Composite Wavelet Bases for Operator Equations ... - Semantic Scholar
Composite Wavelet Bases for Operator Equations Wolfgang Dahmen, Reinhold Schneider November 29, 1996 Abstract
This paper is concerned with the construction of biorthogonal wavelet bases de ned on a union of parametric images of the unit d-cube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo-di erential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases de ned on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales as well as appropriate moment conditions.
Key Words: Biorthogonal wavelets, norm equivalences, boundary element methods, composite multiresolution, multiscale methods for partial di erential equations
AMS subject classi cation: 65Y20, 68Q25, 65F35, 45L10, 65M99, 76D07
1 Motivation and Background 1.1 Introductory Remarks
The fact that, roughly speaking, the representation of certain (elliptic) operators and their inverses relative to appropriate wavelet bases are nearly sparse or even diagonal have initiateded numerous investigations of wavelet based methods for the numerical solution of various types of operator equations. For instance, the above mentioned near diagonality or more rigorously the fact that Sobolev norms are equivalent for a certain range to weighted sequence norms of wavelet coecients give rise to (asymptotically optimal) preconditioning techniques [DK, DPS, J, O]. The observation made in [BCR] that for certain integral operators wavelet discretizations lead to fast matrix-vector multiplication has since initiated a number of further attempts to develop schemes of this type for the numerical treatment of equations involving operators with global kernels. A few comments on the actual need of such further investigations are in order. The approach proposed in [BCR] hinges on the availability of an appropriate wavelet basis. 1
For periodic problems as considered in [BCR, DPS1, DPS2] a variety of such bases is indeed known. Furthermore, when, unlike the situation considered in [BCR], the order of the operator is di erent from zero, which is for instance the case for the single layer potential or the hypersingular operator, the need for preconditioning enters the solution process putting further constraints on the bases. Finally, the issue of accuracy deserves some additional attention. [BCR] addresses the question of realizing a matrixvector multiplication at the expense of roughly at most N log N operations, N being the size of the matrix, within some prescribed accuracy tolerance which at this point is independent of N though. However, when the matrix-vector multiplication becomes an ingredient of some (stable) numerical scheme for the solution of a corresponding operator equation it would not make sense to choose much smaller than the overall discretization error. Likewise when is large compared to the dicretization error the resolution of the scheme would be wasted and one could have used coarser discretizations and hence smaller systems in the rst place. Thus from a principal point of view, should be related to N and the basic question may then be formulated as follows: Given some multiscale basis, is it possible to replace the sti ness matrix relative to this basis by some sparse matrix which is still well conditioned and has at most O(N ) nonvanishing entries such that the solution of the sparse system still exhibits the same asymptotic convergence to the solution of the operator equation as the solution to the full discrete system? Meanwhile this question can, in principle, be answered for a wide range of operator equations including those of order di erent from zero and for various types of numerical schemes [DPS1, DPS2, DPS, PS, PSS, S]. It can be shown that depending on the type of operator the various e ects can be balanced in such a way that one ends up with a scheme which is asymptotically optimal in the above sense provided that the wavelet basis satis es a number of conditions pertaining mainly to norm equivalences, regularity and moment conditions. In particular, it turns out that orthogonal bases are in many situations (whenever the order of the operator is less than one) not optimal. This suggests resorting to biorthogonal bases which permit realizing a suciently high order of vanishing moments relative to the accuracy of the trial spaces. All these requirements on the wavelet bases are relatively easy to ful l when the underlying domain is the full Euclidean space IRd or the torus. However, the situation changes drastically when dealing with more realistic domain geometries in which case very little is known about appropriate bases. Thanks to numerous studies of wavelets on an interval [AHJP, CDV, CQ, DKU] wavelet bases on cubes are also available. In this paper we shall show that suitable biorthogonal multiresolution spaces on cubes which satisfy certain boundary conditions can be composed via parametric lifting to bases on unions of parametric images of cubes. The resulting composite multiresolution spaces and wavelet bases are very appropriate for domain decomposition techniques. The layout of this paper is as follows. We conclude this introductory section with some comments of the type of operator equations we have in mind. In particular we specify the requirements on the wavelets which cover a wide range of cases of practical interest. In Section 2 wavelets on the interval are revisited where special attention is paid to 2
boundary conditions. In Section 3 we discuss tensor products of such wavelets and their parametric lifting to composite domains. After establishing direct and inverse estimates we con rm the validity of norm equivalences for the range required for the equations considered before.
1.2 Some Operator Equations
Suppose that is a bounded domain or a manifold which admits the de nition of Sobolev spaces H s for a certain range of s. Speci cally, when s < 0 then H s is to be understood as the dual (H ?s ) of H ?s . When is a bounded domain the de nition of H s may incorporate (homogeneous) boundary conditions. Assume that A is a boundedly invertible operator from H t onto H ?t , i.e.,
kAvkH ?t kvkH t ;
(1.2.1)
where a b means that a < b and b < a and the latter relation says that b is bounded by a constant times a uniformly in any parameters on which a; b may depend. We wish to solve Au = f (1.2.2) for any given f 2 H ?t . Now suppose that = f : 2 rg is a Riesz basis for L2( ). Here the index usually has the form = (j; k) where jj := j refers to the scale which for simplicity will always correspond to a meshsize of order 2?j . The index k will generally again be comprised of several indices expressing the type of the wavelet and the location of its support. Of course, for practical purposes, all basis functions should be local, i.e., diam supp 2?jj . On the coarsest level j0 the functions contain roughly speaking the polynomial parts and correspond to scaling functions. It is known [D2] that the Riesz basis property implies the existence of a dual Riesz ~ i.e., basis , (1.2.3) h ; ~0 i = ;0 ; ; 0 2 r: Correspondingly, let S~j := f ~ : jj j g. We will consider Galerkin schemes based on the trial spaces
Sj := span f : jj j g: The following facts are well-known [DPS].
Requirement I: The scaled sti ness matrices Aj := 2?t(jj+j0j)hA 0 ; i jj;j0jj satis es
cond2(Aj ) = O(1); j 2 IN; 3
(1.2.4)
provided that for some ; ~ > 0 the norm equivalence 0
kvkH s @
X
jjj0
2
D 2sjj
11 E 2 2 A
v; ~
(1.2.5)
holds for s 2 (? ~; ) and the regularity bounds ; ~ are related to the order 2t of the operator A by
; ~ > jtj: (1.2.6)
Note that under these circumstances one has in particular by duality 0
kvkH ?t @
X
jjj0
2?2tjj jhv; ij
11 2 2A
;
which plays a crucial role for the analysis of adaptive Galerkin schemes for equations of the form (1.2.2).
Di erential operators: A typical case arises when is a bounded domain and A = ? or A = ?+ cI, c > 0 in which case t = 1, H 1 = H01( ), H 1( ), respectively. Boundary integral equations: More generally, we are interested in solving (1.2.2)
for operators of the the form
Z
(Au)(x) = K (x; y)u(y) dsy = f; ?
(1.2.7)
where the kernel K (; ) satis es estimates of the type
@x @y K (x; y) c ; dist? (x; y)?(2+2t+j j+j j):
(1.2.8)
Examples are the single or double layer potential with t = ?1=2; 0, respectively, obtained e.g. when transforming an exterior boundary value problem for Laplace's equation into an integral equation. More generally, kernels of this type arise e.g. in connection with computing electrostatic elds, scattering form 3D obstacles, transmission problems, and high quality computer visualization based on the radiosity concept (see e.g. [BGZ, SS]). Again (1.2.1) holds for the respective values of t. The major obstacle, when dealing with operators of the type (1.2.7), is that corresponding sti ness matrices are not sparse. However, it can be shown that the matrices Aj are nearly sparse when (1.2.8) holds and
4
Requirement II (Moment conditions): holds. Let 2 := [0; 1]n . is said to have vanishing moments of order d~ if for jj > j0 there exists a smooth mapping : 2 ! ? such that supp (2) and Z y ( (y)) dy = 0; j j < d~ (1.2.9) where y = y1 1 yn n .
2
Let us assume for simplicity that A is selfadjoint and positive de nite, i.e., hAv; vi kvkH t . To be speci c let us assume further that the spaces Sj are accurate of order d, i.e.,
?jd d inf kvj ? vkL2 (?) < 2 kvkH d(?) ; v 2 H (?):
vj 2Sj
(1.2.10)
The ndings in [DPS2, DPS] may then be summarized as follows.
Theorem 1.2.1 Suppose that A satis es the above assumptions and (1.2.8). Furthermore, assume that and ~ as above satisfy (1.2.5) and (1.2.9) with jtj < ; ~; d~ > d ? 2t;
respectively. Then for each j 2 IN 0 there exists a matrix Acj (which is obtained by replacing small entries in the sti ness matrix Aj relative to j by zero) such that the following properties hold: (i)
?1
c c
= O(1);
Aj Aj j ! 1;
i.e., the compressed matrices are well-conditioned. (ii) The number of nonzero entries of Acj is of the order dim Sj .
(iii) The solution ucj of the compressed system Acj ucj = fj , fj := f2?tjjhf; i : jj j g exhibits still an asymptotically optimal convergence order, i.e. denoting by u the solution of (1.2.7), one has
?j ( ?s) kuk u ? ucj
H s(?) < H (?) 2
for ?d + 2t s < , s d, provided that ? is regular enough for H d (?) to be well-de ned. Thus optimal order 2?j(2d?2t) is preserved.
This Theorem describes the ideal situation and will serve as a guide line for the subsequent constructions. In summary we conclude that wavelet bases satisfying (1.2.5) for ; ~ > 1 would provide a suitable tool for a wide class of problems. In particular, this would cover the case of operators of order ?1. In the latter case it is furthermore important to realize a possibly high order d~ of vanishing moments. 5
1.3 Domain Decomposition and Representation of Geometry
The requirements I,II formulated above are in general very hard to satisfy for arbitrary domains. However, whenever a domain admits a reasonable decomposition into parametric images of cubes we will show that it is possible to construct composit wavelet bases that (nearly) ful ll the above requirements. The following setting covers, in principle, a wide range of bounded domains in Euclidean space as well as closed surfaces imbedded in some higher dimensional Euclidean space. In view of boundary integral equations, two-dimensional surfaces in IR3 deserve special attention and serve in fact as the primary motivation of the subsequent developments. Nevertheless, open manifolds such as bounded domains in Euclidean space, are covered as well. Throughout the paper the manifold ? will be assumed to be a piecewise smooth manifold (with or without boundary) which is at least globally Lipschitz endowed with a metric g. Denoting the surface element given in local coordinates by dsx = q g(x; x)dx1 ^ dx2 we can de ne the inner product Z
hu; vi? := u(x)v(x)dsx ?
(1.3.1)
for the space L2(?). In addition to the space L2(?) we will have to work with Sobolev spaces H s (?) de ned on ?. A natural way to de ne these spaces is to view them as trace spaces. For a certain range of s these spaces can be de ned equivalently via an atlas and partitions of unity. We will always assume to work in that range. Thus for Lipschitz surfaces we cover s 1. For detailed treatments of this subject we refer e.g. to [BGZ]. Here it is important to note that for closed surfaces one has H s (?) = H ?s (?); where H s (?) is the normed dual of H s (?) with respect to the inner product (1.3.1). The construction of depends crucially on the way the manifold ? is represented. Piecewise de ned parametric surface representations appear to o er most practicality and rest on the perhaps best developed concepts in Computer Aided Design. In the sequel we will always assume the following mathematical representation of ?. As above we denote by 2 = (0; 1)2 the unit square which will serve as a xed parameter domain and set N [ ? = ?i ; ?i = i(2); i = 1; : : :; N; (1.3.2) i=1
where for the i : IRn ! IRn0 are smooth functions chosen in such a way that ? has a certain desired degree of global smoothness. Note that the patches ?i are not supposed to overlap ?i \ ?j = ;; i 6= j; (1.3.3) m i.e., the di erent patches do not intersect. ? is said to be a C -surface if there exist local C m reparametrizations, i.e., for any neighborhood N ?, say, such that N \ ?i 6= ;, N \ ?j 6= ;, there exists an ane map , a neighborhood M 2 [ (2) and a function 2 C m(M) such that j2\M = i; j(2)\M = j ; (1.3.4) some n n0
6
where is some regular reparametrization of (2). For our purposes it will be convenient to work with the following equivalent formulation. For any two i ; j such that := ?i \ ?j 6= ; there exists a congruence j;i : 2 ! 2 and a regular reparametrization j;i of 2 such that
j;i(?i 1 ()) = (?j 1()); @ i j?i 1()= @ j j;i i;j j?i 1 (); j j := 1 + + n m: (1.3.5) In practical realizations for most of the patches the local reparametrizations can be chosen as the identity which means that (up to the congruences j;i ) the parametrizations join in a way that their coordinate functions are C m. The only places where nontrivial reparametrizations have to be employed is near singular vertices by which we mean vertices sharing a number of patches which is di erent from four. Depending on the genus of the surface such singular vertices may always have to occur. An interesting alternative is to enforce componentwise di erentiability everywhere at the expense of employing degenerate patch representations [R2]. We also remark that in all practical realizations the parametrizations i are actually polynomial or piecewise polynomial with suciently high componentwise smoothness. We have chosen this setting since a variety of practical tools have been developed in the CAD community realizing such surfaces for essentially arbitrary topology. Speci cally, in [HM, R1] a practicable concept for generating C 0, C 1 and C 2 surfaces of arbitrary topology is developed employing only quadrilateral patches as required above. Moreover, these surfaces can be re ned by means of subdivision which therefore ts into the present context. Thus we may view a surface ? of the above type as the true target surface or as an approximation which could be successively improved if necessary.
1.4 Road Map
The above applications give rise to the following wish list: (i) Since the solution of (1.2.2) might be smooth on some of the patches ?i it will be important to realize any desired order d of exactness. Here order d of exactness means that the trial spaces satisfy estimates of the form (1.2.10). (ii) In some applications mentioned above it is important to make the order d~ d of vanishing moments as high as one wishes independently of the exactness order d. This is closely related to the exactness order of the dual multiresolution spaces S~j . (iii) In order to cover all the above mentioned cases (second order di erential operators, single layer, double layer or hypersingular operator) we wish to have the elements of and ~ all be globally continuous. The above setting suggests the following approach. We will rst construct wavelet bases de ned on 2. On account of (ii), the concept of biorthogonal wavelets has to be employed. Such wavelet bases can be constructed by taking tensor products of wavelets 7
on the unit interval [0; 1]. We will therefore make heavy use of some recent results from [DKU] which are to some extent taylored to the present needs. Speci cally, for any d; d~ 2 IN , d + d~ even compactly supported primal and dual wavelets are available. However, since the wavelets on 2 will have to be lifted to the patches ?i through the mappings i (iii) requires paying some attention to piecing the patch bases together. This will be facilitated by realizing certain boundary conditions for the bases on [0; 1]. Finally, the Riesz basis property and related norm equivalences will be established through proving direct and inverse estimates combined with general criteria from [D2].
