Some Remarks on Multiscale Transformations

Some Remarks on Multiscale Transformations, Stability and Biorthogonality Wolfgang Dahmen Abstract.

This paper is concerned with the concepts of stability and biorthogonality for a general framework of multiscale transformations. In particular, stability criteria are derived which do not make use of Fourier transform techniques but rather hinge upon classical Bernstein and Jackson estimates. Therefore they might be useful when dealing with possibly nonuniform discretizations or with bounded domains.

x1. Introduction Let c be some string of data ck ; k 2 I , where I is some ( nite or possibly

in nite) index set. These data could represent grey scale values of a digital image, statistical noisy data, or control points in some curve or surface representation, or approximate solutions of some discretized operator equation. The common ground for these rather dierent interpretations is that these data could be viewed as coecients of some expansion

f=

X k2I

ck 'k ;

(1:1)

where the 'k are (typically scalar-valued shape) functions de ned on some domain (or manifold) (which is topologically equivalent to some bounded or unbounded domain) in IRs. As a familiar simple example one could take 'k as B-splines relative to some knot sequence in an interval . When each ck is a point in IR3 say, f represents a space curve. The ck then already convey explicit geometrical information on the curve or, more precisely, on the location of the points f (x); x 2 . It is well known that this kind of information can Curves and Surfaces II P. J. Laurent, A. Le Mehaute, and L. L. Schumaker (eds.), pp. 1{32. Copyright oc 1991 by AKPeters, Boston. ISBN 0-12-XXXX. All rights of reproduction in any form reserved.

1

be drawn from the data ck under much more general circumstances, namely when the 'k have good localization properties and sum to one. Note that this localization represents the resolution of the underlying object only with respect to a single scale. However, in many applications it is important to extract or exploit information on the data that could be associated with dierent scales ranging from a very coarse to a very ne level. For instance, when dealing with images one would want to separate local from global features. The coecients corresponding to higher scales represent successive updates of increasingly local nature. Moreover, the magnitude of such coecients indicates the signi cance of such updates and therefore forms the basis for a primary objective of multiscale representations, namely data compression. In a similar way they facilitate smoothing noisy data (cf. [19,32]). In fact, these ideas carry much further. It is not only possible to reduce the complexity of images and surfaces or to smooth noisy data but similar principles apply to multiscale representations of operators with gobal Schwartz kernel. Here compression strategies lead to sparse approximations to such operators which in turn can be further processed by ecient iterative solvers (see e.g. [2,13,14,15,16]). When dealing with elliptic operatores, multiscale bases usually form much better approximations to the eigenfunctions of the operator than the nodal bases corresponding to the nest scale only. As a consequence, corresponding stiness matrices can be eciently preconditioned by diagonal scalings [11,20,33]. Aside from compression and preconditioning multiscale representations of data usually convey accurate information about the smoothness or regularity of the underlying continuous object. This fact is for instance exploited (in combination with the above mentioned eects) in [23,24,25] to speed up numerical schemes for hyperbolic conservation laws by signi cantly reducing

ux calculations which in this context form the bulk of computations. The central ingredient of any of the above multiscale applications is an appropriate transformation T relating the ne scale data c to their multiscale representation d c = Td: (1:2) The success of such a scheme then relies mainly on the following two requirements. Firstly, the transformation (1.2) should be ecient. This means it should only require the order of #I oating point operations. Secondly, such transformations should not entail any signi cant loss of accuracy in the data, i.e., the condition numbers should remain uniformly bounded



kTk T?1 = O(1);

#I ! 1;

(1:3)

independent of the size of the problem. This latter requirement is trivially satis ed when T is an orthogonal transformation. This is the case when dealing with wavelet representations, i.e., when d are the coecients of an expansion in terms of (orthogonal) wavelets. However, depending on the setting at hand, the construction of orthogonal multiscale bases by itself could

be a very dicult task and computationally as expensive as the whole problem. This is the case when dealing with unstructured grids or discretizations of bounded domains. Even for uniform grids on all of IRs it is by no means simple to construct compactly supported wavelets. In many applications one therefore has to dispense with orthogonality and be content with stability in the sense of (1.3). This is also the point of view taken in [23,24,25]. There a very general framework is developed which is formulated almost entirely in discrete terms deferring any reference to an underlying system of functions as far as possible. However, when it comes to ascertaining stability (1.3), it seems that one ultimately cannot avoid associating the discrete data with expansions in terms of appropriate basis functions. Nevertheless, the only setting for which this line of arguments has been carried through in a concrete way is the one of classical multiresolution analysis for shift-invariant function spaces on IRs or on the torus [5,17,27,28,29,30]. While stability in the above sense is a trivial attribute of orthogonal wavelets, it takes more eort to be ascertained when dealing with so called biorthogonal wavelets. Sharp stability conditions in connection with biorthogonal wavelets are established in [7]. However, these results rely crucially on Fourier transform techniques which do not apply in many cases of practical interest such as classical nite elements or unstructured grids for bounded domains. The objective of this paper is therefore to discuss some aspects of the issue of stability for multiscale transformations in a suciently general setting covering the above mentioned cases of interest. Several equivalent ways of viewing stability will be pointed out. We are mainly interested in criteria which do not make use of Fourier transforms. In particular, classical Bernstein and Jackson estimates turn out to play a crucial role in this regard. The layout of this paper is as follows. In Section 2 we describe a general format of multiscale transformations and discuss some important requirements imposed by two types of applications. We take the opportunity of reviewing some basic ideas and arguments relating stability to Riesz-bases. The essence of this material is certainly not new. Rephrasing it in the present general context is merely to help focusing on some aspects which might be useful for further progress and to prepare the ground for the development in Section 3. Using standard facts from functional analysis, these concepts are reformulated there in terms of linear projectors associated with nested closed subspaces of a Banach space. This leads to stability estimates for a variety of norms. The main result establishes the equivalence between such norms and multiscale sequence norms expressed in terms of dierences of biorthogonal projectors, assuming only the validity of certain Bernstein and Jackson inequalities. It is interesting to note that such relations are needed for establishing the equivalence between Besov-norms of positive and negative order and certain sequence norms [26] (see Theorem 3.11 below). This is used in [26] for developing multilevel preconditioners in connection by saddle point problems which arise when appending boundary conditions for elliptic problems by means of Lagrange multipliers. We will conclude Section 3 with brie y pointing out how these observations relate to the characterization of biorthogonal Riesz-bases

in the context of classical multiresolution for shift-invariant spaces (cf. [7,8]). We conclude in Section 4 with some comments on the construction of stable biorthogonal bases by means of generalized subdivision schemes.

x2. Multiscale Transformations As in the previous section, c = cj will denote a sequence of data cjk ; k 2 Ij ,

associated with a single discretization level. However, we will deal with a whole hierarchy of discretization levels (of increasing neness) represented by (possibly in nite) index sets Ij ; j 2 IN0 . cj should be viewed as a way of encoding information by means of a single scale of resolution, in brief, single scale data. We wish to reexpress the information carried by such single scale data in terms of multiscale data. Roughly speaking cj is to be split into a blurred version cj?1 and the detail dj?1 lost by this compression. We proceed now describing the form of transformations interrelating both data types. To this end, we will always assume that

Ij+1 ' Ij [ Jj ; Ij \ Jj = ;: (2:1) The data cj are supposed to belong to some ( nite or in nite dimensional) sequence space l(Ij ) endowed with a norm k
kl(Ij). Standard examples are the lp-norms 0 1

kcklp (Ij ) :=

@X

k2Ij

1=p

jck jpA ;

for 1  p  1. L(l(I ); l(I 0 )) denotes the space of bounded linear operators from l(I ) to l(I 0 ) represented by matrices A = (ak;m )k2I 0;m2I (with possibly  (I ) is the dual space of l(I ) relative to the standard complex) entries ak;m . lP form hc; c0 il(I )l(I ) := k2I ck c0k and A is the corresponding adjoint of A. For each level j let

Bj;0 2 L(l(Ij ); l(Ij+1 )); Bj;1 2 L(l(Jj ); l(Ij+1 )):

(2:2)

