + := (maxf xg) d, the cardinal B-spline d ' of order d IN is dened as Thus ' is centered around `(d), i.e., where `(d) :=d mod, and has support d'(x) :=d [ ::: d] ; ;x ;b d c d; + : (3..) d'(x + `(d)) = d '(;x) x IR (3..) supp d ' = (;d + `(d)) (d + `(d)) d =[; d ]:=[` `] (3..3) i.e., d = ` ; ` and `(d) =` + `. Thus, the B-splines of even order are centered around while the ones of odd order are symmetric around. The B-spline d' is renable with nitely supported real mask a = fa k g` k=`, i.e., d'(x) = ` k=` ;d d k + b d c d'(x ; k) =: ` k=` a k '(x ; k): (3..) It has been shown in [CDF] that for each d and any ~ d d, ~ d IN, sothatd + ~ d even, there exists a function d ~d ~' L (IR) with the following properties (see [CDF]): (i) d d ~ ~' has compact support, h i supp d d ~ ~' = ; d ; d ~ ++ `(d) d + d ~ ; + `(d) =[` ; d ~ + ` + d ~ ; ] =: [ ~` ~`]: (3..5) 3

15 (ii) d ~ d ~' is renable with nitely supported mask ~a = f~a kg ~` k= ~`, d ~ d ~'(x) = ~` k= ~` (iii) d ~d ~' has the same symmetry properties as d', i.e., (iv) The functions d ' and d d ~ ~' form a dual pair, i.e., d' d ~d ~'(;k) ~a k d d ~ ~'(x ; k): (3..6) d d ~ ~'(x + `(d)) = d d ~ ~'(;x) x IR: (3..7) IR = k k Z: (3..8) (v) d d ~ ~' is exact of order d, ~ i.e., all polynomials of degree less than d ~ can be represented as linear combinations of the translates d ~d ~'(;k) k Z. (vi) The regularity of d ~ d ~' increases proportionally with ~ d. One easily checks that the symmetry properties (3..), (3..7) have the following discrete counterparts a k = a`(d);k ~a k =~a`(d);k k Z: (3..9) In the following d ~ d will be arbitrary as above but xed so that we can suppress them as indices and write briey ' ~'. The following construction of biorthogonal multiresolution analyses on [ ] follows rst again familiar lines in that we retain translates of dilated scaling functions ' ~' whose supports are fully contained in [ ]. Since by (3..5) the support of ~' is at least as large than that of ', i.e., ~` ` ; ~` ;` (even if ~ d<d), we consider rst the dual collections and x some integer ~` satisfying ~` ~` (3..) so that the indices ~ := f ~` ::: ; ~` ; `(d)g (3..) correspond to translates ~' [ m] whose support is contained in [ ]. To preserve polynomial exactness of degree ~ d ; weneed ~ d additional basis functions near the left and right end of the interval which will be constructed according to the receipe from Section 3.. The corresponding index sets are then ~ L := f~` ; ~ d ::: ~` ; g ~ R := f ; ~` +; `(d) ::: ; ~` + ~ d ; `(d)g: (3..) We have included here a shift by ;`(d) in ~ R to make best possible use of symmetry later. On the primal side we need bases of the same cardinality. Since the degree d; of exactness is in general dierent from ~ d ; the boundary index sets necessarily take the form L := f ~` ; ~ d ::: ~` ; ( ~ d ; d) ; g R := f ; ~` +( ~ d ; d)+; `(d) ::: ; ~` + ~ d ; `(d)g (3..3) so that the interior translates ' [ m] are determined by m where := f ~` ; ( ~ d ; d) ::: ; ~` +( ~ d ; d) ; `(d)g: (3..) Of course, it will always be assumed that is large enough to ensure that ~` ; ~` ; `(d)+( ~ d ; d). By construction we have now ~ := ~ L [ ~ [ ~ R = = L [ [ R : (3..5) Moreover, abbreviating ` := ~` ; ( ~ d ; d) (3..6)