2 Biorthogonal Multiresolution with Boundary Conditions
2.1 Multiresolution Sequences
The common approach to biorthogonal multiresolution on [0; 1] begins with some dual pair (; ~) of re nable functions, i.e., X X (x) = ak (2x ? k); ~(x) = a~k ~(2x ? k); k2ZZ
and
D
; ~( ? k)
k2ZZ
Z
E
IR
= (x)~(x ? k) dx = 0;k; k 2 ZZ: IR
(2.1.1)
The idea is then to construct (the primal) spaces Sj on [0; 1] by taking those translates (2j ?k) which are supported inside [0; 1] supplemented by certain additional linear combinations of the translates overlapping the end points of the interval. These linear combinations are formed in such a way that the resulting span still contains all polynomials of a desired order (see [AHJP, CQ, CDV, DKU]). To our knowledge only in [DKU] the dual multiresolution spaces S~j induced by ~ also exhibit the original order of polynomial exactness which is crucial in the present context. It is well known that the order of polynomial exactness determines the approximation order of the spaces. An important family of initial dual pairs is based on B-splines. Denoting by [x0; : : :; xd]f the dth order divided di erence of f at the points x0; : : : ; xd 2 IR (see e.g. [dB]), the dth order centered cardinal B-spline is de ned by d?1 (2.1.2) (x) = d(x) := d[0; 1; : : : ; d] ? x ? b 2d c ; + where xl+ := (max f0; xg)l and bxc (dxe) is the largest (smallest) integer less (greater) than or equal to x. Thus is centered around `(2d) , i.e., x 2 IR; (2.1.3) d (x + `(d)) = d (?x); where `(d) := d mod 2, and has support h i j k l m supp d = 21 (?d + `(d)); 21 (d + `(d)) = [? 2d ; 2d ] := [`1; `2]; (2.1.4) !
8
i.e., d = `2 ? `1 and `(d) = `1 + `2. Thus, the B-splines of even order are centered around 0 while the ones of odd order are symmetric around 12 . The B-spline d is re nable with nitely supported real mask a = fak g`k2=`1 , i.e., `2 X
!
`2 X d (2 x ? k ) =: ak (2x ? k): (2.1.5) d (x) = d k + b 2d c k=`1 k=`1 It has been shown in [CDF] that for each d and any d~ d, d~ 2 IN , so that d + d~ even, there exists a function d;d~~ with the following properties (see [CDF]): (i) d;d~~ has compact support, i h suppd;d~~ = ? 21 d ? d~ + 1 + 21 `(d) ; 21 d + d~ ? 1 + 21 `(d) = [`1 ? d~ + 1; `2 + d~ ? 1] =: [`~1; `~2]: (2.1.6)
(ii)
21?d
~ is re nable with nitely supported mask a~,
d;d~
~ d;d~(x) = (iii)
`~2 X k=`~1
a~k d;d~~(2x ? k):
(2.1.7)
~ has the same symmetry properties as d, i.e.,
d;d~
~
~
d;d~(x + `(d)) = d;d~(?x);
x 2 IR:
(2.1.8)
(iv) The functions d and d;d~~ form a dual pair, i.e., hd; d;d~~( ? k)iIR = 0;k ; k 2 ZZ:
(2.1.9)
(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 2 ZZ .
(vi) The regularity of d;d~~ increases proportionally with d~. One easily checks that the symmetry properties (2.1.3), (2.1.8) have the following discrete counterparts ak = a`(d)?k ; a~k = ~a`(d)?k ; k 2 ZZ: (2.1.10) In the following d; d~ will be arbitrary as above but xed so that we can suppress them as indices and write brie y ; ~ if there is no risk of confusion. We will brie y recall next from [DKU] pairs of generator bases j ; ~ j which span multiresolution sequences of spaces Sj ([0; 1]), S~j ([0; 1]) which are exact of order d; d~, respectively. These collections have the form 0j = Lj [ Ij [ Rj; ~ 0j = ~ Lj [ ~ Ij [ ~ Rj; (2.1.11) 9
Setting g[j;k] := 2j=2g(2j ?k), the sets Ij , ~ Ij consist of the interior basis functions [j;k], ~[j;k], k 2 Ij ; ~ Ij , respectively, which do not interfere with the end points of the interval. Here ~ Ij := f`~; : : :; 2j ? `~ ? `(d)g; Ij = f`; : : :; 2j ? ` ? `(d)g: (2.1.12) where ` := `~ ? (d~ ? d): (2.1.13) To ensure that the interior functions are indeed fully supported in [0; 1], `~ has to be only boundeded from below by `~ `~2; (2.1.14) (see (2.1.6). Similarly the collections X : k 2 X g; X :k2 ~ Xj = f~j;k ~ Xj g Xj = fj;k j where for X 2 fL; Rg ~ Lj := f`~ ? d;~ : : : ; `~ ? 1g; ~ Rj := f2j ? `~ + 1 ? `(d); : : : ; 2j ? `~ + d~ ? `(d)g; (2.1.15) and Lj = f` ? d; : : : ; ` ? 1g; Rj = f2j ? ` + 1 ? `(d); : : : ; 2j ? ` + d ? `(d)g: (2.1.16) L , ~L are certain xed linear combinations of the translates [j;k] j[0;1], The functions j;k j;k ~[j;k] j[0;1] chosen so as to ensure that the linear spans of the collections 0j and ~ 0j contain all polynomials of degree less than d; d~, respectively. In fact, set L j;` ?d+r :=
j;R2j ?`+d?`(d)?r :=
`?1 X m=?`2 +1
~m;r [j;m] [0;1] ; r = 0; : : : ; d ? 1; (2.1.17)
2j ?X `1 ?1
m=2j ?`?`(d)+1
~Rj;m;r [j;m] [0;1] ; r = 0; : : :; d ? 1;
and likewise on the dual side
~j;L`~?d~+r := ~j;R2j ?`~+d~?`(d)?r :=
`?1 X ~
m=?`~2 +1
m;r ~[j;m] [0;1] ; r = 0; : : : ; d~ ? 1; (2.1.18)
2j ?X `~1 ?1
m=2j ?`~?`(d)+1
Rj;m;r ~[j;m] [0;1] ; r = 0; : : :; d~ ? 1;
where Lj;m;r := 2j R (2j x)r(2j x ? m) dx = R xr (x ? m) dx =: m;r
Rj;m;r
:=
2j
IR R
IR
IR r j j (2 (1 ? x)) (2 x ? m) dx =
10
R
IR
(2j ? x)r(x ? m) dx:
(2.1.19)
Analogously let
~Lj;m;r := 2j R (2j x)r~(2j x ? m) dx = R xr ~(x ? m) dx =: ~m;r IR IR R R r~ j j j R ~j;m;r := 2 (2 (1 ? x)) (2 x ? m) dx = (2j ? x)r~(x ? m) dx:
(2.1.20)
IR
IR
Since
Lj;m;r = m;r ; Rj;m;r = 2j ?m?`(d);r ; r = 0; : : : ; d~ ? 1; ~ Lj;m;r
= ~m;r ;
~Rj;m;r
= ~ 2j ?m?`(d);r ; r = 0; : : : ; d ? 1;
(2.1.21)
it is not hard to verify the following symmetry relations which will be used frequently. Remark 2.1.1 One has
Rj;2j?`+d?`(d)?r (1 ? x) = Lj;`?d+r (x); r = 0; : : : ; d ? 1; ~Rj;2j?`~+d~?`(d)?r (1 ? x) = ~Lj;`~?d~+r (x); r = 0; : : : ; d~ ? 1:
(2.1.22)
and
[j+1;m](x) = [j+1;2j+1?m?`(d)] (1 ? x); = ; ~: (2.1.23) To make sure that the collections of boundary functions Xj , ~ Xj are separated we will assume in the following that l m j log2(`~ + `~2 ? 1) + 1 =: j0 (2.1.24) De ning for any set of functions in L2( )
S () := closL2 (span ) ; one has [DKU]
Proposition 2.1.1 Let 0j ; ~ 0j be given by (2.1.11). (i) The spaces Sj;[0;1] := S (0j ) and S~j;[0;1] := S (~ 0j ) are nested, i.e.,
Sj;[0;1] Sj+1;[0;1]; S~j;[0;1] S~j+1;[0;1]; j j0:
(2.1.25)
(ii) The spaces Sj;[0;1], S~j;[0;1] are exact of order d, d~, respectively, i.e.,
d([0; 1]) Sj;[0;1]; d~([0; 1]) S~j;[0;1]; j j0:
11
(2.1.26)
The nestedness of the spaces S (0j ) and S (~ 0j ) follows from the fact that also the boundary functions satisfy two-scale re nement relations whose exact format is given in [DKU]. Of course, the re nement lters of the boundary functions di er from those of the interior functions. However, there are only niteley many of them, namely d; d~ for each end of the interval, respectively, and the lter coecients are independent of the level j . Nevertheless, in absence of translation invariance, it will be extremely convenient to view the collections 0j or ~ 0j as (column) vectors whose entries are the respective basis functions. More generally, we will extend this convention in a canonical way to any other collections of functions or in some Hilbert space H with inner product h; i which will arise below. Speci cally, h; i := (h; i)2; 2 will denote a matrix. Thus the collection on the right side is treated as a row vectors. Accordingly, h; i is a column or row vector when or , respectively, consist of only one element. Likewise the fact that, due to re nability, each j;k can be written as a linear combinations of elements in 0j+1 can be conveniently expressed by a matrix relation between 0j and 0j+1 (and likewise for the collections ~ 0j ) of the following form ~ 0j;0: 0j T = (0j+1)T M0j;0; (~ 0j )T = (~ 0j+1)T M (2.1.27) ~ 0j;0 consists of the lter or mask coecients of the ith Thus the ith column of M0j;0, M element of 0j , ~ 0j , respectively. As mentioned above the dependence of the re nement ~ j;0'on j is very weak in the sense that there are only nitely many matrices M0j;0, M di erent coecients, whose numbering but not their values depend on j . In fact, the re nement matrices have a stationary interior block which grows with j and an upper left and lower right block of xed size which corresponds to the boundary functions. Moreover, these blocks are symmetric in that the lower right block is obtained from the upper left one by reversing the order of rows and columns which is an immediate consequence of Remark 2.1.1. Again see [DKU] for details. Due to the boundary modi cations, the collections 0j and ~ 0j are no longer biorthogonal. However, it has been shown in [DKU] that these collections can always be biorthogonalized, a fact we shall make essential use of. In [DKU] this has been realized by a change of basis in ~ 0j . Here we will depart somewhat from this strategy in order to deal with additional requirements concerning boundary conditions. To describe this, note rst that by construction the interior functions in Ij and ~ Ij are still biorthogonal. Therefore they will be left essentially untouched and it suces to con ne the change of bases to the collections of boundary functions. Since generally d~ d we will always tacitly assume in the sequel that the primal collections Xj , X 2 fL; Rg, are extended by the corresponding number d~ ? d interior functions to match the size of ~ Xj . To formulate the main observation we recall from [DKU2, DS] the following facts. Proposition 2.1.2 The matrices TX := hXj ; ~ Xj i 12
are independent of j . Moreover, one has
det TX 6= 0; det T0X 6= 0;
(2.1.28)
where T0X is the submatrix of TX which is obtained by discarding the rst row and column. Proposition 2.1.3 There exist d~ d~ matrices CX , C~ X , for X 2 fL; Rg, independent of j j0 such that the collections X;j := CX Xj ; ~ X;j := C~ X;j ~ Xj ; (2.1.29) satisfy
hX;j ; ~ X;j i = I; X 2 fL; Rg:
Denoting the sets of new boundary functions as X;j = fj;k : k 2 Xj g; ~ X;j = f~j;k : k 2 Xj g; and likewise one has for that
(2.1.30)
j;k := [j;k]; k 2 Ij ; ~j;k := ~[j;k]; k 2 ~ Ij ; j := L;j [ Ij [ R;j ; ~ j := ~ L;j [ ~ Ij [ ~ R;j ;
(2.1.31)
hj ; ~ j i[0;1] = I:
(2.1.32)
Moreover, the following boundary conditions hold j;k (0) = ~j;k (0) = j;k (1) = ~j;k (1) = 0; k 2 j n f` ? d; 2j ? `(d) ? ` + dg; (2.1.33) while
j;`?d (0) = j;2j?`(d)?`+d (1) = 2j=2; ~j;`~?d~(0) =
~j;2j?`(d)?`~+d~(1) = 2j=2det T0L=det TL:
(2.1.34)
Proof: By biorthogonality of the interior functions, (2.1.33) follows from (2.1.30). Since by (2.1.29),
hX;j ; ~ X;j i[0;1] = CX hXj ; ~ Xj i[0;1]C~ TX ; (2.1.35) the independence of CX , C~ X of j is a consequence of the above remarks on the matrices TX . Furthermore, by symmetry (see Remark 2.1.1),
TR = TlL;
(2.1.36)
where for any matrix M the matrix which is obtained by reversing the order of rows and columns of M is denoted by Ml. 13
Thus it suces to determine CL; C~ L since CR = ClL ; C~ R = C~ lL :
(2.1.37)
To this end, consider the singular value decomposition
TL = UT V; where U; V are orthogonal matrices and is a diagonal matrix containing the singular values i of TL ordered according to size i i+1. By (2.1.28), all singular values are strictly greater than zero. One easily checks that for any invertible matrix R the matrices CL := R?1=2U; C~ L := R?T ?1=2V; (2.1.38) satisfy CLTLC~ TL = I; (2.1.39) which, on account of (2.1.35), means that the corresponding sets X;j ; ~ X;j are biorthogonal (see (2.1.30)). Thus it remains to choose R so that the boundary conditions (2.1.33) hold. Let Lj(0) denote the vector whose entries are the elements of Lj evaluated at zero. By construction one has (2.1.40) Lj(0) = ~ Lj(0) = 2j=2e1; where (e1)i = 1;i is the rst coordinate vector. By de nition (2.1.29) and (2.1.40), one has to nd a constant b such that L;j (0) = CLLj(0) = 2j=2R?1=2U1 = 2j=2e1; and
~ L;j (0) = C~ L~ Lj(0) = 2j=2R?T ?1=2V1 = b2j=2e1; where U1; V1 is the rst column of U; V, respectively. Thus the desired boundary conditions are equivalent to R?1=2U1 = e1; (2.1.41) and ?1=2V1 = bRT e1: (2.1.42) Now we make the ansatz ! 1 )T ?1=2 a ( V R := UT 1=2 ; 0 where a is some constant and U0 is the submatrix of U obtained by discarding the rst column. Since U is an orthogonal matrix, it is clear that
R?1=2U1 = a e1: Since
T?L 1 = VT ?1U 14
(2.1.43)
we conclude that
:= T?L 1 1;1 ; and note that, by (2.1.36), 6= 0. >From the de nition of R, (2.1.42) and (2.1.43) we infer b = = det T0L=det TL: (2.1.44)
Remark 2.1.2 It immediately follows from (2.1.37) and Remark 2.1.1 that the biorthog-
onalized boundary functions in X;j ; ~ X;j inherit the symmetry properties from Remark 2.1.1. It is now easy to derive from the re nement relations of the collections 0j ; ~ 0j the re nement matrices for the biorthogonalized bases j ; ~ j (see Remark 3.3.1 below or [DKU]). >From Remark 2.1.2, (2.1.37) and (2.1.10) one concludes that Remark 2.1.3 the re nement matrices Mj;0; M~ j;0 in ~ j;0; Tj = Tj+1 Mj;0; ~ Tj = ~ Tj+1M (2.1.45) also satisfy
Mlj;0 = Mj;0; M~ lj;0 = M~ j;0:
Moreover, one infers from (2.1.33), (2.1.34) that
~ j;0)k;k0 = (Mj;0)k;k0 = 0 for (M
(
k = ` ? d; k0 6= k; j +1 k = 2 ? `(d) ? ` + d; k0 = 6 k;
(2.1.46) (2.1.47)
that is, the re nement relation of basis functions vanishing at the boundary involve only basis functions on the next ner scale which also vanish at the boundary.