In view of (2.1), the composed mapping

Bj := (Bj;0 ; Bj;1) (2:3) is to be viewed as an element of L(l(Ij )  l(Jj ); l(Ij+1 )), and in all relevant examples one may identify l(Ij )  l(Jj ) with l(Ij+1 ) so that Bj actually can

be interpreted as a change of bases in l(Ij+1). Apparently this corresponds to a step of a typical pyramid type scheme arising in connection with wavelet decompositions, in that Bj;0 cj+1 stands for a blurred version of cj+1 while Bj;1cj+1 represents the detail lost by this compression. The only dierence is that here the transformations are allowed to depend on the level j in order to deal with bounded domains or unstructured grids. Thus setting

cj?1 := Bj?1;0 cj ; dj?1 := Bj?1;1 cj ; j 2 IN;

(2:4)

leads to the following scheme

Bj?1;0 Bj?2;0 Bj?3;0 cj ! cj?1 ! cj?2 !


c0 Bj?1;1 Bj?2;1 Bj?3;1 & & &


dj?1 dj?2


d0 (2:5) which transforms the single scale data cj into multiscale data d(j) := (c0 ; d0; d1; : : : ; dj?1 )T : (2:6) To reverse the transformation (2.5), i.e., to recover cj from d(j), let Aj;0 2 L(l(Ij ); l(Ij+1 )); Aj;1 2 L(l(Jj ); l(Ij+1 )) (2:7) satisfy or equivalently

Aj;0 Bj;0 + Aj;1 Bj;1 = I;

(2:8)

Bj Aj = I:

(2:9) Here Aj = (Aj;0 ; Aj;1 ) and I denotes the identity in the space l(Ij+1 ); l(Ij ), respectively. The inverse operation to (2.5) can then be described as follows:

A0;0 A1;0 A2;0 Aj?1;0 c0 ! c1 ! c2 !


! cj A0;1 A1;1 A2;1 Aj?1;1 % 1 % 2 %


% 0 d d d

(2:10)

Alternatively, de ning   ^Aj := Aj 0 2 L(l(Ij ) nm?=1j l(Jm ); l(Ij+1 ) nm?=1j+1 l(Jm)); (2:11) 0 I we can write (cf. (2.6)) cn = Tnd(n); (2:12) where Tn := A^ n?1


A^ 0: (2:13) Of course, the particular structure of the transformations (2.5) and (2.10) is motivated by viewing cj as the coecients of some expansion

fj =

X

k2Ij

cjk 'jk

relative to some basis j = f'jk : k 2 Ij g of a space Sj while d(j) are the coecients of the same function fj relative to another basis. Such an underlying system of bases may never enter the computations explicitly but it turns out to be crucial for at least two reasons. It should be relevant for the type of information to be extracted from both representations. It enters when trying to prove various important properties of the transformation detailed below. To be more speci c, let S = fSj gj2IN be a sequence of strictly nested closed subspaces Sj of L2( ). Suppose that each Sj is generated by an l2stable collection of functions . Here l2 -stable means that there exist positive constants c1; c2 such that 0



X

c1 kckl (Ij ) 

ck 'jk

k2I 2

j

L2 ( )

 c2 kckl (Ij ) 2

(2:14)

holds uniformly in c 2 l2(Ij ) and j 2 IN0. We will also use the notation

0Z 11=p kf kLp ( ) := @ jf (x)jp dxA

for 1  p  1 (with the standard interpretation for p = 1), and (f; g) :=

Z

f (x)g(x) dx:



Here and in the sequel we will assume that

j

'k L ( )  1;

(2:15)

2

where `' is to express a relation of the type (2.14) to hold uniformly with respect to all the involved parameters. Remark 2.1. The nestedness of the Sj combined with l2-stability implies the existence of some matrix Aj;0 = (ajk;m )k2Ij ;m2Ij 2 L(l2 (Ij ); l2(Ij+1 )) such that X j j+1 am;k 'm : (2:16) 'jk = +1

m2Ij+1

Aj;0 is sometimes referred to as re nement or subdivision matrix. It is

perhaps worth describing two typical examples.

Example 2.2. Suppose is some polygonal domain in IR2 and let Tj denote the triangulation of obtained by subdividing each triangle in Tj?1 into four congruent triangles introducing the midpoints of the edges in Tj?1 as new vertices. Here one can take Ij as the set of vertices in Tj and 'jk as the unique continuous piecewise linear functions relative to Tj satisfying 'jk (m) = 2j m;k ; m; k 2 Ij : It is easy to see that the j are uniformly stable in the sense of (2.14) and that (2.15) is valid (see e.g. [12]). Moreover, one readily checks that

'jk =

X

m2Ij+1

2?j?1 'jk (m)'jm+1;

i.e., ajm;k := 2?j?1 'jk (m).

Example 2.3. Let = IRs and let M be some xed s  s matrix with integer entries whose eigenvalues are all strictly larger than one (cf. e.g. [21]). Suppose ' 2 L2(IRs) is (a0 ; M )-re nable, i.e., there exists some mask a0 = fa0k gk2ZZs such that X 0 ak '(Mx ? k); x 2 IRs; a:e:: (2:17) '(x) = k2ZZs Thus here one has Ij = M ?j ZZs and

Aj;0 = A0 = (a0m?Mk )m;k2ZZs :

(2:18) Now suppose one wants to reexpress an element in Sj by successively updating lower-level information. To this end, one has to build a multiscale basis by successively completing lower level bases. More precisely, given Aj;0 , i.e., j satisfying (2.16), one has to nd Aj;1 2 L(l2 (Jj ); l2(Ij+1 )); Bj;0 2 L(l2 (Ij ); l2(Ij+1 )); Bj;1 2 L(l2 (Jj ); l2(Ij+1 )); (2:19) such that the functions j k=

satisfy Thus

'jk+1 =

X

m2Ij+1

X m2Ij

ajm;k 'jm+1; k 2 Jj ;

bjk;m 'jm +

X m2Jj

bjk;m mj :

j := f kj : k 2 Jj g completes j to a basis for Sj+1. Following [3] one can state

(2:20) (2:21) (2:22)

Proposition 2.4. Let the matrices Aj;e ; Bj;e ; e 2 f0; 1g satisfy the assumptions of Remark 2.1 and (2.19). Assume that j and j+1 are l2-stable. Then the following statements are equivalent: (i) j , de ned by (2.20) and (2.22) is l2 -stable and satis es (2.21). (ii) The matrices Aj ; Bj , de ned as in (2.3), satisfy

Aj Bj = Bj Aj = I:

(2:23)

Thus under either condition one has

Sj+1 = Sj

M

Wj+1;

(2:24)

where Wj+1 is the (L2 -closure of the) linear span of j . Noting that (2.23) is equivalent to

Bj;e0 Aj;e = e;e0 I; e; e0 2 f0; 1g; Aj;0 Bj;0 + Aj;1 Bj;1 = I; and recalling (2.8), Proposition 2.4 yields Remark 2.5. One has

X

k2In

cnk'nk =

X

k2I0

c0 ' 0

k k+

nX ?1 X j =0 k2Jj

djk

j k

(2:25)

(2:26)

if and only if cn = Tnd(n), where d(n) and Tn are de ned by (2.6) and (2.13), respectively. Thus Tn is the mapping that takes the coecients relative to the multiscale basis ?1 j (2:27) n = 0 [nj=0 into the coecients relative to the single scale basis n. There are two issues of central practical importance that immediately come to mind. Complexity: One would prefer the basis functions in j and j to be all compactly supported. This leaves the chance of nding Aj that are uniformly sparse by which we mean that all rows and columns contain only a uniformly bounded nite number of nonzero entries. If #Ij behaves essentially like a xed fraction of #Ij+1 for all j this would ensure the transformation (2.10) to be ecient in that it requires only O(#Ij ) operations. Likewise, one would want (2.5) to be ecient in the same sense. In this context the following should be kept in mind. Remark 2.6. Usually the matrices Aj;0 are given along with the bases j . One then has to nd suitable (sparse) matrices Aj;1 such that Aj := (Aj;0 ; Aj;1 ) is invertible. A characterization of all possible such completions