16 we observe that L = f` ; d ::: `; g R = f ; ` +; `(d) ::: ; ` + d ; `(d)g (3..7) = f` ::: ; ` ; `(d)g: Note also that by (3..5) and (3..), ` ~`;( ~ d;d) =;` +` ; so that the functions ' [ m] m, are, under the above assumption on, indeed supported in ( ), i.e., supp ' [ k] [ ] k supp ~' [ k] [ ] k ~ : (3..8) The next step is to construct modied basis functions near the end points of the interval for the primal and dual side. To this end, it will be convenient to abbreviate ~ m r := ~' r (m) m r := ' r (m): (3..9) Since obviously and likewise R R L m r := ( x) r '( x ; m) dx = x r '(x ; m) dx IR R IR ; R R m r := r ( ; x) '( x ; m) dx = ( ; x) r '(x ; m) dx IR IR R R ~ L m r := ( x) r ~'( x ; m) dx = x r ~'(x ; m) dx IR R IR ; R ~ R m r := r ( ; x) ~'( x ; m) dx = ( ; x) r ~'(x ; m) dx IR IR (3..) (3..) we conclude on one hand that L m r ~' [ m] (x) = (r+=) x r m Z mz ~ L m r ' [ m](x) = (r+=) x r m Z mz On the other hand, noting that by (3..), Z x r '(x ; m) dx = Z IR IR IR (3..) also reveals that ( ; x) r '( ; x ; m) dx = R m r ~' [ m] (x) = (r+=) ( ; x) r r = ::: d ~ ; (3..) ~ R m r ' [ m](x) = (r+=) ( ; x) r r = ::: d; : Z ( ; x) r '(x ; ( ; m)+`(d)) dx L m r = m r R m r = ;m;`(d) r r = ::: ~ d ; ~ L m r =~ m r ~ R m r =~ ;m;`(d) r r = ::: d; : (3..3) Furthermore, recalling (3..5), let ~`; ~ m r L := ~' r(m) m r L := ;= Employing (3..9) and (3..3), one veries that L m r = ;= ;d m; ~` e;`(d) q= ; ~`;`(d)+ ;d m;` e;`(d) ~ m r L = ;= q= ;`;`(d)+ q=d m; ~` e R q r ~a + ;m;`(d);q ~ R q r a + ;m;`(d);q: q r ~a m;q : (3..) (3..5) 5

17 Thus dening m r R := ;= R q r~a m;q ~ R m r := ;= ~ R q r a m;q (3..6) q= ;`;`(d)+ q= ; ~`;`(d)+ one obtains R m r = L + ;m;`(d) r ~ R m r = ~ L + ;m;`(d) r : (3..7) We can now follow the lines of Section 3. to dene according to (3..3) L `;d+r := R ;`+d;`(d);r := `; m=;`+ ;`; m= ;`;`(d)+ ~ m r ' [ m] [ ] r = ::: d; ~ R m r ' [ m] [ ] r = ::: d; (3..8) and likewise on the dual side ~ L ~`; ~ d+r := ~ R ; ~`+ ~ d;`(d);r := ~`; m=; ~`+ ; ~`; m= ; ~`;`(d)+ m r ~' [ m] [ ] r = ::: ~ d ; R m r ~' [ m] [ ] r = ::: ~ d ; (3..9) while k := ' [ k] k ~ k := ~' [ k] k ~ : (3..3) In the sequel we will always assume that l m log ( ~` + ~` ; ) + =: (3..3) so that the supports of the left and right end functions do not overlap (but see Remark 5.). The following symmetry relations will be used frequently. Remark 3. One has for x [ ] R ;`+d;`(d);r ( ; x) = L `;d+r(x) r = ::: d; ~ R ; ~`+ ~ d;`(d);r ( ; x) = ~ L ~`; ~ d+r (x) r = ::: ~ d ; : (3..3) and [+ m] (x) = [+ +;m;`(d)]( ; x) = ' ~': (3..33) Proof: The relation (3..33) follows directly from (3..). Moreover, by (3..), (3..3), one has ;`; R ;`+d;`(d);r ( ; x) = = = m= ;`+;`(d) ;`; m= ;`+;`(d) `; m=+`;`(d) ~ ;m;`(d) r = '( ( ; x) ; m) [ ] ~ ;m;`(d) r = '( x ; ( ; m ; `(d))) [ ] ~ m r = '( x ; m) [ ] : 6