We conclude this section with a simple example.
Piecewise constants, d = 1: First note that, in view of (2.1.6), in this case we can choose `~ = d~ so that ` = 1 = d and Lj = f0g; ~ Lj = f0; : : :; d~ ? 1g. This means that the primal spaces result from simply
restricting the integer translates of scaling functions to [0; 1]. One easily deduces from (2.1.1) that the entries of the d~ d~ matrix TL are for d~ 3 given by (2.1.48) (TL)k;k0 = k;k0 ; k; k0 = 0; : : : ; d~ ? 1: Moreover, kZ+1 k0 +1 k0 +1 k;k0 = xk0 dx = (k + 1)k0 + 1? k : (2.1.49) k Thus ~?1 TL = (Pk0 (k))dk;k (2.1.50) 0 =0 ; 15
where
!
r r+1 X xi: (2.1.51) Pr (x) r +1 1 ((x + 1)r+1 ? xr+1) = i i=0 In this case d = 1, d + d~ even, the above explicit form of TL can, of course, also be used to verify (2.1.36) directly.
2.2 Composite Bases on Curves
A boundary integral equation may be formulated on a curve with corners. Thus in absence of a regular parametrization one may have to split the curve into several segments each being regular. We wish to demonstrate next how to join multiresolution spaces de ned on such adjacent curve segments such that the resulting functions on the whole curve are still continuous. The same principle will apply later to tensor product constructions on cubes. The basic idea is quite simple. Suppose we have two regular parametrizations ; : [0; 1] ! IRm, m 1 with
(1) = (0);
(2.2.1)
so that ? = ([0; 1]) [ ([0; 1]) is a continuous (piecewise smooth) curve in IRm. It will be convenient to introduce a geometric mesh by setting 8 > k = ` ? d; < 0; 1 ; k = 2j ? `(d) ? ` + d; (2.2.2) q(k) := > : ?j 2 k; k 6= ` ? d; 2j ? `(d) ? ` + d; and
2j; := (q(j )); 20j; := (q(j n @ j )); 2 f; g; 2j := 2j; [ 2j; ; (2.2.3) where @ := f` ? d; 2j ? `(d) ? ` + dg. By (2.2.1), the point (q(2j ? `(d) ? ` + d)) = (1) = (0) = (q(` ? d)) 2 2j; \ 2j can be identi ed. Now de ne
j; (x) := j;k (?1(x)); = (q(k)) 2 20j; ; 2 f; g; and Similarly let
(
?1 ([0; 1]); j;(1)(x) := 2j ?`((d)??1`(+xd))(; (x)); xx 22 ([0 ; 1]): `?d
~j;(1)(x) :=
(
1~ 2 2j ?`~(d)?`~+d~( 1 ~ ( ?1 ( )) 2 `~?d~
?1(x)); x 2 ([0; 1]); x ; x 2 ([0; 1]);
16
(2.2.4) (2.2.5)
Remark 2.2.1 By (2.1.33), (2.1.34), the collections j = f j; : 2 2j g; ~ j = f ~j; : 2 2j g;
(2.2.6)
belong to C (?) and are biorthogonal
h?j ; ~ ?J i? = I;
(2.2.7)
where here for f; g 2 L2(?) Z1
Z1
0
0
hf; gi? := f ((x))g((x))dx + f ((x))g((x))dx
(2.2.8)
de nes an inner product which is equivalent to the canonical one on ? in the sense that corresponding norms are equivalent.
Clearly, the bases j ; ~ j are still re nable. We shall identify next the corresponding global re nement matrices. They are given by (M?j;0); = (Mj;0)k;k0 ; = (q(k0)) 2 2j+1; ; = (q(k)) 2 2j; :
(2.2.9)
In fact, note that by (2.1.33), (2.1.34), (M?j;0); = 0 if 2 20j+1; ; 2 20j;0 ; 6= 0 2 f; g:
(2.2.10)
Thus only those entries of M?j;0 are di erent from zero for which both indices ; are contained in the same closed arc. Since (1) = (0) belong to both arcs we have to con rm that the above de nition (2.2.9) is consistent. This is indeed the case, because by Remark 2.1.3, (Mj;0)`?d;`?d = (Mj;0)2j+1 ?`(d)?`+d;2j ?`(d)?`+d : ~ ?j;0 is slightly di erent: Due to the di erent normalization of ~j;(1), the form of M 8 > > > > > > > > >
(M > > > > 1 ~? > = (0); = (q(k)) 2 20j+1; ; > > 2 (Mj;0)k;`~?d~; > : 1 ~ ? = (1); = (q(k)) 2 20j+1; : 2 (Mj;0)k;2j ?`(d)?`~+d~; (2.2.11) we will employ similar principles below in the tensor product case. ~ ?j;0); = (M
17
3 Wavelets and Boundary Conditions 3.1 Biorthogonal Wavelets and Symmetry
While it is fairly easy to construct composite biorthogonal generator bases ?j ; ~ ?j as indicated above, the question arises how to generate also biorthogonal wavelet bases de ned on ? which are also continuous on ?. First we recall from [DKU2] that for any d + d~ even d~ d and corresponding biorthogonal generator bases j ; ~ j as above one can construct wavelet bases
j = f!j;k : k = 1; : : : 2j g; ~ j = f!~j;k : k = 1; : : :; 2j g; (3.1.1) which are biorthogonal
h j ; ~ j0 i[0;1] = j;j0 I; j; j 0 j0:
(3.1.2)
Moreover, expressing the corresponding two-scale relations in matrix form as ~ j;1;
Tj = Tj+1Mj;1; ~ Tj = ~ Tj+1 M (3.1.3) one has
(3.1.4) M~ Tj;e Mj;e0 = e;e0 I; e; e0 2 f0; 1g: ~ j;e contain only a uniformly bounded numEach column of any of the matrices Mj;e; M
ber of nonzero coecients forming the lters of the wavelets and generator basis functions whose supports therefore satisfy diamsupp!j;k ; diamsupp~!j;k ; diamsuppj;k ; diamsupp~j;k 2?j : (3.1.5)
~ j;1 constructed in [DKU] do not necessarily However, the particular matrices Mj;1; M share the same symmetry properties as the re nement matrices in Remark 2.1.3. One can check that the construction in [DKU2] does give Mlj;1 = Mj;1; M~ lj;1 = M~ j;1; (3.1.6) when d is even. Let us point out rst how to arrange (3.1.6) also in the remaining cases d odd. Proposition 3.1.1 Given Mj;0; M~ j;0 from (2.1.45) and M0j;1; M~ 0j;1 the particular matrices constructed in [DKU2] satisfying (3.1.4) as above. Then there always exist ma~ j;1 which also satisfy (3.1.4) and inherit the same sparseness properties trices Mj;1; M 0 0 ~ j;1, so that of Mj;1 ; M ~ j;0; M ~ j;1)k = O(1); k(Mj;0; Mj;1)k; k(M (3.1.7) while in addition
Mlj;1 = Mj;1; M~ lj;1 = M~ j;1: 18
(3.1.8)
Proof: Let us denote by I! the permutation matrix whose only nonzero entries are on the antidiagonal and have the value 1. Thus for any matrix M one has Ml = I!MI! where I! is always assumed to have the right size without further speci cation. Since (I!)2 = I one easily veri es that for any two matrices A; B of appropriate sizes (Al)T = (AT )l; (AB)l = AlBl; (A?1)l = (Al)T : (3.1.9) ~ 0j;1 consisting of the rst 2j?1 Now let Nj;1; N~ j;1 denote the submatrix of M0j;1; M columns, respectively (recall that these matrices have always 2j columns). Let Hj;1 := (Nj;1; Nlj;1); H~ j;1 := (N~ j;1; N~ lj;1); and note that by (3.1.4), (2.1.46) and (3.1.9), M~ Tj;0Nj;1 = 0; (M~ lj;0)T Nlj;1 = (M~ Tj;0Nj;1)l = 0: Applying the same reasoning to Mj;0; H~ j;1 provides M~ Tj;0Hj;1 = 0; MT0;1H~ j;1 = 0:
(3.1.10)
The matrices Hj;1; H~ j;1 have still full rank. To see this, consider the collection Tj := Tj+1Hj;1; and note that by (3.1.10), ~ Tj;0Hj;1 = 0: h~ j ; j i[0;1] = M~ Tj;0h~ j+1 ; j+1i[0;1] = M Thus it suces to show that the elements of j are linearly independent. Since by construction, Hj;1 = Hlj;1 one has (Hj;1)k;2j?1+r = (Hj;1)2j+1 ?`(d)?k;2j?1 +1?r ; r = 1; : : : ; 2j?1: Combining this with the symmetry relations in Remarks 2.1.1, 2.1.2, straightforward calculations yield
j;2j?1 +k (x) = j;2j?1+1?k (1 ? x); k = 1; : : : ; 2j?1 : (3.1.11) Moreover, by de nition, j;k = !j;k , k = 1; : : : ; 2j?1. Thus j = f!j;k ; !j;k (1 ? ) : k = 1; : : : ; 2j?1 g: Now suppose that cT j = 0. Then hcT j ; !~j;k i[0;1] = ck (3.1.12) for k = 1; : : : ; m, where m is the rst column in Mj;1 (and hence in Hj;1), which corresponds to the stationary interior masks so that !j;m is fully supported in [0; 1]. If j is large enough the support of !~j;m will not overlap any of the supports of the functions 19
!j;k (1 ?), k = 1; : : :; 2j?1 so that (3.1.12) follows indeed from biorthogonality and gives ck = 0, for k = 1; : : : ; m. Likewise testing with the functions !~ j;k (1 ?) for k = 1; : : : ; m ensures that also ck = 0 for k = 2j ; : : : ; 2j ? m + 1. Therefore
cT j
=
j ?m 2X
k=m+1
ck j;k = 0
(3.1.13)
is a linear combination of functions which are supported in (0; 1). Now note from the construction in [DKU2] that for d odd one has that the rst column of Nlj;1 has the same support as the 2j?1 + 1st column of M0j;1. Thus the functions in the above linear combination all have pairwise di erent supports. Therefore (3.1.13) implies that c = 0, which con rms our claim. Now consider ~ Tj;1 ! N T H~ j;1Hj;1 = ~ l T (Nj;1; Nlj;1) (Nj;1) ~ Tj;1Nlj;1 ! I N = =: Kj : (N~ lj;1)T Nj;1 I Since by (3.1.9), the matrix Kj has the form where
N~ Tj;1Nlj;1 = (N~ lj;1)T Nj;1 l ; ! I A j K j = Al I ; j !
!
l Aj = a0l 00 ; Alj = 00 a0 ;
and the block a is independent of j . By the above remarks Kj must be invertible. Noting that
Alj Aj
!
!
l ?1 l = a0a 00 ; (I ? Alj Aj )?1 = (I ? a0 a) 0I ;
one easily con rms that
I 0 0 01 C B l ?1 l l ?1 K?j 1 = BB@ 00 I +?(aI(?I ?alaa)a?)1ala aa(I(I??aalaa)?)1 00 CCA : 0 0 0 I Thus both Kj and K?j 1 are banded with band width independent of j . Moreover, one easily checks from the de nition of Kj and (3.1.9) that Klj = Kj ; (K?j 1)l = K?j 1: (3.1.14) 0
20
Thus de ning,
Mj;1 := Hj;1K?j 1; M~ j;1 := H~ j;1;
(3.1.15) we readily infer from (3.1.10) and (3.1.14) that (3.1.4) holds. Moreover, since by construction H~ lj;1 = H~ j;1 and Hlj;1 = Hj;1 the assertion follows from (3.1.14) and (3.1.9). In the sequel we will always refer to the symmetric version when dealing with (3.1.4) and (3.1.3).
3.2 Some Preliminary Remarks on Boundary Conditions
Returning to the composite generator bases constructed above a little thought reveals that dim S (?j ) = 2 dim j ? 1; (3.2.1) while (3.2.2) dim S (?j+1) ? dim S (?j ) = 2 dim S ( j ): Thus glueing wavelets which do not vanish at 0 or 1 together, as in (2.2.4), would result in a de cient complement space. On the other hand, if all the wavelets in j vanished at 0 and 1 the above lifting technique ?j := j ?1 [ j ?1 would readily produce wavelets de ned on all of ? which are continuous and biorthogonal relative to the inner product (2.2.8). The following observation is an immediate consequence of the boundary relations (2.1.33) and (2.1.34). Remark 3.2.1 The wavelet !j;k vanishes at 0 (1) if and only if the rst (last) entry in the k-th column of Mj;1 is zero. An analogous statement holds for ~ j .
Remark 3.2.2 Suppose that 0j is a basis for some complement of S (j ) in S (j+1) and at least one element ! 2 0j does not vanish at zero. Then there exists no basis of the complement S ( 0j ), that has all elements vanish at zero.
Proof: If there were a basis 00j of S ( 0j ) whose elements vanish at zero the set f!g[ 00j would be still linearly independent, since ! is the only function which does not vanish at zero, but still, by assumption, f!g [ 00j S ( 0j ), which is a contradiction.