Aj;1 is given in [3]. Since the inverse of a sparse matrix is generally not sparse it is then usually hard to nd such Aj;1 that Bj is sparse as well. Stability: Aside from computational complexity, the stability of the transformation Tn is of central importance. More precisely, setting ?1 Jj ; Jn := I0 [nj=0

one has to ensure that for cn = Tnd(n)



kcn kl (In) 

d(n)

l (Jn) ; 2

2

uniformly in n 2 IN, which is equivalent to

kTnk ; T?n 1 = O(1); n ! 1:

(2:28)

Here k
k denotes the operator norm induced by the underlying sequence or vector norm. The following well-known fact indicates how to go about the stability question. Remark 2.7. Suppose the matrices Aj and Bj are associated with j and j through (2.16), (2.20), (2.21), and let Tn be de ned by (2.13). If the j and j are orthonormal bases, then the Tn are uniformly stable, i.e., (2.28) holds. In fact, when dealing with orthonormal bases the transformations Tn are clearly unitary matrices so that kTnk = kT?n 1k = 1. Since in this case the inverse is the adjoint matrix, sparsity of the Aj already implies sparsity of the Bj . Thus if one could construct an orthonormal multiscale basis consisting of compactly supported functions, one would simultaneously solve the complexity and stability problem. However, the concrete construction of such bases suitable for practical purposes seems to be con ned so far to the classical concept of orthonormal wavelets on IRs [17,18,27], i.e., the underlying sequence S consists of shift-invariant spaces generated by dilates and integer translates of a single re nable function ' as described in Example 2.3. Even then one seems to be essentially con ned to tensor products of univariate wavelets. On the other hand, it is also well-known that one needs actually much less. In fact, suppose Aj have been found such that (2.21) holds, and the kj are only orthogonal between levels, i.e., ( kj ; ml ) = 0 for j 6= l and all k 2 Jj ; m 2 Jl:

(2:29)

Since by Proposition 2.4 the j are also l2-stable, one gets for cn = Tnd(n)

P

and fn = k2In cnk'nk

kcn k2l (In) 2

?1 X X 0 0 nX

2

djk  kfn kL ( ) = ck 'k +

k2I j =0 k2Jj

2

X

2 nX ?1

X

+

djk kj

=

c0k '0k

L ( ) k2I L ( ) j =0 k2Jj nX ?1

(n)

2

2 2 j 0



 c l (I ) + d l (Jj ) = d 2

0

0

2

2

0

2 j k

L2 ( )

2

l2 (Jn )

2

j =0

:

(2:30)

Orthogonality between levels, which is much easier to realize than full orthogonality (see e.g. [1,5,22,29,30]), clearly suces to ensure stability, but the matrices Bj controlling the scheme (2.5) are then generally not sparse. Nevertheless, we have repeated the standard reasoning (2.30) only to emphasize that the key to stability estimates lies in switching to function norms, i.e., in making explicit use of the underlying bases j ; j . In fact, replacing the equal sign in (2.30) (which is due to the Pythagorean Theorem) by another `' yields a bit more exibility which can actually help to remedy the above drawback. To describe this, let := 0

1 [ j =0

j :

(2:31)

Moreover, in the following it will be convenient to set

J?1 := I0 ; ?1 := 0; d?1 := c0 ; and

J :=

1 [ j =?1

Jj :

The following proposition summarizes essentially familiar facts but will serve as a motivation for the subsequent discussion. Proposition 2.8. Suppose the j ; j 2 IN0, are uniformly stable and that S is dense in L2( ), i.e., [j2IN Sj = L2( ). Then Tn are uniformly stable, i.e., (2.28) holds, if and only if is a Riesz-basis of L2( ), i.e., every f 2 L2( ) has a unique expansion 0

f=

1 X X j =?1 k2Jj

djk (f )

j k

(2:32)

which converges in L2( ) such that

01 11=2 X X kf kL ( )  @ jdjk (f )j2 A : 2

(2:33)

j =?1 k2Jj

Proof: Suppose rst that (2.28) holds, and let fn = Pk2In cnk'nk denote the orthogonal projection of f 2 L2( ) to Sn, where we assume that f does not vanish identically. Thus, since S is dense, one has for n suciently large that 2 kfnk  kf kL ( )  kfn kL ( ) ? kf ? fnkL ( )  21 kfnkL ( ) : 2

2

Now let

2

2

(2:34)

cn = Tnd(n):

(2:35) By the uniform stability of the n, we infer from (2.34) and (2.35) that there exists some constant c independent of n such that for n suciently large

kf k  c d(n) : L2 ( )

T?n 1

l (Jn) 2

Thus when (2.28) holds, one concludes that



lim sup

d(n)

l (J )  c kf kL ( ) ; n!1

2

n

(2:36)

2

where c is independent of n and f . Since therefore

kfn ? fn+mkL ( )  c

d(n) ? d(n+m)

l (J ) ; 2

2

it is easy to derive from (2.36) the existence of an expansion (2.32) as well as the estimate 0 11=2

c kf kL ( )  @ 2

1 X X

j =?1 k2Jj

jdjk (f )j2 A :

Likewise we deduce from (2.34) and (2.35) that



kf kL ( )  c kTnk

d(n)

l (J ) 2

2

n

which proves (2.33). The converse is trivial. We conclude this section by brie y reviewing two types of applications of the above transformations. The transformation from single to multiscale data (2.5) is needed explicitly when decomposing a signal or image in order to smooth or compress the d(n). In addition, (2.10) is used for reconstruction

from the compressed multiscale data. There is another type of application where the object to be analysed is not given explicitly, but is sought for instance as the solution of an operator equation

Au = f

(2:37)

(with suitable side conditions incorporated in (2.37)) on some domain . Now given such a sequence S of trial spaces, a Galerkin scheme for determining an approximate solution

un =

X

k2In

cnk'nk 2 Sn

of (2.37) requires solving the linear system

An cn = f n;

(2:38)

where fkn := (f; 'nk) and

An = ((A'nk; 'nm ) )k;m2In

is the nodal or single-scale stiness matrix. When n consists of compactly supported functions and A is an elliptic dierential operator of order r the matrix An is sparse but its spectral condition numbers typically grow like O(#Inr=s ). This makes iterative solvers prohibitively slow. On the other hand, direct solvers cannot be applied because the size of such problems is typically very large, and direct solvers usually do not preserve sparsity. In contrast, stiness matrices with respect to appropriate multiscale bases are much better behaved in that when subjected to a simple diagonal scaling resulting condition numbers remain uniformly bounded independent of the size of the problem. As a consequence, iterative schemes such as the conjugate gradient method produce a solution with desired accuracy at the expense of O(#In) operations. Note that (2.38) is equivalent to

A n d(n) = Tnf ; where

A n := TnAn Tn

is the stiness matrix relative to the multiscale basis n. As mentioned before, for suitable choices of Wj+1 or equivalently of Aj;1 , the matrix A n can easily be preconditioned by a diagonal scaling [11]. When A is an operator with global Schwartz kernel, one can show that in contrast to An the matrix A n can be approximated by a sparse matrix [2,14,15]. Both tasks as well as the transformation to the single scale representation (which is needed at the end) require only the process (2.10) involving the matrices Aj .