18 Since by (3..3) ` ; `(d) =;` we obtain (3..3). Dening now let and similarly Finally, dene Proposition 3.3 k := k k, fl Rg, ' [ k] k, := f k : k g (3..3) ~ := f ~ L k : k ~ L g[f~' [ k] : k ~ g[f ~ R k : k ~ R g: (3..35) S := S( ) S ~ := S(~ ): (3..36) (i) The spaces S and ~ S are nested, i.e., (ii) The spaces S, ~ S are exact of order d, ~ d,respectively, i.e., S S + ~ S ~ S+ : (3..37) d ([ ]) S ~ d ([ ]) ~ S : (3..38) Proof: As for (i), we have toshow that the elements of the collections ~ are all renable. Since by (3..), '( x ; m) = ` k=` a k '( + x ; (m + k)) = m+` k=m+` a k;m '( + x ; k) and since ` + ` theright hand side involves for m only summands for k + the interior functions are obviously renable. On account of (3..), the same holds for the dual side so that assertion (i) follows as soon as we have conrmed renability of the functions k ~ k, k ~, fl Rg. To this end, Lemma 3. immediately yields the renement relations for L k, k L, ~ L k, k ~ L. In fact, we infer from Lemma 3. and (3..9), (3..) that `+`; L `;d+r(x) = ;(r+=) L + `;d+r(x)+ + `+`; m=`+` m=` ~ L m r ' [+ m](x) ~ m r ' [+ m] (x)! (3..39) and ~ L ~`; ~ d+r (x) = ~ L + ~`; ~ d+r (x)+ ~`+~`; + ~`+ ~`; m= ~`+ ~` m= ~` L m r ~' [+ m] (x): m r ~' [+ m] (x) A (3..) By Remark 3. and (3..7), one obtains L `;d+r(x) = + ;`;`(d) R + +;`+d;`(d);r ( ; x)+ + + ;`;`(d);` m= + ;`;`(d);`+ m= + ;`;`(d);`+ ~ R + m r ' [+ m]( ; x) A ~ R m r ' [+ m]( ; x): (3..) 7

19 At the right end,wethus obtain for r = ::: d; the renement relations R ;`+d;`(d);r (x) and for r = ::: ~ d ; + ;`;`(d) =;(r+=) R + + ;`+d;`(d);r (x)+ + + ;`;`(d);` m= + ;`;`(d);`+ m= + ;`;`;`(d)+ + ~ R ; ~`+ d;`(d);r ~ (x) ~ ; ~`;`(d) =;(r+=) R + + ; ~`+ d;`(d);r ~ (x)+ + + ; ~`;`(d); ~` m= + ; ~`;`(d); ~`+ ~ R + m r ' [+ m](x) ~ R m r ' [+ m](x) (3..) m= + ; ~`; ~`;`(d)+ R + m r ~' [+ m](x) R m r ~' [+ m] (x): (3..3) The proof of (ii) follows standard lines. It is short enough to be included. First note that by (3..), r r ~ R m r = (r;i) (;) i ~ m i : (3..) i Thus (3..), (3..) and (3..8) provide for any r f ::: d; g and x [ ] i= (r+=) ( ; x) r = = = ;`; m=;`+ `; m=;` r i= ~ R m r ' [ m](x) r i=! r ;`;`(d) (r;i) (;) i ~ m i ' [ m] (x)+ ~ R m r i ' [ m](x) + R ;`+d;`(d);r (x) r (r;i) (;) i L `;d+i(x)+ i m m=` ~ R m r ' [ m](x) + R ;`+d;`(d);r (x) which conrms the rst part of (3..38). The rest is completely analogous. 3.3 Biorthogonalization By construction, the spanning sets and ~ from (3..3) and (3..35) have equal cardinality. It remains to verify next that these sets of functions are linearly independent and, which is a stronger property, that and ~ can be biorthogonalized. Thus, we seek coecients e k m, k m ~, fl Rg, suchthat the functions ~ k := e ~ k m m k ~ (3.3.) satisfy Dening the generalized gramian m ~ k ~ k = k k k k ~ fl Rg: (3.3.) [ ] ; := ( k ~ m) [ ] k m ~ (3.3.3) 8