In fact, more can be said. Remark 3.2.3 There exist no biorthogonal wavelet bases j ; ~ j for the spaces S (j ), S (~ j ) as above such that all elements of j vanish at 0 and 1. Proof: Suppose there exist biorthogonal wavelet bases j;0; ~ j for the spaces j ; ~ j above, such that all the elements in j;0 vanish at zero and one. The fact that the spaces S (j ); S (~ j ) satisfy certain direct and inverse estimates combined with the biorthogonality implies, on account of the results in [D2], that the sets [ [ j0 [ j;0; ~ j0 [ ~ j ; j j0
j j0
21
form biorthogonal Riesz-bases for L2([0; 1]). On the other hand, (2.1.33) and (2.1.34) also imply that
j;0 S (j+1;0); j j0; where j;0 := j n fj;`?d ; 2j?`(d)?`+d g spans a subspace of S (j ) satisfying homogeneous boundary conditions. Clearly, S (j;0) still satis es the same inverse estimates. Moreover, it is not hard to show that it also satis es a direct estimate of the form ?sj inf kf ? gj kL2 ([0;1]) < 2 kf kH s([0;1]); gj 2S (j;0 )
for 0 s < 1=2, where H s([0; 1]) denotes the Sobolev space of (noninteger) order s on [0; 1]. Therefore the results in [D2] still imply that also the reduced collection j0 ;0 [
[
j j0
j;0;
is a Riesz-basis. Therefore the function j0;`?d has a representation in this latter basis S contradicting the fact that the collection j0 [ jj0 j;0 is also a Riesz basis. On the other hand, the number of all wavelets that do not vanish at the end of the interval can always be limited. Remark 3.2.4 Given j , ~ j and associated wavelet bases j , ~ j satisfying (3.1.2). Then there always exists another pair of biorthogonal wavelet bases 0j , ~ 0j still satisfying (3.1.2), where only one element of 0j does not vanish at zero and one, respectively. Moreover, one still has that 0 ) 2?j . diam(supp !j;k Proof: By assumption, j satis es (3.1.3). We wish to perform a change of basis 0 !j;k
=
j
2 X
k0 =1
bk;k0 !j;k0
which in matrix form reads
0j = Bj j = Bj MTj;1j+1:
(3.2.3)
Recall from [DKU2] that Mj;1 has the form
ML Mj;1 =
MIj;1
(3.2.4)
MR 22
;
where the size of the upper left and lower right blocks is independent of j and the interior block MIj;1 is stationary, i.e., it is a nite section of a biin nite matrix of the form (ak?2l)k;l2ZZ whose size is proportional to 2j . All nonzero entries of the rst (last) row of Mj;1 are contained in the corner blocks ML (MR). Let 2 IN be the smallest integer so that the rst (last) row has no nonzero entry with column index larger (smaller) than (2j ? ). Then we choose Bj to have the form 1 B 0 0 L (3.2.5) Bj = B@ 0 I(2j?2) 0 CA ; 0 0 BR where I(r) denotes the r r identity matrix and where the BL, BR are Householder (orthogonal) matrices which re ect the rst column of MTj;1 into the coordinate vector e1 2 ZZ 2j . Thus setting, M0j;1 := Mj;1BTj and M~ 0j;1 := M~ j;1BTj the bases 0j = Bj j and
~ 0j = Bj ~ j are still biorthogonal (3.1.2), (since Bj is an orthogonal matrix) and satisfy ~ 0j;1. By construction, one has S ( j ) = S ( 0j ), S ( ~ j ) = S ( ~ 0j ). (3.1.3) relative to M0j;1, M 0 0 T
j = (Mj;1) j+1 . The assertion follows now from Remark 3.2.2. 0
The above comments show that the task of forming continuous wavelets on the union of several curve segments from bases on the individual segments appears to be a bit more delicate. For the case d = d~ = 2 this was studied in [JL] where successive projections are used. In the following we will present an approach that avoids glueing the wavelets across segment boundaries and where corresponding lters are obtained by local operations. Remark 3.2.5 To this end, we will employ concepts developed in [CDP]. The main idea is to determine rst some initial complement spaces between two successive spaces S (j ) and S (j+1), from which the desired complements spanned by biorthogonal wavelets will be generated with the aid of certain projections. The point is to perform this latter projection on the global composite spaces so that continuous functions are taken into continuous ones. This requires that the initial complements can easily be glued together which here means that they are spanned by functions which vanish at the end points of the interval. Although these initial complements will not correspond to biorthogonal bases they still should exhibit certain stability properties which we brie y describe rst.
3.3 Some Auxiliary Facts
As indicated above, we shall have to manipulate and vary complements between two successive multiresolution spaces. The necessary tools have been developed in [CDP] (see also [Sw]). Since we shall have to apply them several times for di erent settings, we nd it worth brie y stating them here in sucient generality. Thus we will consider for the time being some Hilbert space H with inner product h; i and norm kk = h; i1=2. Later H will be L2(D), where D = [0; 1]; 2; ?. fj gj2IN , j = f j;k : k 2 j g H , will denote a uniformly stable sequence of re nable bases, i.e.,
kck`2 (j ) kcT j k 23
(3.3.1)
uniformly in j and
Tj = Tj+1 Mj;0; j 2 IN 0: (3.3.2) Moreover, denoting by [X; Y ] the space of bounded linear operators from a normed linear space X into the normed linear space Y one has [CDP]
Mj;0 2 [`2(j ); `2(j+1)]; kMj;0k = O(1); j 2 IN 0; where
kMj;0k :=
sup
u2`2 (j );kuk`2 (j ) 1
(3.3.3)
kMj;0uk`2 (j+1 ) :
Although in all our applications the index sets j will be nite we remark that the results remain valid for in nite sets j as well where the corresponding matrix-vector operations are to be understood in the sense of absolute convergence. The notation and conventions made above for H = L2([0; 1]) will be employed in this more general in obious analogy. We are interested in determining some stable basis j = f j;k : k 2 rj g S (j+1) such that
S (j+1) = S (j ) S ( j ):
(3.3.4)
Proposition 3.3.1 ([CDP]) Suppose that fj g is uniformly stable and (3.3.2) holds. Then fj [ j g for j S (j+1 ), satisfying (3.3.4), is uniformly stable if and only if there exist Mj;1 2 [`2(rj ); `2(j+1)]; such that
(3.3.5) Tj = Tj+1 Mj;1; and that Mj = (Mj;0; Mj;1 ) 2 [`2 (j [ rj ); `2 (j+1 )] are invertible and satisfy
kMj k ;
M?j 1
= O(1); j 2 IN 0:
(3.3.6)
Writing M?j 1 = Gj = GGj;j;10 , one obtains the reconstruction formula
Tj+1 = Tj Gj;0 + Tj Gj;1:
(3.3.7)
Given j and Mj;0, any Mj;1 2 [`2(rj ); `2(j+1)] such that (3.3.6) and (3.3.7) hold, is called a stable completion of Mj;0. We wish to determine next particular complements induced by linear projectors. To this end, suppose that j F H , where F is some subspace of H , and that j F , the dual of F relative to the dual pairing induced by h; i, such that
hj ; j i = I; j 2 IN: Hence
Pj f := hf; j ij = 24
X
k2j
hf; j;k i j;k
(3.3.8) (3.3.9)
are linear projectors onto S (j ). Moreover assume that the matrices hj+1 ; j i de ne uniformly bounded mappings in [`2(j ); `2(j+1)], i.e.,
khj+1 ; j ik = O(1); j 2 IN:
(3.3.10)
We are interested in nding uniformly stable bases for
W (j ) := (Pj+1 ? Pj )S (j+1):
(3.3.11) Theorem 3.3.1 ([CDP]) Suppose that j ; j satisfy (3.3.8), (3.3.10) and that M j;1 j := (Mj;0; M j;1) let G j = are some uniformly stable completions of Mj;0 . For M Gj;0 = M ?j 1 . Then G j;1 Mj;1 := (I ? Mj;0hj+1 ; j iT )M j;1 (3.3.12) are also uniformly stable completions of the Mj;0 and the inverses are given by Gj;0 = G j;0 + hj+1 ; j iT M j;1G j;1; Gj;1 = G j;1: (3.3.13) The collections span the spaces W (j ), i.e., .
Tj = Tj+1 Mj;1
(3.3.14)
W (j ) = S ( j ):
(3.3.15)
We will apply also the following special case where j = ~ j H is also re nable, i.e., ~ j;0: ~ Tj = ~ Tj+1 M (3.3.16) Clearly hj ~ j i = I (3.3.17) implies (3.3.18) M~ Tj;0Mj;0 = I:
Corollary 3.3.1 ([CDP]) Let fj g, f~ j g, Mj;0, M~ j;0 be related as above. Suppose j;1 is some stable completion of Mj;0 and that G j = M ?j 1. Then that M Mj;1 := (I ? Mj;0M~ Tj;0)M j;1 (3.3.19) is also a stable completion and Gj = M?j 1 has the form ~ Tj;0! M (3.3.20) Gj = : Gj;1 Moreover, the collections
j := MTj;1j+1 ; ~ j = G j;1~ j+1 25
(3.3.21)
form biorthogonal systems D
E
j ; ~ j = I;
Thus setting
E
D
D
E
j ; ~ j = j ; ~ j = 0:
Qj f := hf; ~ j ij ;
one has
(3.3.22) (3.3.23)
(Qj+1 ? Qj )S (j+1) = S ( j ); (Qj+1 ? Qj )S (~ j+1) = S ( ~ j ):
(3.3.24)
Note that, in view of the nestedness of the spaces, (3.3.22) implies that the collections [ [ = 0 [ j ; ~ := ~ 0 [ ~ j j 2IN 0
are biorthogonal
j 2IN 0
D
E j ; ~ j0 = j;j0 I: (3.3.25) Here we have set for simplicity r?1 := 0; ?1;k = 0;k; ~?1;k = ~0;k. Finally, we will frequently make use of the following fact from [DKU2], which interrelates the re nement relations and stable completions relative to di erent generator bases.
Remark 3.3.1 Suppose that Cj is a nonsingular (#j ) (#j ) matrix and let j = Cj 0j : (3.3.26) Assume that M0j;0 is the re nement matrix for 0j and M0j;1 is a stable completion of M0j;0. Then one has Tj = Tj+1Mj;0 (3.3.27) where
Moreover,
Mj;0 = C?j+1T M0j;0CTj :
(3.3.28)
Mj;1 = C?j+1T M0j;1
(3.3.29)
is the corresponding stable completion and
M?j 1
?T G0 CT ! C = j 0 j;0T j+1 =: Gj : Gj;1Cj+1
3.4 Initial Complements with Boundary Conditions
(3.3.30)
We return now to multiresolution on [0; 1]. Combining biorthogonality and compact support, one can show that the collections j ; ~ j are uniformly stable in the sense of (3.3.1) [CDP, DKU]. We ultimately plan to apply Corollary 3.3.1 to construct biorthogonal wavelets on composite domains such as the curve considered above. In order to be able to patch complement bases de ned for the individual component spaces continuously together, we are lead by the observations in Section 3.1 to construct 26
j;1 of M0j;0 such that the (uniformly stable) rst some initial stable completions M complement bases j;1 ( j )T = (0j+1)T M (3.4.1) satisfy
j (0) = j (1) = 0: (3.4.2) 0 Here we denote at this point by Mj;0 the re nement matrices of the collections 0j prior to biorthogonalization (see (2.1.27)). We will consider rst a special case.
The piecewise linear case, d = 2: In the rst case of interest, d = 2, this is very
easy, namely the so called hierarchical bases do the job. Again for ` = 1; `~ = d~ one has j = f0; : : :; 2j g. Thus the collections
j := fj0 +1;2k?1 : k = 1; : : : ; 2j g (3.4.3)
correspond to taking the ne scale generators at the points in 2?j?1 j+1 n 2?1j . Obviously one has !j;k (0) = ! j;k (1) = 0; k = 1; : : : ; 2j : (3.4.4) Recall that in this case the basis 0j from (2.1.11) consists of the restrictions to the hat functions to [0; 1]. The corresponding re nement matrices are of the form 0
M0j;0 =
B B B B B B B B B B B B B B B B B B B B B @
p12 0 p1 p1 2 2 2 2 0 p12 0 p1 2
0 ...
0
2
0 ...
:::
0 0 0
p1 2 2
::: ...
::: ::: ::: 0 :::
p1 2 2
0
1
C C C C C C C C C C C C C C C C C C C 1 C p C 2 2 A
... 0
:
(3.4.5)
p1
2
The stable completion which corresponds to the hierarchical complement basis in (3.4.3) is given as 1 0 0 0 0 : : : 0 C B C B 1 0 0 C B C B C B 0 0 0 C B C B . . . . C B . . . . . . . C B . C: (3.4.6) Mj;1 = BB C 0 1 0 C B C B C B 0 0 0 C B C B C B 0 0 1 A @ 0 ::: 0 0 0 27
Since obviously, j0 +1;2k+1 = !j;k ; k = 0; : : : ; 2j ? 1; 0 2j;k ? 12 j0 +1;2k?1 + j0 +1 k + 1 p 0 1 ? 2 !j;k?1 ? 21 !j;k ; k = 1; : : : ; 2j ? 1; = 2j;k p p = 2j;0 0 ? 12 !j;0 0; j0 +1;2j+1 = 2j;0 2j ? 21 !j;0 2j ?1;
j0 +1;2k = j0 +1;0
p
(3.4.7)
?j 1 are readily identi ed as the corresponding blocks G j;0, G j;1 of the inverse G j = M 0 B B B B B B B B B @
G j;0 = and
p
2 0 0 0 p 0 0 2 0 ... p 2 0 ::: 0
0
? 12 1 ? 12 0 0 : : : B B B 0 0 ? 12 1 ? 21 B B
G j;1 = BBB ... B B @
Since obviously,
... 0 :::
::: ... ... 0 0
0 0
p0
2
1 C C C C C C C C C A
;
(3.4.8)
1
0 C ... C C C ... C C: C C ? 12 0 0 CCA ? 21 1 ? 12
(3.4.9)
(3.4.10) M j
;
G j
= O(1); j 2 IN 0; we infer form Proposition 3.3.1 that the bases 0j [ j are uniformly stable. When d > 2 the situation is a little more involved and we will apply the tools from the previous section. To this end, let j;`?d (f ) := 2?j=2f (0); j;2j ?`(d)?`+d (f ) := 2?j=2 f (1); (3.4.11) and set ~ j;0 := ~ j n f~j;`~?d~; ~j;2j ?`(d)?`~+d~g: (3.4.12) Lemma 3.4.1 The collections of functionals j := fj;`?d ; j;2j ?`(d)?`+d g [ ~ j;0 (3.4.13) satisfy (3.3.8) hj ; j i[0;1] = I; 28
as well as (3.3.10). De ning W (j ) as in (3.3.11) relative to the projectors Pj f := hf; j i[0;1]j onto S (j ), one has that g(0) = g(1) = 0; g 2 W (j ): (3.4.14) Proof: The biorthogonality of j and j is an immediate consequence of (2.1.32) and the boundary conditions (2.1.33), (2.1.34) combined with the de nition (3.4.13). The relation (3.4.14) follows from the de nition of the Pj and again (2.1.33). To identify stable bases j for the complements W (j ) of S (j ) in S (j+1), we will employ Theorem 3.3.1. As the initial stable completion we take here Mj;1 from (3.1.3) which corresponds to the compactly supported biorthogonal wavelet bases j con~ Tj;0; Gj;1 = structed in [DKU2]. Thus the corresponding inverses are given by Gj;0 = M M~ Tj;1. Since the Mj , Gj have a uniformly bounded number of non-zero entries per row and column which are also uniformly bounded, it is clear that (3.3.6) holds, so that the Mj;1 are indeed stable completions in the above sense. Lemma 3.4.2 Let j be de ned by (3.4.13). The matrices M j;1 := (I ? Mj;0hj+1 ; j iT[0;1])Mj;1 (3.4.15) are also uniformly stable completions of the Mj;0 and the inverses are given by G j;0 = Gj;0 + hj+1; j iT[0;1]Mj;1Gj;1; G j;1 = Gj;1: (3.4.16) Moreover, the collections j;1
Tj := Tj+1M (3.4.17) are uniformly stable bases of the spaces W (j ), W (j ) = S ( j ); (3.4.18) so that, by Lemma 3.4.1,
j (0) = j (1) = 0: (3.4.19) Proof: Since the elements of the bases j ; ~ j all have compact support the matrices hj+1 ; j iT have in each row and column only a uniformly bounded nite number of nonvanishing entries which are also uniformly bounded. In fact, de ning as in the proof of Proposition 2.1.3 0 det T (3.4.20) := det TL ; L we obtain by (2.1.34), 1=2 j;`?d (~j+1;k ) = 21=2 `~?d;k ~ ; j;2j ?`(d)?`+d (~j +1;k ) = 2 2j+1 ?`(d)?`~+d;k ~ ; while by (3.1.4) and (3.4.13), for k 2 j+1; k0 2 j n f` ? d; 2j ? `(d) ? ` + dg hj+1; j i[0;1] k;k0 = hj+1;k ; ~j;k0 i[0;1] = (M~ j;0)k;k0 :
Hence the hypotheses of Theorem 3.3.1 are satis ed, which con rms (3.4.18). The rest follows from Lemma 3.4.1. 29
Remark 3.4.1 Employing (3.1.9) and Remark 2.1.3 one easily con rms that the ma j ; G j all satisfy trices M M lj = M j ; G lj = G j :
3.5 Continuous Wavelets on Composite Curves
We wish to apply the above results to the construction of globally continuous biorthogonal wavelets on curves. Since it suces to consider just two adjacent arcs we return to the situation described in Section 2.2. For
rj := f1; : : :; 2j g let and set De ning
w(k) := 2?j (k ? 21 ); k 2 rj ;
rj; := (w(rj )); 2 f; g:
:= !j;k ; = (w(k)) 2 rj; ; (3.5.1) we immediately infer from (3.4.19) that these functions are continuous on ? and span ?j;1 is also a complement of S (j ) in S (j+1 ). The corresponding stable completion M j;1 from (3.4.17) satis es easily identi ed. In fact, we infer from Remark 3.2.1 that M j;1)`?d;k = (M j;1)2j+1 ?`(d)?`+d;k = 0; k 2 rj : (M j;
As a consequence there is no consistency problem when setting j;1)k0;k ; = (q(k0)) 2 2j+1; ; = (w(k)) 2 rj; ; 2 f; g: ?j;1); = (M (M (3.5.2) Proposition 3.5.1 For M?j;0; M~ ?0;j and M ?j;1 given by (2.2.9), (2.2.11) and (3.5.2) let
M?j;1 := (I ? M?j;0(M~ ?j;0)T )M ?j;1: Then the collections
(3.5.3)
Tj := Tj+1M?j;1 (3.5.4) S form biorthogonal wavelet bases, i.e., := j0 [ jj0 j form a Riesz basis for L2 (?). For the identi cation of the corresponding dual wavelets and their lters we refer to the discussion of corresponding construction for higher dimensional manifolds below.