The above observations show that uniform stability of multiscale transformations is intimately tied to identifying an underlying Riesz-basis. In [8] Riesz-bases are constructed through so called biorthogonal wavelets. Moreover, it is shown there that both the matrices Aj and Bj can be chosen to be sparse. Unfortunately, the approach in [7,8] relies crucially on Fourier transform techniques which are not applicable in the present general setting. The remainder of this paper is therefore devoted to a discussion of various aspects of biorthogonality and to appropriate reformulations. This will ultimately lead us to a dierent type of criteria that ensure certain norm equivalences covering (2.33) as a special case.

x3. Riesz-Bases and Biorthogonality

After interrelating stability and Riesz-bases, we will now tie this into the concept of biorthogonality (see e.g. [8] for the case of shif-invariant spaces). We adhere to the notation of the previous section and suppose that de ned in (2.31) forms a Riesz-basis of L2( ). Thus every f 2 L2 ( ) has an expansion (2.32) satisfying (2.33). Hence the mapping f ! djk (f ) is a bounded linear functional on L2( ). Therefore there exists a ~kj 2 L2( ) such that and therefore where

djk (f ) = (f; ~kj )

(3:1)

( kj ; ~ml ) = j;l k;m ; (j; k); (l; m) 2 J ;

(3:2)

J := f(j; k) : k 2 Jj ; j = ?1; : : : ; 1g:

~ Thus the Riesz-basis leads in a natural way to a biorthogonal system ( ; ). Before exploring the properties of such a system in more detail we observe that the above reasoning immediately carries over to a more general setting which may be described as follows. Let F be a Banach space endowed with a norm k
kF . Let F 0 be its dual (or conjugate space), i.e., the space of all bounded linear functionals on F . Denoting the action of f 0 2 F 0 on f 2 F by f 0 (f ) = hf; f 0 i, the space F 0 becomes of course also a Banach space under the norm kf 0 kF 0 = sup jhf; f 0 ij: (3:3) kf kF =1 It will be convenient to assume (f 0 )(f ) = hf; f 0 i when working with the

eld of complex numbers. Furthermore, let l(J ) be the space of all sequences c = fcjk : (j; k) 2 J g such that kckl(J ) is nite where k
kl(J ) is (quasi-) monotone in the following sense. There exists a positive constant such that for all c 2 l(J ) and all c~ such that for some J~  J ,

(j c if j ~ck = k 0 if

(j; k) 2 J~; (j; k) 2 J n J~

one has

k~ckl(J )  kckl(J ) : (3:4) Examples are F = Lp( ), l(J ) = lp(J ); 1  p  1. A collection = f kj : (j; k) 2 J g may be called a Riesz-basis of F if every f 2 F admits a unique strongly

converging expansion in F and if

X

j k

(3:5)

kd(f )kl(J )  kf kF :

(3:6)

f=

(j;k)2J

djk (f )

Since (3.4) implies, in particular, that



jdjk j  f(j;k);(l;m)g(l;m)2J l(J ) kdkl(J ) ;

(3:7)

the same reasoning as above yields a collection ~ = f ~kj : (j; k) 2 J g of bounded linear functionals satisfying h kj ; ~ml i = j;lk;m ; (j; k); (l; m) 2 J :

(3:8)

In fact, a straightforward duality argument based on (3.3) combined with (3.6) and (3.7) yields

j



~k F 0  c f(j;k);(l;m)g(l;m)2J l(J )

(3:9)

for some constant c which does not depend on j; k. Substituting the expansion (3.5) into hf; f 0 i and using duality shows that every f 0 2 F 0 has a unique expansion

f0 =

X

(j;k)2J

h kj ; f 0 i ~kj :

(3:10)

To see how to interpret the convergence on the right hand side of (3.10) let for f 2 F ; f 0 2 F 0

Qnf :=

X

(j;k)2Jn

hf; ~kj i kj ; Q0nf 0 :=

X

(j;k)2Jn

Remark 3.1. Let Qn; Q0n be de ned by (3.11).

h kj ; f 0 i ~kj :

(3:11)

(i) Qn; Q0n are linear projectors and are dual to each other, i.e., hQn f; f 0 i = hf; Q0n f 0 i.

(ii) The Qn ; Q0n are uniformly bounded, and kQnkF = kQ0nkF 0 ;

(3:12)

where k
kF ; k
kF 0 denote also the respective operator norms induced by the norms on F and F 0, respectively. (iii) The right hand side of (3.10) is weak -convergent. Proof: (i) is obvious. As for (ii) note that assuming only the strong convergence in (3.5), the uniform boundedness of the Qn follows from the Uniform Boundedness Principle, while (3.12) is a well-known consequence of duality. Here one doesn't even have to resort to the Uniform Boundedness Principle since the uniform boundedness of the Qn is also an immediate consequence of (3.4) and (3.6). Since jhf; f 0 ? Q0nf 0 ij = jhf ? Qnf; f 0 ij  kf 0 kF 0 kf ? Qn f kF (iii) follows as well. Throughout the following we will assume that F is re exive. To see that then the convergence in (3.10) is even strong, recall the classical Lebesgue estimate: Remark 3.2. For any sequence Q of bounded linear projectors Qj with ranges Sj  F , one has

kQnf ? f kF  (1 + kQnkF ) fninf kf ? fnkF 2Sn

(3:13)

Now let S~ denote the sequence of ranges S~j  F 0 . The S~j are closed since the Q0j are bounded projections. Now let

V :=

[

j 2IN0

S~j ;

and suppose that V 6= F 0 . Thus there exists some f 0 2 F 0 n V and some f 2 F = F 00 such that

hf; f 0 i = 1; hf; g0 i = 0 for all g0 2 V: (3:14) Since Q0n f 0 2 V we have hf; Q0n f 0 i = 0 for all n 2 IN0. By Remark 3.1 (iii) this gives hf; f 0 i = 0 contradicting (3.14). This shows

[

j 2IN0

S~j = F 0 :

(3:15)

Thus recalling Remark 3.1 (ii) and applying Remark 3.2 to Q0 yields Remark 3.3. The expansion (3.10) converges strongly. P Let hd; ciJ := (j;k)2J djk cjk and let l0 (J ) be the corresponding dual of l(J ).

Remark 3.4. Under the above assumptions one has



kf 0 kF 0 

d~

l0(J ) ;

(3:16)

where d~jk := h kj ; f 0 i. Proof: By (3.5) and (3.6), one has for djk (f ) := hf; ~kj i and h kj ; f 0 i := d~jk

kf 0 kF 0 = =

sup jhf; f 0 ij =

kf kF =1

sup j

X

kf kF =1 (j;k)2J

sup jhd(f ); d~ (f 0 )iJ j 

kf kF =1

djk (f )h kj ; f 0 ij

sup jhd; d~ (f 0 )ij;

kdkl(J ) =1

which proves the claim. The above observations show that a Riesz basis leads in a natural way ~ whose more detailed properties can to a biorthogonal pair of bases ( ; ) be desribed in terms of the operators Qn and their duals. These properties may now easily be complemented by the following simple consequences which parallel the classical framework of biorthogonal wavelets. Remark 3.5. Under the above assumptions one has (i) The ranges Sn; S~n of Qn ; Q0n, respectively, are closed linear subspaces of F ; F 0, respectively, satisfying

S0  S1  : : :  Sn  : : : F S~0  S~1 


 S~n  : : : F 0 : Moreover,

[ j 2IN0

Sj = F ;

[ j 2IN0

S~j = F 0 ;

(3:17) (3:18) (3:19)

where the closures are taken with respect to the respective norms. (ii) The mappings Qn ? Qn?1 and Q0n ? Q0n?1 are also projectors with ranges Wn; W~n, respectively, and

Sn = Sn?1  Wn; S~n = S~n?1  W~ n: Moreover, the spaces Wn; W~ n are biorthogonal relative to h
;
i, i.e.,

h(Qn ? Qn?1 )f; (Q0j ? Q0j?1 )f 0 i = 0;

f 2 F ; f 0 2 F 0 ; n 6= j; (3:20)

or equivalently, (Qn ? Qn?1 )(Qj ? Qj?1 ) = j;n(Qn ? Qn?1); j; n 2 IN0:

(3:21)

However, in practical applications, a Riesz-basis is usually not given. Rather the discussion in the previous section shows that one has to construct one in order to ascertain stability. Instead the starting point is usually an ascending sequences S of strictly nested closed subspaces Sj of F

S0  S1  : : :  S n  : : : F which is dense

[ j 2IN0

Sj = F :

(3:22) (3:23)

One then has to search for suitable complements Wj of Sj?1 in Sj such that a stable basis j of Wj becomes part of a Riesz-basis. Moreover, Remark 3.5 suggests to realize these complements Wj by means of dierences of suitable linear projectors Qj onto Sj , namely Wj = (Qj ? Qj?1 )Sj . Remark 3.5 also shows that these complements should have a special biorthogonality structure. This can be conveniently expressed by a simple property of the Qj which is, in particular, easily con rmed to hold for the above projectors de ned in (3.11). The following well-known facts are relevant in this context. Remark 3.6. Let S satisfy (3.22) and let Q be an associated sequence of linear projectors Qj from F onto Sj . Let Q0 denote the sequence of dual projectors Q0j with ranges S~. The following facts are equivalent: (i) Qj Qn = Qj j  n: (3:24) (ii) The Qj ?Qj?1 (and hence the Q0j ?Q0j?1 ) are also projectors and therefore satisfy (3.21) or equivalently