20 the matrix E ; := e k m k m ~ satises (3.3.) if and only if E T = ;; fl Rg: (3.3.) Thus, we have to conrm that ; is always nonsingular. We will have frequent opportunity to exploit the symmetry relations (3..3) and (3..33) in order to relate quantities indexed by L to those indexed by R. It will therefore be convenient to denote for any matrix M by M l the matrix which is obtained by reversing the order of rows and columns of M. The main result of this section can be formulated as follows. Theorem 3. The matrices ; are always nonsingular. Moreover, they have uniformly bounded condition numbers. That is, the matrices ; L are independent of, while which means (; R ) k m =(; L ) ;`(d);k ;`(d);m, k m ~ R. ; L = ; L (3.3.5) ; R = ; l L (3.3.6) Proof: By (3..8) we have forr = ::: d; and k = ::: ~ d ; `;d+r ~ L ~`; ~ d+k [ ] = = `; ~`; =;`+ =; ~`+ `; ~`; =;`+ =; ~`+ Similarly one obtains for r = d ::: ~ d ;, k = ::: ~ d ;, `;d+r ~ L ~`; d+k ~ = [ ] ~`; =; ~`+ k Z ~ r k ; '[ ] ~' [ ] [ ] ~ r k Since for and ;` + ` ;, ; ~` + ~` ; Z '(x ; )~'(x ; )dx = Z Z '(x ; )~'(x ; )dx: '(x ; (` ; d + r)) ~'(x ; )dx: '(x ; )~'(x ; )dx (3.3.7) (3.3.5) follows, while (3.3.6) is an immediate consequence of the symmetry relations (3..3), (3..33). Thus, it remains to conrm that ; L is nonsingular. Let us consider rst the special case d =,i.e., '(x) = [ ).Here` =,` = and (3..8) gives `; (x) = `; m= ~ m ' [ m] (x) = R since, by (..9), ~ m = ~'(x ; m)dx =. Therefore, one has IR `; m= ' [ m] (x) (3.3.8) `; ~ L ~`; d+k ~ [ ] = = = `; ~ L ~`; d+k ~ = IR ~`; m=; ~`+ `; Z = IR m k `; = ~`; m=; ~`+ ; '[ ] ~' [ m] IR = `; m k ; `; ~' [ m] IR = k x k '(x ; )dx = `; (( +) k+ ; k+ )= `k+ k + k + = 9

21 i.e., Moreover, since s = ' [ s], s = ` ::: ~` ;, we have ~ L ~`; d+k ~ [ ] Thus ; L takes for d = the form `; ~ L ~`; d+k ~ = `k+ [ ] k + k = ::: d ~ ; : (3.3.9) ; L = `. = = ~`; m=; ~`+ m k ; '[ ] ~' [ m] IR = k = Z + x k dx k + (( +)k+ ; k+ ) = ` : : : ~` ; : ` (`+) ;`. ~`;( ~`;) `3 ` ~d 3 ~d (`+) 3 ;`3 3 ~`3;( ~`;) 3 3 (`+) ~d ;` ~d ~d. ~` ~d ;( ~`;) ~ d ~d Adding the rst row to the second one, adding the result to the third row and so on produces the matrix ` ` +. ~` ` (`+). ~` `3 ` ~d 3 ~d (`+) 3 3 ~`3 3 (`+) ~ d ~d Dividing the ith row by ` + i ; and multiplying then the i-th column of the resulting matrix by i nally produces a Vandermonde matrix which is nonsingular. This conrms the claim for d =,any ~ d such that d + ~ d is even, and any ~` satisfying (3..). Now suppose that d. We will use induction on d. Tothisend,itwillbeusefultokeep track of the dependence of the various entities on the parameters d d ` ~ ~`. Therefore, we write L k (x) =L k (x d ~ d `) ~ m r =~ m r (d ~ d)=. ~` ~d ~d Z IR C A : C A : x r d d ~ ~'(x ; m)dx: Rewriting formula (3..) in [DKU] in present terms (see also [DS]) yields the relations while d dx L `;d+r(x d ~ d `)= 8 >< >: ; d;' [ `;`(d;)] (x) r =, (r L `;d+r;`(d;) (x d ; d ~ + `; `(d ; )) ;~`; r (d ~ d) d; ' [ `;`(d;)] ) r = ::: d;, (3.3.) d dx k = ; d;' [ k;`(d;)] ; d; ' [ k+;`(d;)] k = ` ` + ::: (3.3.) These relations are obtained by straightforward calculations with the aid of ~ m r (d ~ d) ; ~ m; r (d ~ d)=r~ m;`(d;) r; (d ; ~ d +) r = ::: d; which in turn follow from the denition and d dx d '(x) = d; '(x ; `(d ; )) ; d; '(x + `(d ; ) ; ) (see [DS] for more details). Indicating also the dependence of ; L on d d ~ by writing ; L (d d), ~ let us assume that ; L (s q) is nonsingular for all s<dand all admissible q such thats + q is even. By (3..8) and the denition of k, ~ L k (3..8), (3..9) and (3..) we have `;d+r ~ L ~`; d+k ~ = `;d+r m k ~' [ m] (3.3.) [ ] m=; ~`+ [ ] = = k ; `;d+r () k [ ] k = ::: ~ d ; :