30
3.6 Tensor Products
The next step is to take tensor products of the univariate constructions. This follows mostly canonical lines and one only has to x some notation. We will apply the following rules: Super- or subscripts 2 indicate quantities de ned on the unit n-cube 2 = [0; 1]n, usually obtained as tensor products. For instance, 2j = j j . Likewise, k is to be understood as a multiindex k = (k1; : : :; kn ) as soon as it is associated with a multivariate quantity. The wavelets require now a further index e 2 f0; 1gn , namely 2 (x) = ! !j;e;k j;e1 ;k1 (x1) !j;en ;kn (xn ); where ( 0; 2 !j;e;k = !j;k ;;kk 22 rj ;; ifif ee = = 1; and
j;k
j
rj := f1; : : : ; 2j g:
2 , i.e., not to use It will thererfore sometimes be convenient to write !j;20;k instead of j;k an extra notation for the scaling functions. The matrices M2j ; G2j are now naturally blocked into components M2j;e ; G2j;e, e 2 f0; 1gn , where (3.6.1) (M2j;e)k;k0 = (Mj;e1 )k1 ;k10 (Mj;en )kn ;kn0 ; and M2j;1 is comprised of all the components M2j;e ; e 2 f0; 1gn n f0g, while M2j;0 is the re nement matrix of 2j . Tensor products of functionals de ned on univariate functions are canonically de ned by their action on the respective variables. Interpreting the collections 2j in this sense, we will make crucial use of the projectors
Pj2f := hf; 2j i2 2j
(3.6.2)
onto the spaces S (2j ). To describe their relevant properties, let us denote by
@ 2j := fk 2 2j : ki 2 f` ? d; 2j ? `(d) ? ` + dg; for some 1 i ng and analogously by @ ~ j the set of indices associated with the boundary of 2.
Lemma 3.6.1 The projectors Pj2 have the following properties. (i) Whenever k 2 2j belongs to @ 2j , then the quantities hf; 2j;k i2 depend only on
values of f restricted to the intersection of all faces of 2 which k is associated with. In particular, one has that Pj2 interpolates at the vertices of 2, i.e.,
(Pj2 f )(e) = f (e); e 2 f0; 1gn : (ii) When d > n=2 the projectors Pj2 have the following approximation properties
kf ? Pj2 f kL2 (2) n=2, and any p 2 j;2, one has kf ? Pj2f k2L2 (j;k ) < kf ? pk2L2 (j;k ) + kPj2(f ? p)kL2 (j;k ) 2 2 (3.6.5) kf ? pkL2 (j;k ) + kmax 0 2 jj;k0 (f ? p)j ; j;k
2k where we have used that kj;k 1. Recall that the functionals 2j;k are tensor L2 (2) < products of L2-inner products and (scaled) point evaluations. Taking the respective scalings into account, it is easy to see that (3.6.6) jj;k0 (f )j n=2 the Sobolev imbedding theorem ensures that kf ? pkL1 (2)
This paper is concerned with the construction of biorthogonal wavelet bases de ned on a union of parametric images of the unit d-cube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo-di erential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases de ned on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales as well as appropriate moment conditions.
Key Words: Biorthogonal wavelets, norm equivalences, boundary element methods, composite multiresolution, multiscale methods for partial di erential equations
AMS subject classi cation: 65Y20, 68Q25, 65F35, 45L10, 65M99, 76D07
1 Motivation and Background 1.1 Introductory Remarks
The fact that, roughly speaking, the representation of certain (elliptic) operators and their inverses relative to appropriate wavelet bases are nearly sparse or even diagonal have initiateded numerous investigations of wavelet based methods for the numerical solution of various types of operator equations. For instance, the above mentioned near diagonality or more rigorously the fact that Sobolev norms are equivalent for a certain range to weighted sequence norms of wavelet coecients give rise to (asymptotically optimal) preconditioning techniques [DK, DPS, J, O]. The observation made in [BCR] that for certain integral operators wavelet discretizations lead to fast matrix-vector multiplication has since initiated a number of further attempts to develop schemes of this type for the numerical treatment of equations involving operators with global kernels. A few comments on the actual need of such further investigations are in order. The approach proposed in [BCR] hinges on the availability of an appropriate wavelet basis. 1
For periodic problems as considered in [BCR, DPS1, DPS2] a variety of such bases is indeed known. Furthermore, when, unlike the situation considered in [BCR], the order of the operator is di erent from zero, which is for instance the case for the single layer potential or the hypersingular operator, the need for preconditioning enters the solution process putting further constraints on the bases. Finally, the issue of accuracy deserves some additional attention. [BCR] addresses the question of realizing a matrixvector multiplication at the expense of roughly at most N log N operations, N being the size of the matrix, within some prescribed accuracy tolerance which at this point is independent of N though. However, when the matrix-vector multiplication becomes an ingredient of some (stable) numerical scheme for the solution of a corresponding operator equation it would not make sense to choose much smaller than the overall discretization error. Likewise when is large compared to the dicretization error the resolution of the scheme would be wasted and one could have used coarser discretizations and hence smaller systems in the rst place. Thus from a principal point of view, should be related to N and the basic question may then be formulated as follows: Given some multiscale basis, is it possible to replace the sti ness matrix relative to this basis by some sparse matrix which is still well conditioned and has at most O(N ) nonvanishing entries such that the solution of the sparse system still exhibits the same asymptotic convergence to the solution of the operator equation as the solution to the full discrete system? Meanwhile this question can, in principle, be answered for a wide range of operator equations including those of order di erent from zero and for various types of numerical schemes [DPS1, DPS2, DPS, PS, PSS, S]. It can be shown that depending on the type of operator the various e ects can be balanced in such a way that one ends up with a scheme which is asymptotically optimal in the above sense provided that the wavelet basis satis es a number of conditions pertaining mainly to norm equivalences, regularity and moment conditions. In particular, it turns out that orthogonal bases are in many situations (whenever the order of the operator is less than one) not optimal. This suggests resorting to biorthogonal bases which permit realizing a suciently high order of vanishing moments relative to the accuracy of the trial spaces. All these requirements on the wavelet bases are relatively easy to ful l when the underlying domain is the full Euclidean space IRd or the torus. However, the situation changes drastically when dealing with more realistic domain geometries in which case very little is known about appropriate bases. Thanks to numerous studies of wavelets on an interval [AHJP, CDV, CQ, DKU] wavelet bases on cubes are also available. In this paper we shall show that suitable biorthogonal multiresolution spaces on cubes which satisfy certain boundary conditions can be composed via parametric lifting to bases on unions of parametric images of cubes. The resulting composite multiresolution spaces and wavelet bases are very appropriate for domain decomposition techniques. The layout of this paper is as follows. We conclude this introductory section with some comments of the type of operator equations we have in mind. In particular we specify the requirements on the wavelets which cover a wide range of cases of practical interest. In Section 2 wavelets on the interval are revisited where special attention is paid to 2
boundary conditions. In Section 3 we discuss tensor products of such wavelets and their parametric lifting to composite domains. After establishing direct and inverse estimates we con rm the validity of norm equivalences for the range required for the equations considered before.
1.2 Some Operator Equations
Suppose that is a bounded domain or a manifold which admits the de nition of Sobolev spaces H s for a certain range of s. Speci cally, when s < 0 then H s is to be understood as the dual (H ?s ) of H ?s . When is a bounded domain the de nition of H s may incorporate (homogeneous) boundary conditions. Assume that A is a boundedly invertible operator from H t onto H ?t , i.e.,
kAvkH ?t kvkH t ;
(1.2.1)
where a b means that a < b and b < a and the latter relation says that b is bounded by a constant times a uniformly in any parameters on which a; b may depend. We wish to solve Au = f (1.2.2) for any given f 2 H ?t . Now suppose that = f : 2 rg is a Riesz basis for L2( ). Here the index usually has the form = (j; k) where jj := j refers to the scale which for simplicity will always correspond to a meshsize of order 2?j . The index k will generally again be comprised of several indices expressing the type of the wavelet and the location of its support. Of course, for practical purposes, all basis functions should be local, i.e., diam supp 2?jj . On the coarsest level j0 the functions contain roughly speaking the polynomial parts and correspond to scaling functions. It is known [D2] that the Riesz basis property implies the existence of a dual Riesz ~ i.e., basis , (1.2.3) h ; ~0 i = ;0 ; ; 0 2 r: Correspondingly, let S~j := f ~ : jj j g. We will consider Galerkin schemes based on the trial spaces
Sj := span f : jj j g: The following facts are well-known [DPS].
Requirement I: The scaled sti ness matrices Aj := 2?t(jj+j0j)hA 0 ; i jj;j0jj satis es
cond2(Aj ) = O(1); j 2 IN; 3
(1.2.4)
provided that for some ; ~ > 0 the norm equivalence 0
kvkH s @
X
jjj0
2
D 2sjj
11 E 2 2 A
v; ~
(1.2.5)
holds for s 2 (? ~; ) and the regularity bounds ; ~ are related to the order 2t of the operator A by
; ~ > jtj: (1.2.6)
Note that under these circumstances one has in particular by duality 0
kvkH ?t @
X
jjj0
2?2tjj jhv; ij
11 2 2A
;
which plays a crucial role for the analysis of adaptive Galerkin schemes for equations of the form (1.2.2).
Di erential operators: A typical case arises when is a bounded domain and A = ? or A = ?+ cI, c > 0 in which case t = 1, H 1 = H01( ), H 1( ), respectively. Boundary integral equations: More generally, we are interested in solving (1.2.2)
for operators of the the form
Z
(Au)(x) = K (x; y)u(y) dsy = f; ?
(1.2.7)
where the kernel K (; ) satis es estimates of the type
@x @y K (x; y) c ; dist? (x; y)?(2+2t+j j+j j):
(1.2.8)
Examples are the single or double layer potential with t = ?1=2; 0, respectively, obtained e.g. when transforming an exterior boundary value problem for Laplace's equation into an integral equation. More generally, kernels of this type arise e.g. in connection with computing electrostatic elds, scattering form 3D obstacles, transmission problems, and high quality computer visualization based on the radiosity concept (see e.g. [BGZ, SS]). Again (1.2.1) holds for the respective values of t. The major obstacle, when dealing with operators of the type (1.2.7), is that corresponding sti ness matrices are not sparse. However, it can be shown that the matrices Aj are nearly sparse when (1.2.8) holds and
4
Requirement II (Moment conditions): holds. Let 2 := [0; 1]n . is said to have vanishing moments of order d~ if for jj > j0 there exists a smooth mapping : 2 ! ? such that supp (2) and Z y ( (y)) dy = 0; j j < d~ (1.2.9) where y = y1 1 yn n .
2
Let us assume for simplicity that A is selfadjoint and positive de nite, i.e., hAv; vi kvkH t . To be speci c let us assume further that the spaces Sj are accurate of order d, i.e.,
?jd d inf kvj ? vkL2 (?) < 2 kvkH d(?) ; v 2 H (?):
vj 2Sj
(1.2.10)
The ndings in [DPS2, DPS] may then be summarized as follows.
Theorem 1.2.1 Suppose that A satis es the above assumptions and (1.2.8). Furthermore, assume that and ~ as above satisfy (1.2.5) and (1.2.9) with jtj < ; ~; d~ > d ? 2t;
respectively. Then for each j 2 IN 0 there exists a matrix Acj (which is obtained by replacing small entries in the sti ness matrix Aj relative to j by zero) such that the following properties hold: (i)
?1
c c
= O(1);
Aj Aj j ! 1;
i.e., the compressed matrices are well-conditioned. (ii) The number of nonzero entries of Acj is of the order dim Sj .
(iii) The solution ucj of the compressed system Acj ucj = fj , fj := f2?tjjhf; i : jj j g exhibits still an asymptotically optimal convergence order, i.e. denoting by u the solution of (1.2.7), one has
?j ( ?s) kuk u ? ucj
H s(?) < H (?) 2
for ?d + 2t s < , s d, provided that ? is regular enough for H d (?) to be well-de ned. Thus optimal order 2?j(2d?2t) is preserved.