Wj = (Qj ? Qj?1 )Sj = (Qj ? Qj?1 )Sn = (Qj ? Qj?1 )F ; j  n: (3:25) (iii) S~ is also nested S~j  S~j+1; j 2 IN0 : (3:26) (iv) The corresponding complements W~ j satisfy the biorthogonality relation (3.20). Thus all the algebraic properties identi ed above are equivalent to the commutator relation (3.24). We will therefore try to go the other (hopefully more practical) way around starting o with a dense nested sequence S and an associated sequence Q of uniformly bounded projectors satisfying (3.24). We wish to explore to what extent and under which circumstances the above facts about Riesz-bases can be redeveloped along this line. Speci cally, de ning for convenience Q?1 = 0, under the assumption (3.24), the telescoping expansion f=

X

j 2IN0

(Qj ? Qj?1 )f

(3:27)

readily provides then the components fj = (Qj ? Qj?1 )f of f 2 F from Wj , provided the expansion (3.27) converges in a suitable sense. Of course, Remark 3.1 suggests that Q should be uniformly bounded, i.e., kQnkF = O(1); n ! 1. Then the Lebesgue estimate (3.13) again yields Remark 3.7. If S is dense and Q is uniformly bounded, then the right hand side of (3.27) converges strongly to f for every f 2 F . Moreover, if S is in addition nested and Q satis es (3.24) then Q0 is also strongly convergent. The argument is completely analogous to the one in Remark 3.1 and Remark 3.3. (3.27) may be viewed as a basis free analog to (3.5). What is missing is the relation of (3.27) to a suitable discrete norm. Of course, in general, this requires identifying suitable bases in the rst place. However, for the most important applications we have in mind this can actually be deferred a bit. In fact, take F = L2( ); l(J ) = l2(J ). If we had any uniformly l2-stable bases j of Wj at hand, the relation (3.6) would be equivalent to

kf kL ( ) 2

0 11=2 X  @ k(Qj ? Qj?1 )f k2L ( )A



nj2IN o : =

k(Qj ? Qj?1 )f kL ( ) j2IN

l (IN ) 2

0

2

0

2

(3:28)

0

Thus the issue of stability is ultimately reduced to the following task: Given a dense and nested S construct an associated uniformly bounded sequence Q of projectors satisfying (3.24) such that (3.28) holds. We will show next that relations of the type (3.28) and even more general variants do indeed hold if one quanti es the strong convergence of the Q and Q~ a bit and if some mild regularity assumptions are imposed on S and S~. To describe this, let us assume in the following that there exists some one parameter family of nonnegative subadditive functionals !F (
; t); t  0, with the following properties: There exists a positive constant c such that

!F (f; t)  c kf kF ; lim ! (f; t) = 0 t!0 F +

f 2 F: f 2 F:

(3:29) (3:30)

!F (f1 + f2; t)  !F (f1 ; t) + !F (f2 ; t); f1 ; f2 2 F : (3:31) Moreover, for each  > 0 there exists some 0 > 0 such that for all t > 0, f 2F !F (f; t)  0 !F (f; t): (3:32) Thus !F has the properties of a modulus of smoothness. One can quantify now the approximation properties of a given dense sequence S of closed subspaces of F by requiring that a Jackson estimate ?j ) inf k f ? f k  c ! ( f;  (3:33) j F F f 2S j

j

holds for some xed  > 1. In addition we will have to impose a mild regularity assumption on S in terms of the following inverse or Bernstein estimate

?




!F (fj ; t)  c minf1; tj g kfj kF (3:34) which is to hold for some > 0 and all fj 2 Sj , uniformly in j 2 IN0. These notions are illustrated by the following example

Example 3.8. Let be some domain in IRs and let F = Lp( ); F 0 = Lp0 ( ) for 1 < p < 1; p1 + p10 = 1: Let j
j denote a norm on IRs and let

(3:35)

Xl  l  l?j l (?1) f (x + jh) (
h f )(x) := j j =0

be the lth order dierence operator in the direction h. With

h;l := fx 2 : x + jh 2 ; j = 0; : : : lg; the lth order Lp-modulus of continuity of f is de ned as



!l(f; t; )p := sup
lh f Lp ( h;l ) : jhjt

In this case

!Lp ( )(f; t) := !l(f; t; )p (3:36) is known to satisfy (3.30), (3.31), and (3.32), and hence is an admissible choice. For instance, (3.33) is valid for (3.36) with l  1 when Sj is spanned by compactly supported functions which sum to one on . In general, (3.33) is known as a Whitney-type estimate. It is typically valid for some l 2 IN when Sj contains all polynomials of degree less than l and is spanned by compactly supported functions such that the diameter of the supports is uniformly proportional to ?j . For various versions of dierent moduli see e.g. [10]. Likewise such an inverse estimate for (3.36) is known to hold for essentially all nite element spaces, spline spaces, or multiresolution settings of interest for some l 2 IN and some > 0. Usually it is established rst for the basis functions and then extended to all elements in Sj by stability. Furthermore, the above moduli serves to de ne subspaces of F for which the above direct and inverse estimates can be made more precise. For xed  > 1 and 0 < r < , 1  q  1 let

jf jqBqr (F ) :=

X

j 2IN0

jrq !F (f; ?j )q :

Then Bqr (F ) is the space of all functions f 2 F for which

kf kBqr (F ) := kf kF + jf jBqr (F ) is nite. In the special setting of Example 3.8 and !F de ned by (3.36), r ( ) when r < minfl; g. It is known that under Bqr (F ) is the Besov space Bp;q suitable regularity assumptions on one has r ( ) = W r ( ); r > 0; r 62 IN; B2r;2( ) = H r ( ); r > 0; Bp;p p

where H r ( ); Wpr ( ) denote the Sobolev-spaces of order r in L2( ); Lp ( ), respectively (see e.g. [31]). Let us mention next a few consequences of the basic inequalities (3.33) and (3.34). Remark 3.9. Let S satisfy (3.34). Then for each xed r < and 1  q  1 one has Sj  Bqr (F ); j 2 IN0 ; and there exists a constant c independent of 0 < t  r such that

kfj kBqt (F )  c jt kfj kF

(3:37)

holds for all fj 2 Sj uniformly in j 2 IN0. Proof: By (3.34), one has for any fn 2 Sn

jfn jqBqr(F ) =

1 X

j =0 n X

 c 

jrq !F (fn ; ?j )q jrq kfn kqF +

j =0 c nrq kfnkq

F;

1 X j =n+1

jrq (n?j)q kfnkqF

where c depends only on the constant in (3.34) and on ?r. Since one trivially has kfnkBqt (F )  kfn kBqr(F ) for t  r, the assertion follows.

Remark 3.10. Suppose S satis es (3.33) and let Q be any associated se-

quence of uniformly bounded linear projectors. Then

kQj f ? f kF  c !F (f; ?j ); (3:38) where c is independent of j 2 IN0 and f 2 F . Moreover, there exists a constant c such that for all j 2 IN0, 0 < t  r < and all f 2 Bqt (F ) kQj f ? f kF  c ?tj kf kBqt (F ) :

(3:39)

Proof: (3.38) is an immediate consequence of the Lebesgue estimate (3.13) and the uniform boundedness of Q. (3.39) follows from (3.38) and the fact that

11=q 0 X itq !F (f; ?i )q A = ?tj kf kBqt (F ) !F (f; ?j )  ?jt @ i2IN0

(3:40)

holds for all 0  t < and f 2 Bqt (F ). Such estimates can be extended by interpolation and more can be said about bounding the operators Qn on the spaces Bqt (F ). Since this will be used later, we pause here to point out a convenient tool for deriving such estimates. To this end, we will need the following result which was established in [11] for the case (3.35), (3.36). The same arguments work to prove the present somewhat more general version. Theorem 3.11. Let S be a nested and dense sequence of closed subspaces of F and Q an associated sequence of uniformly bounded projectors. Assume that S satis es the Jackson estimate (3.33) and the Bernstein estimate (3.34) for some > 0. Then for all 0 < r < the norm k
kBqr (F ) is equivalent to