22 Thus ; L is nonsingular if and only if the `;d+r, r = ::: ~ d ;, induce functionals which are total R over d ~. By (3.3.) and (3.3.), we have for any P d ~ and x ^P (x) := P (t)dt ~, d+ d ( `;d+r P) [ ] = ; dx L `;d+r( d d `) ~ ^P = 8 >< >: d;' [ `;`(d;)] ^P ~`; r (d ~ d) ; r [ ] d;' [ `;`(d;)] ^P [ ] r =, [ ] L `;(d;)+r;;`(d;) (d ; ~ d + `; `(d ; )) ^P d;' [ `;d+r;`(d;)] ; d; ' [ `;d+r+;`(d;)] ^P Thus, ( `;d+r P) [ ] =,r = ::: d ~ ;, implies in view of (3.3.3) that d; `;`(d;);(d;)+r ^P = r = ::: d ~ [ ] [ ] [ ] (3.3.3) r = ::: d;, r = d ::: ~ d ;. where d; `;`(d;);(d;)+r := ( L `;`(d;);(d;)+r (d ; ~ d + `; `(d ; )) r = ::: d; d;' [ `;`(d;);(d;)+r] r = d ; ::: ~ d: By our induction assumption and the previous remark this means that ^P, and hence P. This completes the proof. Now let where ~ k are dened for k ~ summarized as follows. ~ = f ~ k : k ~ g (3.3.), fl Rg, by (3.3.) with (3.3.). The above ndings can be Corollary 3.5 The following holds: (i) The collections, ~ dened by (3..3) and (3.3.) are biorthogonal. (ii) dim S =dim~ S =# =d + ; `(d) ; ` += ; (` ; d) ; `(d)+: (3.3.5) (iii) The basis functions have small support, i.e., (iv) The bases f g, f~ g are uniformly stable. diam(supp k ) diam(supp ~ k ) : (3.3.6) (v) The proectors are uniformly bounded. Q v := (v ~ ) [ ] Q v := (v ) [ ] ~ (3.3.7) (vi) The spaces ~ S = S( ~ ) are nested and exact of order ~ d. Proof: (i) follows from (3.3.), (3.3.) and Theorem 3., while (ii) is an immediate consequence of (i) and (3..7). (iii) results from (3..3), (3..5) and the denitions (3..8), (3..9), (3..3). Combining (iii) with the fact that the entries of the matrices E are independent of, (3.3.5) yields k k k k ~ k k < k : (3.3.8) Thus (iv) follows, in view of (i), (iii) and (3.3.8), from Lemma. (i). Finally, (v) is a consequence of (iv) and Remark.3 and (vi) follows from Proposition 3.3 and (3.3.).

23 Although the proof of Theorem 3. makes crucial use of the fact that the primal multiresolution is generated by B-splines it is perhaps worth pointing out that, in principle, the argument can be extended to other cases as well. For instance, consider a Daubechies scaling function d of sucient regularity and order d of exactness. There is a canonical way of generating a family of dual pairs d;r d+r essentially by dierentiation and integration described e.g. in [CF, DKU, Le, U]. One could then employintegration by parts as in the above proof but so that one ends up with a Gramian matrix whose regularity follows from the linear independence of the involved functions. Since the Daubechies scaling functions lack the above nice symmetry properties so that both ends of the interval need separate treatment and since the role of the B-splines as generators will be crucial also later for the construction of biorthogonal wavelets we will not pursue this issue here any further. 3. Direct and Inverse Estimates, Norm Equivalences Combining Corollary 3.5 with Lemma. (ii) provides the following results. Corollary 3.6 One has inf v V kv ; v k L([ ]) < ;s kvk H s ([ ]) v Hs ([ ]) (3..) where s d V = S, ~d V = S ~. (3..) As mentioned before, the Sobolev regularity of ~' is proportional to ~ d. It is actually strictly positive as soon as ~' L (IR) [V]. Let := supfs : ' H s (IR)g = d ; ~ := supfs :~' Hs (IR)g: The following fact follows from [D3]. Proposition 3.7 The inverse estimate kv k H s ([ ]) < s kv k L([ ]) v V (3..3) holds where s< V = S, ~ V = S ~. (3..) Combining Corollaries 3.5 and 3.6 and Proposition 3.7 with Theorem. provides Corollary 3.8 Let the Q be dened by (3.3.7). Then one has (with Q ; := ) = s k(q ; Q ; )vk L ([ ]) = kvkh s ([ ]) s [ d], kvk (H ;s ([ ])) s (;~ ). (3..5) 3.5 Renement Matrices We conclude this section with identifying the renement matrices corresponding to and ~. This will be of crucial importance later on for the identication of stable bases for the complements (Q ;Q ; )S.