This Theorem describes the ideal situation and will serve as a guide line for the subsequent constructions. In summary we conclude that wavelet bases satisfying (1.2.5) for ; ~ > 1 would provide a suitable tool for a wide class of problems. In particular, this would cover the case of operators of order ?1. In the latter case it is furthermore important to realize a possibly high order d~ of vanishing moments. 5
1.3 Domain Decomposition and Representation of Geometry
The requirements I,II formulated above are in general very hard to satisfy for arbitrary domains. However, whenever a domain admits a reasonable decomposition into parametric images of cubes we will show that it is possible to construct composit wavelet bases that (nearly) ful ll the above requirements. The following setting covers, in principle, a wide range of bounded domains in Euclidean space as well as closed surfaces imbedded in some higher dimensional Euclidean space. In view of boundary integral equations, two-dimensional surfaces in IR3 deserve special attention and serve in fact as the primary motivation of the subsequent developments. Nevertheless, open manifolds such as bounded domains in Euclidean space, are covered as well. Throughout the paper the manifold ? will be assumed to be a piecewise smooth manifold (with or without boundary) which is at least globally Lipschitz endowed with a metric g. Denoting the surface element given in local coordinates by dsx = q g(x; x)dx1 ^ dx2 we can de ne the inner product Z
hu; vi? := u(x)v(x)dsx ?
(1.3.1)
for the space L2(?). In addition to the space L2(?) we will have to work with Sobolev spaces H s (?) de ned on ?. A natural way to de ne these spaces is to view them as trace spaces. For a certain range of s these spaces can be de ned equivalently via an atlas and partitions of unity. We will always assume to work in that range. Thus for Lipschitz surfaces we cover s 1. For detailed treatments of this subject we refer e.g. to [BGZ]. Here it is important to note that for closed surfaces one has H s (?) = H ?s (?); where H s (?) is the normed dual of H s (?) with respect to the inner product (1.3.1). The construction of depends crucially on the way the manifold ? is represented. Piecewise de ned parametric surface representations appear to o er most practicality and rest on the perhaps best developed concepts in Computer Aided Design. In the sequel we will always assume the following mathematical representation of ?. As above we denote by 2 = (0; 1)2 the unit square which will serve as a xed parameter domain and set N [ ? = ?i ; ?i = i(2); i = 1; : : :; N; (1.3.2) i=1
where for the i : IRn ! IRn0 are smooth functions chosen in such a way that ? has a certain desired degree of global smoothness. Note that the patches ?i are not supposed to overlap ?i \ ?j = ;; i 6= j; (1.3.3) m i.e., the di erent patches do not intersect. ? is said to be a C -surface if there exist local C m reparametrizations, i.e., for any neighborhood N ?, say, such that N \ ?i 6= ;, N \ ?j 6= ;, there exists an ane map , a neighborhood M 2 [ (2) and a function 2 C m(M) such that j2\M = i; j(2)\M = j ; (1.3.4) some n n0
6
where is some regular reparametrization of (2). For our purposes it will be convenient to work with the following equivalent formulation. For any two i ; j such that := ?i \ ?j 6= ; there exists a congruence j;i : 2 ! 2 and a regular reparametrization j;i of 2 such that
j;i(?i 1 ()) = (?j 1()); @ i j?i 1()= @ j j;i i;j j?i 1 (); j j := 1 + + n m: (1.3.5) In practical realizations for most of the patches the local reparametrizations can be chosen as the identity which means that (up to the congruences j;i ) the parametrizations join in a way that their coordinate functions are C m. The only places where nontrivial reparametrizations have to be employed is near singular vertices by which we mean vertices sharing a number of patches which is di erent from four. Depending on the genus of the surface such singular vertices may always have to occur. An interesting alternative is to enforce componentwise di erentiability everywhere at the expense of employing degenerate patch representations [R2]. We also remark that in all practical realizations the parametrizations i are actually polynomial or piecewise polynomial with suciently high componentwise smoothness. We have chosen this setting since a variety of practical tools have been developed in the CAD community realizing such surfaces for essentially arbitrary topology. Speci cally, in [HM, R1] a practicable concept for generating C 0, C 1 and C 2 surfaces of arbitrary topology is developed employing only quadrilateral patches as required above. Moreover, these surfaces can be re ned by means of subdivision which therefore ts into the present context. Thus we may view a surface ? of the above type as the true target surface or as an approximation which could be successively improved if necessary.
1.4 Road Map
The above applications give rise to the following wish list: (i) Since the solution of (1.2.2) might be smooth on some of the patches ?i it will be important to realize any desired order d of exactness. Here order d of exactness means that the trial spaces satisfy estimates of the form (1.2.10). (ii) In some applications mentioned above it is important to make the order d~ d of vanishing moments as high as one wishes independently of the exactness order d. This is closely related to the exactness order of the dual multiresolution spaces S~j . (iii) In order to cover all the above mentioned cases (second order di erential operators, single layer, double layer or hypersingular operator) we wish to have the elements of and ~ all be globally continuous. The above setting suggests the following approach. We will rst construct wavelet bases de ned on 2. On account of (ii), the concept of biorthogonal wavelets has to be employed. Such wavelet bases can be constructed by taking tensor products of wavelets 7
on the unit interval [0; 1]. We will therefore make heavy use of some recent results from [DKU] which are to some extent taylored to the present needs. Speci cally, for any d; d~ 2 IN , d + d~ even compactly supported primal and dual wavelets are available. However, since the wavelets on 2 will have to be lifted to the patches ?i through the mappings i (iii) requires paying some attention to piecing the patch bases together. This will be facilitated by realizing certain boundary conditions for the bases on [0; 1]. Finally, the Riesz basis property and related norm equivalences will be established through proving direct and inverse estimates combined with general criteria from [D2].
2 Biorthogonal Multiresolution with Boundary Conditions
2.1 Multiresolution Sequences
The common approach to biorthogonal multiresolution on [0; 1] begins with some dual pair (; ~) of re nable functions, i.e., X X (x) = ak (2x ? k); ~(x) = a~k ~(2x ? k); k2ZZ
and
D
; ~( ? k)
k2ZZ
Z
E
IR
= (x)~(x ? k) dx = 0;k; k 2 ZZ: IR
(2.1.1)
The idea is then to construct (the primal) spaces Sj on [0; 1] by taking those translates (2j ?k) which are supported inside [0; 1] supplemented by certain additional linear combinations of the translates overlapping the end points of the interval. These linear combinations are formed in such a way that the resulting span still contains all polynomials of a desired order (see [AHJP, CQ, CDV, DKU]). To our knowledge only in [DKU] the dual multiresolution spaces S~j induced by ~ also exhibit the original order of polynomial exactness which is crucial in the present context. It is well known that the order of polynomial exactness determines the approximation order of the spaces. An important family of initial dual pairs is based on B-splines. Denoting by [x0; : : :; xd]f the dth order divided di erence of f at the points x0; : : : ; xd 2 IR (see e.g. [dB]), the dth order centered cardinal B-spline is de ned by d?1 (2.1.2) (x) = d(x) := d[0; 1; : : : ; d] ? x ? b 2d c ; + where xl+ := (max f0; xg)l and bxc (dxe) is the largest (smallest) integer less (greater) than or equal to x. Thus is centered around `(2d) , i.e., x 2 IR; (2.1.3) d (x + `(d)) = d (?x); where `(d) := d mod 2, and has support h i j k l m supp d = 21 (?d + `(d)); 21 (d + `(d)) = [? 2d ; 2d ] := [`1; `2]; (2.1.4) !
8
i.e., d = `2 ? `1 and `(d) = `1 + `2. Thus, the B-splines of even order are centered around 0 while the ones of odd order are symmetric around 12 . The B-spline d is re nable with nitely supported real mask a = fak g`k2=`1 , i.e., `2 X
!
`2 X d (2 x ? k ) =: ak (2x ? k): (2.1.5) d (x) = d k + b 2d c k=`1 k=`1 It has been shown in [CDF] that for each d and any d~ d, d~ 2 IN , so that d + d~ even, there exists a function d;d~~ with the following properties (see [CDF]): (i) d;d~~ has compact support, i h suppd;d~~ = ? 21 d ? d~ + 1 + 21 `(d) ; 21 d + d~ ? 1 + 21 `(d) = [`1 ? d~ + 1; `2 + d~ ? 1] =: [`~1; `~2]: (2.1.6)
(ii)
21?d
~ is re nable with nitely supported mask a~,
d;d~
~ d;d~(x) = (iii)
`~2 X k=`~1
a~k d;d~~(2x ? k):
(2.1.7)
~ has the same symmetry properties as d, i.e.,
d;d~
~
~
d;d~(x + `(d)) = d;d~(?x);
x 2 IR:
(2.1.8)
(iv) The functions d and d;d~~ form a dual pair, i.e., hd; d;d~~( ? k)iIR = 0;k ; k 2 ZZ:
(2.1.9)
(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 2 ZZ .
(vi) The regularity of d;d~~ increases proportionally with d~. One easily checks that the symmetry properties (2.1.3), (2.1.8) have the following discrete counterparts ak = a`(d)?k ; a~k = ~a`(d)?k ; k 2 ZZ: (2.1.10) In the following d; d~ will be arbitrary as above but xed so that we can suppress them as indices and write brie y ; ~ if there is no risk of confusion. We will brie y recall next from [DKU] pairs of generator bases j ; ~ j which span multiresolution sequences of spaces Sj ([0; 1]), S~j ([0; 1]) which are exact of order d; d~, respectively. These collections have the form 0j = Lj [ Ij [ Rj; ~ 0j = ~ Lj [ ~ Ij [ ~ Rj; (2.1.11) 9
Setting g[j;k] := 2j=2g(2j ?k), the sets Ij , ~ Ij consist of the interior basis functions [j;k], ~[j;k], k 2 Ij ; ~ Ij , respectively, which do not interfere with the end points of the interval. Here ~ Ij := f`~; : : :; 2j ? `~ ? `(d)g; Ij = f`; : : :; 2j ? ` ? `(d)g: (2.1.12) where ` := `~ ? (d~ ? d): (2.1.13) To ensure that the interior functions are indeed fully supported in [0; 1], `~ has to be only boundeded from below by `~ `~2; (2.1.14) (see (2.1.6). Similarly the collections X : k 2 X g; X :k2 ~ Xj = f~j;k ~ Xj g Xj = fj;k j where for X 2 fL; Rg ~ Lj := f`~ ? d;~ : : : ; `~ ? 1g; ~ Rj := f2j ? `~ + 1 ? `(d); : : : ; 2j ? `~ + d~ ? `(d)g; (2.1.15) and Lj = f` ? d; : : : ; ` ? 1g; Rj = f2j ? ` + 1 ? `(d); : : : ; 2j ? ` + d ? `(d)g: (2.1.16) L , ~L are certain xed linear combinations of the translates [j;k] j[0;1], The functions j;k j;k ~[j;k] j[0;1] chosen so as to ensure that the linear spans of the collections 0j and ~ 0j contain all polynomials of degree less than d; d~, respectively. In fact, set L j;` ?d+r :=
j;R2j ?`+d?`(d)?r :=
`?1 X m=?`2 +1
~m;r [j;m] [0;1] ; r = 0; : : : ; d ? 1; (2.1.17)
2j ?X `1 ?1
m=2j ?`?`(d)+1
~Rj;m;r [j;m] [0;1] ; r = 0; : : :; d ? 1;
and likewise on the dual side
~j;L`~?d~+r := ~j;R2j ?`~+d~?`(d)?r :=
`?1 X ~
m=?`~2 +1
m;r ~[j;m] [0;1] ; r = 0; : : : ; d~ ? 1; (2.1.18)
2j ?X `~1 ?1
m=2j ?`~?`(d)+1
Rj;m;r ~[j;m] [0;1] ; r = 0; : : :; d~ ? 1;
where Lj;m;r := 2j R (2j x)r(2j x ? m) dx = R xr (x ? m) dx =: m;r
Rj;m;r
:=
2j
IR R
IR
IR r j j (2 (1 ? x)) (2 x ? m) dx =
10
R
IR
(2j ? x)r(x ? m) dx:
(2.1.19)
Analogously let
~Lj;m;r := 2j R (2j x)r~(2j x ? m) dx = R xr ~(x ? m) dx =: ~m;r IR IR R R r~ j j j R ~j;m;r := 2 (2 (1 ? x)) (2 x ? m) dx = (2j ? x)r~(x ? m) dx:
(2.1.20)
IR
IR
Since
Lj;m;r = m;r ; Rj;m;r = 2j ?m?`(d);r ; r = 0; : : : ; d~ ? 1; ~ Lj;m;r
= ~m;r ;
~Rj;m;r
= ~ 2j ?m?`(d);r ; r = 0; : : : ; d ? 1;
(2.1.21)
it is not hard to verify the following symmetry relations which will be used frequently. Remark 2.1.1 One has
Rj;2j?`+d?`(d)?r (1 ? x) = Lj;`?d+r (x); r = 0; : : : ; d ? 1; ~Rj;2j?`~+d~?`(d)?r (1 ? x) = ~Lj;`~?d~+r (x); r = 0; : : : ; d~ ? 1:
(2.1.22)
and
[j+1;m](x) = [j+1;2j+1?m?`(d)] (1 ? x); = ; ~: (2.1.23) To make sure that the collections of boundary functions Xj , ~ Xj are separated we will assume in the following that l m j log2(`~ + `~2 ? 1) + 1 =: j0 (2.1.24) De ning for any set of functions in L2( )
S () := closL2 (span ) ; one has [DKU]
Proposition 2.1.1 Let 0j ; ~ 0j be given by (2.1.11). (i) The spaces Sj;[0;1] := S (0j ) and S~j;[0;1] := S (~ 0j ) are nested, i.e.,
Sj;[0;1] Sj+1;[0;1]; S~j;[0;1] S~j+1;[0;1]; j j0:
(2.1.25)
(ii) The spaces Sj;[0;1], S~j;[0;1] are exact of order d, d~, respectively, i.e.,
d([0; 1]) Sj;[0;1]; d~([0; 1]) S~j;[0;1]; j j0:
11
(2.1.26)