0 11=q X kf kF ;q;r := @ jrq k(Qj ? Qj?1 )f kqF A ; j 2IN0

(3:41)

where as before Q?1 = 0. Remark 3.12. Let S ; Q satisfy the assumptions of Theorem 3.11. (i) Then for   t  r < one has

kfnkBqt (F )  c n(t? ) kfnkBq (F )

fn 2 Sn; n 2 IN0 ;

where c is independent of n; t;  . (ii) If in addition to the above assumptions Q satis es (3.24) then

kQnkBqt (F ) = O(1);

n ! 1:

(3:42)

(iii) For   t < there exists a positive constant c such that for all f 2 Bqt (F )

kQn f ? f kBq (F )  c n( ?t) kf kBqt (F ) ; n 2 IN0:

(3:43)

Proof: Since the Qn are projectors onto Sn, Theorem 3.11 yields for fn 2 Sn kfnkqBq (F )  c

n X j =0

jq k(Qj ? Qj?1 )fn kqF

which, on account of (3.39), gives (i). Since by (3.24)

(

for j > n; (Qj ? Qj?1 )Qn = 0 Qj ? Qj?1 for j  n; and similarly

(

for j  n; (Qj ? Qj?1 )(Qn f ? f ) = 0 (Qj?1 ? Qj )f for j > n; the remainder of the assertions is easily derived from Theorem 3.11. Returning now to the issue of establishing an equivalence of the type (3.28), one should note that, in general, one cannot pass r to zero to recover k
kF by the right hand side expression in (3.41). Nevertheless, the following main result of this section tells us under which circumstances this is possible. Theorem 3.13. Let S be nested and dense in a Hilbert space F . Let Q be an associated uniformly bounded sequence of projectors satisfying the biorthogonality relation (3.24) and let S~ consist of the ranges of the dual sequence Q0 . Assume that S and S~ both satisfy Jackson and Bernstein estimates (3.33) and (3.34) (with respect to the same !F = !F 0 ) for some > 0. Then one has 0 11=2

kf kF  @

X

j 2IN0

k(Qj ? Qj?1 )f k2F A :

(3:44)

The proof of this result can be found in [9]. Relations of the form (3.44) are employed in [26] for establishing the equivalence of Besov-norms of positive and negative order with certain discrete norms of the form (3.41) also for negative r. This in turn is used there in the derivation of multilevel preconditioners for saddle point problems which arise when appending boundary conditions in elliptic problems by Lagrange multipliers. A few comments on the assumptions in Theorem 3.13 are in order. In view of Remark 3.9, the spaces Sj and S~j both must possess a little more regularity than just belonging to F , F 0 , respectively, so that by Remark 3.12,

kQn kBqt (F ) = O(1); kQ0n kBqt0 (F 0) = O(1); n ! 1;

(3:45)

holds for t < . By duality this implies that the Qn ; Q0n possess uniformly bounded extensions to somewhat larger spaces, namely that

kQ0nk(Bqt (F ))0 = O(1); kQn k(Bqt0 (F 0))0 = O(1); n ! 1 is also valid for t < . Playing the duality game again also yields

(3:46)

Remark 3.14. Under the assumptions in Theorem 3.13, one has kQnf ? f k(Bqt0 (F 0))0  c ?nt kf kF ; f 2 F ; n 2 IN0 ;

(3:47)

if and only if

kQ0n f 0 ? f 0 kF 0  c ?nt kf 0 kBqt0 (F 0) ; f 0 2 Bqt0 (F 0 ); n 2 IN0:

(3:48)

So far all the assumptions are symmetric with respect to S and S~. Of course, some properties on the dual side such as uniform boundedness of Q0 are already implied. Therefore the question arises as to what extent assumptions on S and Q alone already suce since this is usually the part which is better controlled. Remark 3.14 says that convergence of Q in some `negative norm' already implies an estimate of the type (3.39) for Q0 Thus, for each g0 2 Bqt0 (F 0 ) one has inf kf 0 ? fn0 kF 0  kf 0 ? g0 kF 0 + kg0 ? Q0ng0 kF 0

fn0 2S~n

 c (kf 0 ? g0kF 0 + ?nt kg0 kBqt0 (F 0)):

Since g0 was arbitrary, one obtains inf kf 0 ? fn0 kF 0  c K (f 0 ; ?n; t; F 0 ; Bqt0 (F 0 ));

fn0 2S~n

where

K (f; ; t; U; V ) := ginf (kf ? gkU + t kgkV ) 2V

(3:49)

is called K -functional. In the special case described in Example 3.8 (see (3.36)), it is known (cf. [10]) and the literature quoted there) that, under mild assumptions concerning the regularity of the boundary of , one has for instance K (
; ; l; L2 ( ); H l ( ))  !l(
; ; )2 : Hence the Jackson estimate (3.33) for S~ would actually follow in this case from convergence of Q in the dual spaces H ?t . Let us conclude this section with some brief remarks on how the above observations blend in with the results on biorthogonal wavelets [7,8]. In this case S and S~ are generated by the dilates and integer translates of a single re nable function '; '~ as in Example 2.3 which are biorthogonal, i.e., ('; '(
? k))IR = 0;k ; k 2 ZZ. It is known that the integer translates of such functions must sum to a nonvanishing constant which already implies the Jackson estimate (3.33) for l = 1 in (3.36). Moreover, they must possess some extra regularity [7,8]. Using stability this can be exploited to con rm also the inverse estimate (3.34) as in [11]. However, in this special situation much more is known with regard to to the actual construction of the '; '~ and hence of Q. We will brie y address this issue in the following section.

x4. Subdivision

Adhering to the notation in the previous section, let S be again a sequence of closed nested subspaces of F . Moreover, let j be a stable basis of Sj relative to k
kF ; k
kl(Ij ), i.e.,



X

kckl(Ij) 

ck 'jk

k2Ij F

(4:1)

uniformly in j 2 IN0. Any bounded linear projector Qj on Sj then has the form X Qj f = hf; '~jk i'jk ; (4:2) k2Ij

where the '~jk 2 F 0 satisfy

h'jk ; '~jmi = k;m ; k; m 2 Ij : (4:3) Remark 4.1. Suppose that Q is uniformly bounded in F and that the j are uniformly stable. Then ~ j is uniformly stable relative to k
kF 0 ; k
kl0(Ij ). In fact, by stability of j , one obtains



X c0 '~j



k2I k k

j

F0

j Pk2Ij c0k hf; '~jk ij j Pk2Ij c0k ck j  sup  c sup kck kf kF f 2Sj c2l(Ij ) l(Ij ) = c kc0 kl0(Ij ) :

P

Conversely since for fj0 = k2Ij c0k '~jk one has hf; fj0 i = hQj f; fj0 i the uniform boundedness of Q yields jh Qj f; fj0 ij 0 0 kfj kF  c sup kQ f k  c kc0 kl0(Ij ); j F f 2F where we have used as before the stability of the j in the last step. Clearly stability and nestedness of S implies re nability of j with some Aj;0 2 L(l(Ij ); l(Ij+1 )) (cf. (2.16)). Moreover, one easily veri es the following fact. Remark 4.2. Q satis es the commutator relation (3.24) if and only if ~ j are also re nable, i.e., X j j+1 '~jk = bm;k '~m ; (4:4) where

m2Ij+1

bjm;k = h'jm+1; '~jk i: Moreover, with Bj;0 := (bjm;k )m2Ij ;k2Ij one has Bj;0 Aj;0 = I:

(4:5)

+1

(4:6)