24 From Lemma 3. and (3..39) we infer that satises (..3) with M L M := A (3.5.) M R where M L, M R are (d + ` + ` ; ) d blocks of the form (M L ) m k = and by (3..) i.e., 8 >< >: Moreover, A has the form ;(k;`+d+=) k m m k f` ; d ::: `; g = L ;(k;`+d+=) ~ m k;`+d m = ` ::: ` + ` ; k L, (3.5.) ~ m k;`+d L m =` + ` ::: ` +` ; k L, M R = M l L (3.5.3) (M R ) ;`(d);m ;`(d);k =(M L ) m k m = ` ; d ::: ` +` ; k L : (A ) m k = p a m;k ` + ` m ` + + ; (` + `(d)) k : (3.5.) The structure of the renement matrix ~M corresponding to ~ dened in (3..35) is completely analogous and results from replacing ` ` ` d by ~` ~` ~` d, ~ respectively, i.e., ~M L ~M = ~A (3.5.5) ~M R with the ( ~ d + ~` + ~` +) ~ d blocks ( ~ M L) m k = 8 >< >: ;(k;~`+ ~ d+=) k m m k f ~` ; ~ d ::: ~` ; g = ~ L ;(k;~`+ ~ d+=) m k;~`+ ~ d m = ~` ::: ~` + ~` ; k ~ L, L m k; ~`+ ~ d m =~` + ~` ::: ~` + ~` ; k ~ L, (3.5.6) and as well as ~M R = ~ M L l (3.5.7) ( ~ A ) m k = p ~a m;k ~` + ~` m ~` + + ; ( ~` + `(d)) k ~ : (3.5.8) To determine now the renement matrices for the biorthogonalized bases ~ dened in (3.3.) we write the biorthogonalization in the form ~ = ~ C T ~ (3.5.9) 3

25 where ~C ;; L I ( ; ~`+;`(d)) A (3.5.) ; ; R with ; dened by (3.3.3) and I (r) the r r identity matrix. We readily infer now from Remark.8 that Keeping (3.5.5) in mind and splitting ~M R into two blocks such as ~M L = ~M = ~ C ; + ~ M ~ C : (3.5.) D K D = ;(k;~`+ ~ d+=) k m k m ~ L with K dened by (3.5.6) one easily conrms from (3.5.5) and (3.5.) that ~M L ~M = ~A (3.5.) ~M R where now and ~A remains the same as in (3.5.8). ;L D; ~M ; L = L K; ; L ~M R = ~M l L (3.5.3) Biorthogonal Wavelets on [ ]. An Initial Stable Completion The common strategy for constructing now biorthogonal wavelets for a biorthogonal multiresolution as above consists of keeping as many translates [ k], ~ [ k] of the form (..) as possible whose support is suciently inside and complementing this set by a certain nite number of additional functions near the end points [AHJP, CDV, Ma]. These additional functions are, roughly speaking, produced by proecting every second ne scale generator near the end points. Although this may in principle be a feasible approach we still feel somewhat uncomfortable with the reasoning in [AHJP], in particular, with regard to stability of the complement bases. Therefore, we take here a completely dierent route suggested by the general development in Section.3. As a rst step we will construct certain stable complement bases for the spaces S corresponding to a stable completion of the renement matrices of in the sense of Section.3. In a second step these initial complements will be proected into the desired ones employing again the tools from Section.3. Remark. We would like to stress that we do not view the following construction of an initial stable completion merely as an auxiliary ingredient of the nal derivation of biorthogonal wavelets. In fact, the corresponding initial complement bases are interesting in their own right since their elements have small or even minimal support. For instance, in the case d =the interior complement functions correspond to the hierarchical bases from [Y]. Therefore, it may not be surprising that the subsequent proection into biorthogonal bases in Theorem.7 below seems to produce automatically interior wavelets which agree with those derived by [CDF]. The construction of the initial stable completion M of M in (3.5.) consists of several steps, each of which involves dierent matrices which are described most conveniently in a schematic block form. All these matrices will depend only weakly on the scale which means that the entries of the various blocks