The nestedness of the spaces S (0j ) and S (~ 0j ) follows from the fact that also the boundary functions satisfy two-scale re nement relations whose exact format is given in [DKU]. Of course, the re nement lters of the boundary functions di er from those of the interior functions. However, there are only niteley many of them, namely d; d~ for each end of the interval, respectively, and the lter coecients are independent of the level j . Nevertheless, in absence of translation invariance, it will be extremely convenient to view the collections 0j or ~ 0j as (column) vectors whose entries are the respective basis functions. More generally, we will extend this convention in a canonical way to any other collections of functions or in some Hilbert space H with inner product h; i which will arise below. Speci cally, h; i := (h; i)2; 2 will denote a matrix. Thus the collection on the right side is treated as a row vectors. Accordingly, h; i is a column or row vector when or , respectively, consist of only one element. Likewise the fact that, due to re nability, each j;k can be written as a linear combinations of elements in 0j+1 can be conveniently expressed by a matrix relation between 0j and 0j+1 (and likewise for the collections ~ 0j ) of the following form ~ 0j;0: 0j T = (0j+1)T M0j;0; (~ 0j )T = (~ 0j+1)T M (2.1.27) ~ 0j;0 consists of the lter or mask coecients of the ith Thus the ith column of M0j;0, M element of 0j , ~ 0j , respectively. As mentioned above the dependence of the re nement ~ j;0'on j is very weak in the sense that there are only nitely many matrices M0j;0, M di erent coecients, whose numbering but not their values depend on j . In fact, the re nement matrices have a stationary interior block which grows with j and an upper left and lower right block of xed size which corresponds to the boundary functions. Moreover, these blocks are symmetric in that the lower right block is obtained from the upper left one by reversing the order of rows and columns which is an immediate consequence of Remark 2.1.1. Again see [DKU] for details. Due to the boundary modi cations, the collections 0j and ~ 0j are no longer biorthogonal. However, it has been shown in [DKU] that these collections can always be biorthogonalized, a fact we shall make essential use of. In [DKU] this has been realized by a change of basis in ~ 0j . Here we will depart somewhat from this strategy in order to deal with additional requirements concerning boundary conditions. To describe this, note rst that by construction the interior functions in Ij and ~ Ij are still biorthogonal. Therefore they will be left essentially untouched and it suces to con ne the change of bases to the collections of boundary functions. Since generally d~ d we will always tacitly assume in the sequel that the primal collections Xj , X 2 fL; Rg, are extended by the corresponding number d~ ? d interior functions to match the size of ~ Xj . To formulate the main observation we recall from [DKU2, DS] the following facts. Proposition 2.1.2 The matrices TX := hXj ; ~ Xj i 12
are independent of j . Moreover, one has
det TX 6= 0; det T0X 6= 0;
(2.1.28)
where T0X is the submatrix of TX which is obtained by discarding the rst row and column. Proposition 2.1.3 There exist d~ d~ matrices CX , C~ X , for X 2 fL; Rg, independent of j j0 such that the collections X;j := CX Xj ; ~ X;j := C~ X;j ~ Xj ; (2.1.29) satisfy
hX;j ; ~ X;j i = I; X 2 fL; Rg:
Denoting the sets of new boundary functions as X;j = fj;k : k 2 Xj g; ~ X;j = f~j;k : k 2 Xj g; and likewise one has for that
(2.1.30)
j;k := [j;k]; k 2 Ij ; ~j;k := ~[j;k]; k 2 ~ Ij ; j := L;j [ Ij [ R;j ; ~ j := ~ L;j [ ~ Ij [ ~ R;j ;
(2.1.31)
hj ; ~ j i[0;1] = I:
(2.1.32)
Moreover, the following boundary conditions hold j;k (0) = ~j;k (0) = j;k (1) = ~j;k (1) = 0; k 2 j n f` ? d; 2j ? `(d) ? ` + dg; (2.1.33) while
j;`?d (0) = j;2j?`(d)?`+d (1) = 2j=2; ~j;`~?d~(0) =
~j;2j?`(d)?`~+d~(1) = 2j=2det T0L=det TL:
(2.1.34)
Proof: By biorthogonality of the interior functions, (2.1.33) follows from (2.1.30). Since by (2.1.29),
hX;j ; ~ X;j i[0;1] = CX hXj ; ~ Xj i[0;1]C~ TX ; (2.1.35) the independence of CX , C~ X of j is a consequence of the above remarks on the matrices TX . Furthermore, by symmetry (see Remark 2.1.1),
TR = TlL;
(2.1.36)
where for any matrix M the matrix which is obtained by reversing the order of rows and columns of M is denoted by Ml. 13
Thus it suces to determine CL; C~ L since CR = ClL ; C~ R = C~ lL :
(2.1.37)
To this end, consider the singular value decomposition
TL = UT V; where U; V are orthogonal matrices and is a diagonal matrix containing the singular values i of TL ordered according to size i i+1. By (2.1.28), all singular values are strictly greater than zero. One easily checks that for any invertible matrix R the matrices CL := R?1=2U; C~ L := R?T ?1=2V; (2.1.38) satisfy CLTLC~ TL = I; (2.1.39) which, on account of (2.1.35), means that the corresponding sets X;j ; ~ X;j are biorthogonal (see (2.1.30)). Thus it remains to choose R so that the boundary conditions (2.1.33) hold. Let Lj(0) denote the vector whose entries are the elements of Lj evaluated at zero. By construction one has (2.1.40) Lj(0) = ~ Lj(0) = 2j=2e1; where (e1)i = 1;i is the rst coordinate vector. By de nition (2.1.29) and (2.1.40), one has to nd a constant b such that L;j (0) = CLLj(0) = 2j=2R?1=2U1 = 2j=2e1; and
~ L;j (0) = C~ L~ Lj(0) = 2j=2R?T ?1=2V1 = b2j=2e1; where U1; V1 is the rst column of U; V, respectively. Thus the desired boundary conditions are equivalent to R?1=2U1 = e1; (2.1.41) and ?1=2V1 = bRT e1: (2.1.42) Now we make the ansatz ! 1 )T ?1=2 a ( V R := UT 1=2 ; 0 where a is some constant and U0 is the submatrix of U obtained by discarding the rst column. Since U is an orthogonal matrix, it is clear that
R?1=2U1 = a e1: Since
T?L 1 = VT ?1U 14
(2.1.43)
we conclude that
:= T?L 1 1;1 ; and note that, by (2.1.36), 6= 0. >From the de nition of R, (2.1.42) and (2.1.43) we infer b = = det T0L=det TL: (2.1.44)
Remark 2.1.2 It immediately follows from (2.1.37) and Remark 2.1.1 that the biorthog-
onalized boundary functions in X;j ; ~ X;j inherit the symmetry properties from Remark 2.1.1. It is now easy to derive from the re nement relations of the collections 0j ; ~ 0j the re nement matrices for the biorthogonalized bases j ; ~ j (see Remark 3.3.1 below or [DKU]). >From Remark 2.1.2, (2.1.37) and (2.1.10) one concludes that Remark 2.1.3 the re nement matrices Mj;0; M~ j;0 in ~ j;0; Tj = Tj+1 Mj;0; ~ Tj = ~ Tj+1M (2.1.45) also satisfy
Mlj;0 = Mj;0; M~ lj;0 = M~ j;0:
Moreover, one infers from (2.1.33), (2.1.34) that
~ j;0)k;k0 = (Mj;0)k;k0 = 0 for (M
(
k = ` ? d; k0 6= k; j +1 k = 2 ? `(d) ? ` + d; k0 = 6 k;
(2.1.46) (2.1.47)
that is, the re nement relation of basis functions vanishing at the boundary involve only basis functions on the next ner scale which also vanish at the boundary.
We conclude this section with a simple example.
Piecewise constants, d = 1: First note that, in view of (2.1.6), in this case we can choose `~ = d~ so that ` = 1 = d and Lj = f0g; ~ Lj = f0; : : :; d~ ? 1g. This means that the primal spaces result from simply
restricting the integer translates of scaling functions to [0; 1]. One easily deduces from (2.1.1) that the entries of the d~ d~ matrix TL are for d~ 3 given by (2.1.48) (TL)k;k0 = k;k0 ; k; k0 = 0; : : : ; d~ ? 1: Moreover, kZ+1 k0 +1 k0 +1 k;k0 = xk0 dx = (k + 1)k0 + 1? k : (2.1.49) k Thus ~?1 TL = (Pk0 (k))dk;k (2.1.50) 0 =0 ; 15
where
!
r r+1 X xi: (2.1.51) Pr (x) r +1 1 ((x + 1)r+1 ? xr+1) = i i=0 In this case d = 1, d + d~ even, the above explicit form of TL can, of course, also be used to verify (2.1.36) directly.
2.2 Composite Bases on Curves
A boundary integral equation may be formulated on a curve with corners. Thus in absence of a regular parametrization one may have to split the curve into several segments each being regular. We wish to demonstrate next how to join multiresolution spaces de ned on such adjacent curve segments such that the resulting functions on the whole curve are still continuous. The same principle will apply later to tensor product constructions on cubes. The basic idea is quite simple. Suppose we have two regular parametrizations ; : [0; 1] ! IRm, m 1 with
(1) = (0);
(2.2.1)
so that ? = ([0; 1]) [ ([0; 1]) is a continuous (piecewise smooth) curve in IRm. It will be convenient to introduce a geometric mesh by setting 8 > k = ` ? d; < 0; 1 ; k = 2j ? `(d) ? ` + d; (2.2.2) q(k) := > : ?j 2 k; k 6= ` ? d; 2j ? `(d) ? ` + d; and
2j; := (q(j )); 20j; := (q(j n @ j )); 2 f; g; 2j := 2j; [ 2j; ; (2.2.3) where @ := f` ? d; 2j ? `(d) ? ` + dg. By (2.2.1), the point (q(2j ? `(d) ? ` + d)) = (1) = (0) = (q(` ? d)) 2 2j; \ 2j can be identi ed. Now de ne
j; (x) := j;k (?1(x)); = (q(k)) 2 20j; ; 2 f; g; and Similarly let
(
?1 ([0; 1]); j;(1)(x) := 2j ?`((d)??1`(+xd))(; (x)); xx 22 ([0 ; 1]): `?d
~j;(1)(x) :=
(
1~ 2 2j ?`~(d)?`~+d~( 1 ~ ( ?1 ( )) 2 `~?d~
?1(x)); x 2 ([0; 1]); x ; x 2 ([0; 1]);
16
(2.2.4) (2.2.5)
Remark 2.2.1 By (2.1.33), (2.1.34), the collections j = f j; : 2 2j g; ~ j = f ~j; : 2 2j g;
(2.2.6)
belong to C (?) and are biorthogonal
h?j ; ~ ?J i? = I;
(2.2.7)
where here for f; g 2 L2(?) Z1
Z1
0
0
hf; gi? := f ((x))g((x))dx + f ((x))g((x))dx
(2.2.8)
de nes an inner product which is equivalent to the canonical one on ? in the sense that corresponding norms are equivalent.
Clearly, the bases j ; ~ j are still re nable. We shall identify next the corresponding global re nement matrices. They are given by (M?j;0); = (Mj;0)k;k0 ; = (q(k0)) 2 2j+1; ; = (q(k)) 2 2j; :
(2.2.9)
In fact, note that by (2.1.33), (2.1.34), (M?j;0); = 0 if 2 20j+1; ; 2 20j;0 ; 6= 0 2 f; g:
(2.2.10)
Thus only those entries of M?j;0 are di erent from zero for which both indices ; are contained in the same closed arc. Since (1) = (0) belong to both arcs we have to con rm that the above de nition (2.2.9) is consistent. This is indeed the case, because by Remark 2.1.3, (Mj;0)`?d;`?d = (Mj;0)2j+1 ?`(d)?`+d;2j ?`(d)?`+d : ~ ?j;0 is slightly di erent: Due to the di erent normalization of ~j;(1), the form of M 8 > > > > > > > > >
(M > > > > 1 ~? > = (0); = (q(k)) 2 20j+1; ; > > 2 (Mj;0)k;`~?d~; > : 1 ~ ? = (1); = (q(k)) 2 20j+1; : 2 (Mj;0)k;2j ?`(d)?`~+d~; (2.2.11) we will employ similar principles below in the tensor product case. ~ ?j;0); = (M
17
3 Wavelets and Boundary Conditions 3.1 Biorthogonal Wavelets and Symmetry
While it is fairly easy to construct composite biorthogonal generator bases ?j ; ~ ?j as indicated above, the question arises how to generate also biorthogonal wavelet bases de ned on ? which are also continuous on ?. First we recall from [DKU2] that for any d + d~ even d~ d and corresponding biorthogonal generator bases j ; ~ j as above one can construct wavelet bases
j = f!j;k : k = 1; : : : 2j g; ~ j = f!~j;k : k = 1; : : :; 2j g; (3.1.1) which are biorthogonal
h j ; ~ j0 i[0;1] = j;j0 I; j; j 0 j0:
(3.1.2)
Moreover, expressing the corresponding two-scale relations in matrix form as ~ j;1;
Tj = Tj+1Mj;1; ~ Tj = ~ Tj+1 M (3.1.3) one has
(3.1.4) M~ Tj;e Mj;e0 = e;e0 I; e; e0 2 f0; 1g: ~ j;e contain only a uniformly bounded numEach column of any of the matrices Mj;e; M
ber of nonzero coecients forming the lters of the wavelets and generator basis functions whose supports therefore satisfy diamsupp!j;k ; diamsupp~!j;k ; diamsuppj;k ; diamsupp~j;k 2?j : (3.1.5)
~ j;1 constructed in [DKU] do not necessarily However, the particular matrices Mj;1; M share the same symmetry properties as the re nement matrices in Remark 2.1.3. One can check that the construction in [DKU2] does give Mlj;1 = Mj;1; M~ lj;1 = M~ j;1; (3.1.6) when d is even. Let us point out rst how to arrange (3.1.6) also in the remaining cases d odd. Proposition 3.1.1 Given Mj;0; M~ j;0 from (2.1.45) and M0j;1; M~ 0j;1 the particular matrices constructed in [DKU2] satisfying (3.1.4) as above. Then there always exist ma~ j;1 which also satisfy (3.1.4) and inherit the same sparseness properties trices Mj;1; M 0 0 ~ j;1, so that of Mj;1 ; M ~ j;0; M ~ j;1)k = O(1); k(Mj;0; Mj;1)k; k(M (3.1.7) while in addition
Mlj;1 = Mj;1; M~ lj;1 = M~ j;1: 18
(3.1.8)
Proof: Let us denote by I! the permutation matrix whose only nonzero entries are on the antidiagonal and have the value 1. Thus for any matrix M one has Ml = I!MI! where I! is always assumed to have the right size without further speci cation. Since (I!)2 = I one easily veri es that for any two matrices A; B of appropriate sizes (Al)T = (AT )l; (AB)l = AlBl; (A?1)l = (Al)T : (3.1.9) ~ 0j;1 consisting of the rst 2j?1 Now let Nj;1; N~ j;1 denote the submatrix of M0j;1; M columns, respectively (recall that these matrices have always 2j columns). Let Hj;1 := (Nj;1; Nlj;1); H~ j;1 := (N~ j;1; N~ lj;1); and note that by (3.1.4), (2.1.46) and (3.1.9), M~ Tj;0Nj;1 = 0; (M~ lj;0)T Nlj;1 = (M~ Tj;0Nj;1)l = 0: Applying the same reasoning to Mj;0; H~ j;1 provides M~ Tj;0Hj;1 = 0; MT0;1H~ j;1 = 0:
(3.1.10)
The matrices Hj;1; H~ j;1 have still full rank. To see this, consider the collection Tj := Tj+1Hj;1; and note that by (3.1.10), ~ Tj;0Hj;1 = 0: h~ j ; j i[0;1] = M~ Tj;0h~ j+1 ; j+1i[0;1] = M Thus it suces to show that the elements of j are linearly independent. Since by construction, Hj;1 = Hlj;1 one has (Hj;1)k;2j?1+r = (Hj;1)2j+1 ?`(d)?k;2j?1 +1?r ; r = 1; : : : ; 2j?1: Combining this with the symmetry relations in Remarks 2.1.1, 2.1.2, straightforward calculations yield
j;2j?1 +k (x) = j;2j?1+1?k (1 ? x); k = 1; : : : ; 2j?1 : (3.1.11) Moreover, by de nition, j;k = !j;k , k = 1; : : : ; 2j?1. Thus j = f!j;k ; !j;k (1 ? ) : k = 1; : : : ; 2j?1 g: Now suppose that cT j = 0. Then hcT j ; !~j;k i[0;1] = ck (3.1.12) for k = 1; : : : ; m, where m is the rst column in Mj;1 (and hence in Hj;1), which corresponds to the stationary interior masks so that !j;m is fully supported in [0; 1]. If j is large enough the support of !~j;m will not overlap any of the supports of the functions 19
!j;k (1 ?), k = 1; : : :; 2j?1 so that (3.1.12) follows indeed from biorthogonality and gives ck = 0, for k = 1; : : : ; m. Likewise testing with the functions !~ j;k (1 ?) for k = 1; : : : ; m ensures that also ck = 0 for k = 2j ; : : : ; 2j ? m + 1. Therefore
cT j
=
j ?m 2X
k=m+1
ck j;k = 0
(3.1.13)
is a linear combination of functions which are supported in (0; 1). Now note from the construction in [DKU2] that for d odd one has that the rst column of Nlj;1 has the same support as the 2j?1 + 1st column of M0j;1. Thus the functions in the above linear combination all have pairwise di erent supports. Therefore (3.1.13) implies that c = 0, which con rms our claim. Now consider ~ Tj;1 ! N T H~ j;1Hj;1 = ~ l T (Nj;1; Nlj;1) (Nj;1) ~ Tj;1Nlj;1 ! I N = =: Kj : (N~ lj;1)T Nj;1 I Since by (3.1.9), the matrix Kj has the form where
N~ Tj;1Nlj;1 = (N~ lj;1)T Nj;1 l ; ! I A j K j = Al I ; j !