As mentioned before, in practice usually the j satisfying (2.16) are given or chosen. One then has to nd Bj;0 satisfying (4.6), e.g., by completing Aj;0 to an invertible Aj and using (2.23) in Proposition 2.4. It now remains to see whether there exists ~ j  F 0 satisfying (4.4). To this end, let us be more speci c about the setting. In the following let F = Lp( ), l(Ij ) = lp(Ij ) for some 1 < p < 1 so that F 0 = Lp0 ( ) where p1 + p10 = 1 and is (topologically equivalent to) some (possibly unR bounded) domain in IRs . Thus hf; gi = f (x)g(x)dx. Moreover, we will

assume that one can associate with each Ij some partition of into cells Ckj ; k 2 Ij , such that the mesh size max fdiam(Ckj ) : k 2 Ij g = hj tends to zero as j tends to in nity. One typically has hj = ?j for some  > 1. In addition we require these partitions to be quasi-uniform. By this we mean that hj  maxk2Ij diam(Ckj )  mink2Ij diam(Ckj ) uniformly in j . Furthermore, the matrices Aj;0 2 L(l(Ij ); l(Ij+1 )) are supposed to have uniformly bounded norms kAj;0 kL(l(Ij);l(Ij )) := kAj;0k and to be uniformly sparse. In particular, the entries are uniformly bounded. In addition, let kj := fm 2 Ij : ajm;k 6= 0g denote the support of the j th column of Aj;0 and more generally, let kj;l := [m2kj;l? mj+l, kj;0 = kj . Finally, for   Ij let +1

1

() :=

[

k2

Ckj :

Then we assume that for each j 2 IN0 ; k 2 Ij there exists some domain jk 

with diam(jk )  c hj (4:7) such that

(kj;l )  jk for all k 2 Ij ; l 2 IN0: (4:8) These properties are easily veri ed for the stationary matrices Aj;0 = A0 from Example 2.3. when the mask is nitely supported (see e.g. [12]). Now, viewing j as a vector with components 'jk , (2.16) may be rewritten as j = ATj;0 j+1, and for  = fj gj2IN and S as above it will be convenient to abbreviate X j T ck 'k =: j c: 0

k2Ij

Thus re nability of  with respect to (A) = fAj;0gj2IN can be expressed for m > j as Tj c = Tm Ajm?1 c; (4:9) m?1 : l(Ij ) ! l(Im ) where Am j := Am;0


Aj;0 . Obviously, one can view Aj as m ? j steps of a (non-homogeneous) subdivision scheme. From the theory of stationary subdivision schemes one expects a close connection between re nability and convergence of such schemes (cf. e.g. [4]). Since in general j+1 is not obtained from j by dilation there is, in contrast to the situation 0

described in Example 2.3, a certain ambiguity in normalizing j . It is clear that Aj;0 depends on this normalization which therefore has to be treated here with some care when addressing the issue of convergence. To develop a suitable notion of convergence for the present setting let j;p k := (meas (Ckj ))?1=p Ckj so that pj = fj;p k : k 2 Ij g  Lp0 ( ) \ Lp ( ). Now let Cpj := diag ((meas (Ckj ))1=p : k 2 Ij ) and suppose that ^ = f^ j gj2IN , ^ j = f'^jk : k 2 Ij g satis es 0

(Cpj ^ )Tj ej = 1; supp '^jk \ Ckj 6= ;; diam(supp '^jk )  c hj ; k 2 Ij ;

(4:10)

where c is independent of j; k and (ej )k = 1; k 2 Ij . Then one can show that for all f 2 Lp( )

T p0

f ? ^ j j (f ) L ( ) = 0; (4:11) lim j !1 p0

p

j;p0

where j (f ) := fhf; k i : k 2 Ij g. Here ^ need not be re nable. In particular, choosing ^ j = pj and using the uniform stability of pj, it is clear that

p0

= 0; implies f = 0: (4:12) lim  (f ) j !1 j l (I ) p j

We say (A) converges strongly if for every c 2 lp(Ij ) there exists a unique fc 2 Lp( ) such that



p m?1 mlim !1 m (fc ) ? Aj c l (I ) = 0: 0

p m

(4:13)

It is useful to keep in mind that convergence means that





fc ? ^ Tm Ajm?1 c

= 0; lim m!1 L ( ) p

(4:14)

for any c 2 lp(Ij ) whenever ^ satis es (4.10). In fact,





0 m ? 1 T T p ^ ^

fc ? m Aj c Lp ( )  fc ? m m (fc ) Lp ( )

0 +

^ Tm (pm (fc ) ? Ajm?1 c)

L ( ); p

whence the assertion follows upon using (4.10), (4.11), and (4.13). Proposition 4.3. Suppose that under the above assumptions (A) converges strongly. Then for each k 2 Ij , j 2 IN0, there exists a unique 'jk 2 F supported in jk which is re nable, i.e., Tj = Tj+1 Aj;0:

Moreover, the limit of the sequence Amj c in Lp( ) is given by

fc = Tj c:

Proof: Let (ej;k )r = k;r ; k; r 2 Ij , and de ne ^ T m?1 j;k 'jk := mlim !1 m Aj e ;

(4:15) (4:16)

where the limit is to be understood in the above sense. The fact that 'jk is supported in jk can be derived from (4.9). Moreover, since with ^ j = pj

kTj+1Aj;0 ej;k ? 'jk kLp ( )  kTj+1Aj;0 ej;k ? (pm+1)T Amj+1 Aj;0 ej;k kLp ( ) + k(pm+1)T Amj ej;k ? 'jk kLp ( ) ! 0; m ! 1;

re nability of  follows. Now for any neighborhood ? in let ?j := fk 2 Ij : supp 'jk \ ? 6= ;g. For ? suciently large relative to the supports jk , again (4.8) can be used to conclude that for a suitable subdomain ?~ P? one has that fc agrees on ?~ with f~c , where c~ := c j?j . But clearly f~c = k2?j ck 'jk since the sum is nite. This completes the proof. As in the stationary case, there is a partial converse (cf. [4]). To this end, it is reasonable to assume that at least constants can be represented by the elements of j ; j 2 IN0, to guarantee a minimum of approximation power. In fact, in the stationary case described in Example 2.3, re nability of ' is known to imply that its integer translates sum to some nonvanishing constant. Speci cally, it will be convenient to assume that j satis es (4.10). Moreover, if  and ~ are both to satisfy (3.4), they must clearly both be injective. Here  is called injective if for each j 2 IN0 the mapping c ! Tj c is injective on the space of all sequences c on Ij . Also when  and ~ are biorthogonal with all functions satisfying the second part of (4.10) one readily checks that  and ~ are both uniformly stable. These remarks indicate that the assumptions below are in some sense unavoidable in connection with biorthogonality. Proposition 4.4. Suppose that  is injective, uniformly stable, re nable relative to (A) as above, and that j satis es (4.10). Then the subdivision scheme induced by (A) converges strongly. Moreover one has (Cpj+1 )?1 Aj;0 Cpjej = ej+1:

(4:17)

Conversely if (A) converges strongly and (4.17) holds, then the corresponding  whose existence is asserted by Proposition 4.3 satis es (4.10). Proof: By assumption (4.9) and (4.10), one has 1 = (Cpj j )T ej = Tj+1Aj;0 Cpj ej ; 1 = (Cpj+1 j+1 )T ej+1:

By injectivity, the relation (4.17) follows. Setting for any j 2 IN0, c 2 lp(Ij ), fc := Tj c = Tm Ajm?1 c, stability combined with re nability yields, in view of Proposition 4.3,



p0

m(fc ) ? Ajm?1 c

lp (Im)  c

Tm (pm0 (fc ) ? Ajm?1 c)

Lp ( )

0 = c

Tm pm (fc ) ? fc

L ( ) : p

By (4.11), the right hand side tends to zero as m ! 1. Since by (4.17), p T p p T m p j m+1 = 1; Tj Cpj ej = mlim !1(m+1 ) Cm+1 e !1(m+1 ) Aj Cj e = mlim

the rest of the assertion easily follows.

Proposition 4.5. Let  satisfy the assumptions of Proposition 4.4. Assume that there exists a uniformly sparse (B) which converges strongly in Lp0 ( ) and satis es (4.6) for all j 2 IN0. Then there exists ~ j  Lp0 ( ); j 2 IN0; satisfying (4.3) and (4.4). Proof: If (B) converges strongly Proposition 4.3 implies that 0

p T m j;k '~jk := mlim !1(m+1 ) Bj e

is re nable. Now write brie y hj ; ~ Tj i := (h'jk ; '~jmi)k;m2Ij so that

D

j ; ~ Tj

E

= + +

 p0 T m T ~ j ; j ? m+1 Bj

 

 p0 T m  ?
T p m j ? Aj m+1; m+1 Bj ?
T  p0 T m  p m 

Aj

m+1; m+1

Bj :

By convergence of (A) and (B), the rst two summands on the right hand side of the above identity tend to zero as m ! 1, while, by (4.6),

h(Amj )T pm+1; (pm0 +1)T Bmj i = (Amj )T Bmj = (Bmj ) Amj = I: This con rms that (4.3) holds which completes the proof. It is not hard to check that it would actually be sucient to require in Proposition 4.5 that (B) converges weakly, i.e.,

hf; fc ? (pm0 +1)T Bmj ci ! 0; m ! 1; holds for all f 2 Lp( ).

Thus, in general, to assure that the multiscale transformations associated with the cascade schemes (2.5), (2.10) are stable, one would have to prove the convergence of the schemes (A) and (B) where the Aj;0 ; Bj;0 satisfy (4.6). In general this is expected to be a rather dicult task which may strongly depend on the particular case at hand. These observations so far oer only a somewhat more focused formulation of the original problem of stability which, however, might be helpful in concrete cases. When the Aj;0 ; Bj;0 do not depend on j and = IRs one obtains stationary subdivision schemes which are extensively studied in [4]. In this case and for p = 2 necessary and sucient conditions on those Aj;0 = A0 and Bj;0 = B0 are derived in [7] such that biorthogonal systems ; ~ exists which give rise to Riesz-bases. In general, given a biorthogonal pair with associated convergent schemes (A); (B), the construction of the corresponding stable multiscale basis can be summarized as follows. Given completions of Aj;0 ; Bj;0 to Aj ; Bj such that (2.9) holds, the functions X j j+1 ~j X j j+1 j bm;k '~m ; k 2 Jj ; am;k 'm ; k = k= m2Ij+1

m2Ij+1

form biorthogonal Riesz-bases provided that the mappings

Aj 2 L(l(Ij+1 ); l(Ij+1 )); Bj 2 L(l0 (Ij+1 ); l0 (Ij+1 )) are uniformly bounded [3]. In fact, as mentioned before, this latter requirement ensures that the corresponding j ; ~ j are uniformly stable and the equivalence (3.44) takes the form (3.6) since (Qn+1 ? Qn)f =

X

k2Jn

hf; ~n i kn ; (Q0n+1 ? Q0n)f 0 =

X

k2Jn

h n ; f 0 i ~kn

for f; f 0 2 L2( ).

References 1. de Boor, C., R. A. DeVore, and A. Ron, On the Construction of Multivariate (pre{) Wavelets, preprint 1992. 2. Beylkin, G., R. Coifman, and V. Rokhlin, Fast Wavelet Transforms and Numerical Algorithms I, Comm. Pure and Appl. Math. XLIV (1991), 141-183. 3. Carnicer, J. M., W. Dahmen, and J. M. Pe~na, Local Decomposition of Re nable Spaces and Wavelets, Applied and Computational Harmonic Analysis, 3 (1996), 127-153. 4. Cavaretta, A. S., W. Dahmen, and C. A. Micchelli, Stationary Subdivision, Memoirs of the American Math. Soc. Vol. 93, No. 453, 1991. 5. Chui, C. K., J. Stockler, and J. D. Ward, Compactly Supported Box Spline Wavelets, Approximation Theory and its Applications 8 (1992), 77-100.

6. Cohen, A., Rapport Scienti que, Universite Paris IX Dauphine, June 1992. 7. Cohen, A. and I. Daubechies, Non{Separable Bidimensional Wavelet Bases, Preprint AT & T Bell Laboratories, New Jersey, 1991. 8. Cohen, A., I. Daubechies and J.-C. Feauveau, Biorthogonal Bases of Compactly Supported Wavelets, Comm. Pure and Appl. Math. 45 (1992), 485-560. 9. Dahmen, W., Stability of Multiscale Transformations, Journal of Fourier Analysis and Applications, 2 (1996), 341{361. 10. Dahmen, W., R. A. DeVore, and K. Scherer, Multidimensional Spline Approximation, SIAM J. Numer. Anal. 17 (1980), 380-402. 11. Dahmen, W., and A. Kunoth, Multilevel Preconditioning, Numer. Math., 63 (1992), 315{344. 12. Dahmen, W., and C. A. Micchelli, Biorthogonal Wavelet Expansions, in preparation. 13. Dahmen, W., S. Prossdorf, and R. Schneider, Wavelet Approximation Methods for Pseudodierential Equations I: Stability and Convergence, Math. Z., to appear. 14. Dahmen, W., S. Prossdorf, and R. Schneider, Wavelet Approximation Methods for Pseudodierential Equations II: Matrix Compression and Fast Solution, Advances in Computational Mathematics 1 (1993), 259335. 15. Dahmen, W., S. Prossdorf, and R. Schneider, Multiscale Methods for Pseudo-dierential Equations, in Recent Advances in Wavelet Analysis, L.L. Schumaker, G. Webb (eds.), Academic Press, New York, 1994, 191235. 16. Dahmen, W., B. Kleemann, S. Prossdorf, and R. Schneider, A Multiscale Method for the Double Layer Potential Equation on a Polyhedron, in Advances of Computational Mathematics, H.P. Dikshit, C.A. Micchelli (eds.), to appear. 17. Daubechies, I., Orthonormal Bases of Compactly Supported Wavelets, Communications on Pure and Applied Mathematics, 41 (1988), 909-996. 18. Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1992. 19. Donoho, D. L., Nonlinear Solution of Linear Inverse Problems by Waveletvaguelette Decomposition, Technical Report, Department of Statistics, Stanford University, 1992. 20. Jaard, S., Wavelet Methods for Fast Resolution of Elliptic Problems, SIAM J. Numer. Anal. 29 (1992), 965-987. 21. Grochenig, K. H., and W. R. Madych, Haar Bases and Self-ane Tilings of IRn, IEEE Trans. Inform. Th. 38 (2), 1992, 556-568. 22. Jia, R. Q., and C. A. Micchelli, Using the Re nement Equation for the Construction of Pre-wavelets II: Powers of Two, in Curves and Surfaces, P.J. Laurent, A. LeMehaute, L.L. Schumaker (eds.), Academic Press, New York, 1991, 209-246.

23. Harten, A., and I. Yad-Shalom, Fast Multiresolution Algorithms for Matrix-vector Multiplication, ICASE Report No. 92-55, October 1992. 24. Harten, A., Multiresolution Representation of Data, CAM Report 93-13, UCLA, 1993. 25. Harten, A., Multiresolution Algorithms for the Numerical Solution of Hyperbolic Conservation Laws, Courant Math. and Comp. Lab. Report, 1993. 26. Kunoth, A., Multilevel Preconditioning, Ph. D. thesis, FU Berlin, 1994. 27. Meyer, Y., Ondelettes et Operateurs 2: Operateur de Calderon-Zygmund, Hermann, Paris 1990. 28. Mallat, S. G., Multiresolution Approximation and Wavelet Orthonormal Basis of L2 (IR), Trans. Amer. Math. Soc. 315 (1989), 69-87. 29. Micchelli, C. A., Using the Re nement Equation for the Construction of Pre-wavelets, Numerical Algorithms, 1 (1991), 75-116. 30. Riemenschneider, S., and Z. Shen, Wavelets and Pre-wavelets in Low Dimensions, J. Approx. Theory, 71 (1992), 18-38. 31. Triebel, H., Interpolation Theory, Function Spaces and Dierential Operators, North-Holland, Amsterdam, 1978. 32. de Vore, R., B. Jawerth, and V. Popov, Compression of Wavelet Decompositions, American Journal of Math. 114 (1992), 737-785. 33. Yserentant, H., On the Multilevel Splitting of Finite Element Spaces, Numer. Math. 49 (1986), 379-412. Wolfgang Dahmen Institut fu r Geometrie und Praktische Mathematik RWTH Aachen Templergraben 55 52056 Aachen Telefon (49) 241 80 3950 Telefax (49) 241 8888 317 e-mail dahmen@ igpm.rwth-aachen.de