!
l Aj = a0l 00 ; Alj = 00 a0 ;
and the block a is independent of j . By the above remarks Kj must be invertible. Noting that
Alj Aj
!
!
l ?1 l = a0a 00 ; (I ? Alj Aj )?1 = (I ? a0 a) 0I ;
one easily con rms that
I 0 0 01 C B l ?1 l l ?1 K?j 1 = BB@ 00 I +?(aI(?I ?alaa)a?)1ala aa(I(I??aalaa)?)1 00 CCA : 0 0 0 I Thus both Kj and K?j 1 are banded with band width independent of j . Moreover, one easily checks from the de nition of Kj and (3.1.9) that Klj = Kj ; (K?j 1)l = K?j 1: (3.1.14) 0
20
Thus de ning,
Mj;1 := Hj;1K?j 1; M~ j;1 := H~ j;1;
(3.1.15) we readily infer from (3.1.10) and (3.1.14) that (3.1.4) holds. Moreover, since by construction H~ lj;1 = H~ j;1 and Hlj;1 = Hj;1 the assertion follows from (3.1.14) and (3.1.9). In the sequel we will always refer to the symmetric version when dealing with (3.1.4) and (3.1.3).
3.2 Some Preliminary Remarks on Boundary Conditions
Returning to the composite generator bases constructed above a little thought reveals that dim S (?j ) = 2 dim j ? 1; (3.2.1) while (3.2.2) dim S (?j+1) ? dim S (?j ) = 2 dim S ( j ): Thus glueing wavelets which do not vanish at 0 or 1 together, as in (2.2.4), would result in a de cient complement space. On the other hand, if all the wavelets in j vanished at 0 and 1 the above lifting technique ?j := j ?1 [ j ?1 would readily produce wavelets de ned on all of ? which are continuous and biorthogonal relative to the inner product (2.2.8). The following observation is an immediate consequence of the boundary relations (2.1.33) and (2.1.34). Remark 3.2.1 The wavelet !j;k vanishes at 0 (1) if and only if the rst (last) entry in the k-th column of Mj;1 is zero. An analogous statement holds for ~ j .
Remark 3.2.2 Suppose that 0j is a basis for some complement of S (j ) in S (j+1) and at least one element ! 2 0j does not vanish at zero. Then there exists no basis of the complement S ( 0j ), that has all elements vanish at zero.
Proof: If there were a basis 00j of S ( 0j ) whose elements vanish at zero the set f!g[ 00j would be still linearly independent, since ! is the only function which does not vanish at zero, but still, by assumption, f!g [ 00j S ( 0j ), which is a contradiction.
In fact, more can be said. Remark 3.2.3 There exist no biorthogonal wavelet bases j ; ~ j for the spaces S (j ), S (~ j ) as above such that all elements of j vanish at 0 and 1. Proof: Suppose there exist biorthogonal wavelet bases j;0; ~ j for the spaces j ; ~ j above, such that all the elements in j;0 vanish at zero and one. The fact that the spaces S (j ); S (~ j ) satisfy certain direct and inverse estimates combined with the biorthogonality implies, on account of the results in [D2], that the sets [ [ j0 [ j;0; ~ j0 [ ~ j ; j j0
j j0
21
form biorthogonal Riesz-bases for L2([0; 1]). On the other hand, (2.1.33) and (2.1.34) also imply that
j;0 S (j+1;0); j j0; where j;0 := j n fj;`?d ; 2j?`(d)?`+d g spans a subspace of S (j ) satisfying homogeneous boundary conditions. Clearly, S (j;0) still satis es the same inverse estimates. Moreover, it is not hard to show that it also satis es a direct estimate of the form ?sj inf kf ? gj kL2 ([0;1]) < 2 kf kH s([0;1]); gj 2S (j;0 )
for 0 s < 1=2, where H s([0; 1]) denotes the Sobolev space of (noninteger) order s on [0; 1]. Therefore the results in [D2] still imply that also the reduced collection j0 ;0 [
[
j j0
j;0;
is a Riesz-basis. Therefore the function j0;`?d has a representation in this latter basis S contradicting the fact that the collection j0 [ jj0 j;0 is also a Riesz basis. On the other hand, the number of all wavelets that do not vanish at the end of the interval can always be limited. Remark 3.2.4 Given j , ~ j and associated wavelet bases j , ~ j satisfying (3.1.2). Then there always exists another pair of biorthogonal wavelet bases 0j , ~ 0j still satisfying (3.1.2), where only one element of 0j does not vanish at zero and one, respectively. Moreover, one still has that 0 ) 2?j . diam(supp !j;k Proof: By assumption, j satis es (3.1.3). We wish to perform a change of basis 0 !j;k
=
j
2 X
k0 =1
bk;k0 !j;k0
which in matrix form reads
0j = Bj j = Bj MTj;1j+1:
(3.2.3)
Recall from [DKU2] that Mj;1 has the form
ML Mj;1 =
MIj;1
(3.2.4)
MR 22
;
where the size of the upper left and lower right blocks is independent of j and the interior block MIj;1 is stationary, i.e., it is a nite section of a biin nite matrix of the form (ak?2l)k;l2ZZ whose size is proportional to 2j . All nonzero entries of the rst (last) row of Mj;1 are contained in the corner blocks ML (MR). Let 2 IN be the smallest integer so that the rst (last) row has no nonzero entry with column index larger (smaller) than (2j ? ). Then we choose Bj to have the form 1 B 0 0 L (3.2.5) Bj = B@ 0 I(2j?2) 0 CA ; 0 0 BR where I(r) denotes the r r identity matrix and where the BL, BR are Householder (orthogonal) matrices which re ect the rst column of MTj;1 into the coordinate vector e1 2 ZZ 2j . Thus setting, M0j;1 := Mj;1BTj and M~ 0j;1 := M~ j;1BTj the bases 0j = Bj j and
~ 0j = Bj ~ j are still biorthogonal (3.1.2), (since Bj is an orthogonal matrix) and satisfy ~ 0j;1. By construction, one has S ( j ) = S ( 0j ), S ( ~ j ) = S ( ~ 0j ). (3.1.3) relative to M0j;1, M 0 0 T
j = (Mj;1) j+1 . The assertion follows now from Remark 3.2.2. 0
The above comments show that the task of forming continuous wavelets on the union of several curve segments from bases on the individual segments appears to be a bit more delicate. For the case d = d~ = 2 this was studied in [JL] where successive projections are used. In the following we will present an approach that avoids glueing the wavelets across segment boundaries and where corresponding lters are obtained by local operations. Remark 3.2.5 To this end, we will employ concepts developed in [CDP]. The main idea is to determine rst some initial complement spaces between two successive spaces S (j ) and S (j+1), from which the desired complements spanned by biorthogonal wavelets will be generated with the aid of certain projections. The point is to perform this latter projection on the global composite spaces so that continuous functions are taken into continuous ones. This requires that the initial complements can easily be glued together which here means that they are spanned by functions which vanish at the end points of the interval. Although these initial complements will not correspond to biorthogonal bases they still should exhibit certain stability properties which we brie y describe rst.
3.3 Some Auxiliary Facts
As indicated above, we shall have to manipulate and vary complements between two successive multiresolution spaces. The necessary tools have been developed in [CDP] (see also [Sw]). Since we shall have to apply them several times for di erent settings, we nd it worth brie y stating them here in sucient generality. Thus we will consider for the time being some Hilbert space H with inner product h; i and norm kk = h; i1=2. Later H will be L2(D), where D = [0; 1]; 2; ?. fj gj2IN , j = f j;k : k 2 j g H , will denote a uniformly stable sequence of re nable bases, i.e.,
kck`2 (j ) kcT j k 23
(3.3.1)
uniformly in j and
Tj = Tj+1 Mj;0; j 2 IN 0: (3.3.2) Moreover, denoting by [X; Y ] the space of bounded linear operators from a normed linear space X into the normed linear space Y one has [CDP]
Mj;0 2 [`2(j ); `2(j+1)]; kMj;0k = O(1); j 2 IN 0; where
kMj;0k :=
sup
u2`2 (j );kuk`2 (j ) 1
(3.3.3)
kMj;0uk`2 (j+1 ) :
Although in all our applications the index sets j will be nite we remark that the results remain valid for in nite sets j as well where the corresponding matrix-vector operations are to be understood in the sense of absolute convergence. The notation and conventions made above for H = L2([0; 1]) will be employed in this more general in obious analogy. We are interested in determining some stable basis j = f j;k : k 2 rj g S (j+1) such that
S (j+1) = S (j ) S ( j ):
(3.3.4)
Proposition 3.3.1 ([CDP]) Suppose that fj g is uniformly stable and (3.3.2) holds. Then fj [ j g for j S (j+1 ), satisfying (3.3.4), is uniformly stable if and only if there exist Mj;1 2 [`2(rj ); `2(j+1)]; such that
(3.3.5) Tj = Tj+1 Mj;1; and that Mj = (Mj;0; Mj;1 ) 2 [`2 (j [ rj ); `2 (j+1 )] are invertible and satisfy
kMj k ;
M?j 1
= O(1); j 2 IN 0:
(3.3.6)
Writing M?j 1 = Gj = GGj;j;10 , one obtains the reconstruction formula
Tj+1 = Tj Gj;0 + Tj Gj;1:
(3.3.7)
Given j and Mj;0, any Mj;1 2 [`2(rj ); `2(j+1)] such that (3.3.6) and (3.3.7) hold, is called a stable completion of Mj;0. We wish to determine next particular complements induced by linear projectors. To this end, suppose that j F H , where F is some subspace of H , and that j F , the dual of F relative to the dual pairing induced by h; i, such that
hj ; j i = I; j 2 IN: Hence
Pj f := hf; j ij = 24
X
k2j
hf; j;k i j;k
(3.3.8) (3.3.9)
are linear projectors onto S (j ). Moreover assume that the matrices hj+1 ; j i de ne uniformly bounded mappings in [`2(j ); `2(j+1)], i.e.,
khj+1 ; j ik = O(1); j 2 IN:
(3.3.10)
We are interested in nding uniformly stable bases for
W (j ) := (Pj+1 ? Pj )S (j+1):
(3.3.11) Theorem 3.3.1 ([CDP]) Suppose that j ; j satisfy (3.3.8), (3.3.10) and that M j;1 j := (Mj;0; M j;1) let G j = are some uniformly stable completions of Mj;0 . For M Gj;0 = M ?j 1 . Then G j;1 Mj;1 := (I ? Mj;0hj+1 ; j iT )M j;1 (3.3.12) are also uniformly stable completions of the Mj;0 and the inverses are given by Gj;0 = G j;0 + hj+1 ; j iT M j;1G j;1; Gj;1 = G j;1: (3.3.13) The collections span the spaces W (j ), i.e., .
Tj = Tj+1 Mj;1
(3.3.14)
W (j ) = S ( j ):
(3.3.15)
We will apply also the following special case where j = ~ j H is also re nable, i.e., ~ j;0: ~ Tj = ~ Tj+1 M (3.3.16) Clearly hj ~ j i = I (3.3.17) implies (3.3.18) M~ Tj;0Mj;0 = I:
Corollary 3.3.1 ([CDP]) Let fj g, f~ j g, Mj;0, M~ j;0 be related as above. Suppose j;1 is some stable completion of Mj;0 and that G j = M ?j 1. Then that M Mj;1 := (I ? Mj;0M~ Tj;0)M j;1 (3.3.19) is also a stable completion and Gj = M?j 1 has the form ~ Tj;0! M (3.3.20) Gj = : Gj;1 Moreover, the collections
j := MTj;1j+1 ; ~ j = G j;1~ j+1 25
(3.3.21)
form biorthogonal systems D
E
j ; ~ j = I;
Thus setting
E
D
D
E
j ; ~ j = j ; ~ j = 0:
Qj f := hf; ~ j ij ;
one has
(3.3.22) (3.3.23)
(Qj+1 ? Qj )S (j+1) = S ( j ); (Qj+1 ? Qj )S (~ j+1) = S ( ~ j ):
(3.3.24)
Note that, in view of the nestedness of the spaces, (3.3.22) implies that the collections [ [ = 0 [ j ; ~ := ~ 0 [ ~ j j 2IN 0
are biorthogonal
j 2IN 0
D
E j ; ~ j0 = j;j0 I: (3.3.25) Here we have set for simplicity r?1 := 0; ?1;k = 0;k; ~?1;k = ~0;k. Finally, we will frequently make use of the following fact from [DKU2], which interrelates the re nement relations and stable completions relative to di erent generator bases.
Remark 3.3.1 Suppose that Cj is a nonsingular (#j ) (#j ) matrix and let j = Cj 0j : (3.3.26) Assume that M0j;0 is the re nement matrix for 0j and M0j;1 is a stable completion of M0j;0. Then one has Tj = Tj+1Mj;0 (3.3.27) where
Moreover,
Mj;0 = C?j+1T M0j;0CTj :
(3.3.28)
Mj;1 = C?j+1T M0j;1
(3.3.29)
is the corresponding stable completion and
M?j 1
?T G0 CT ! C = j 0 j;0T j+1 =: Gj : Gj;1Cj+1
3.4 Initial Complements with Boundary Conditions
(3.3.30)
We return now to multiresolution on [0; 1]. Combining biorthogonality and compact support, one can show that the collections j ; ~ j are uniformly stable in the sense of (3.3.1) [CDP, DKU]. We ultimately plan to apply Corollary 3.3.1 to construct biorthogonal wavelets on composite domains such as the curve considered above. In order to be able to patch complement bases de ned for the individual component spaces continuously together, we are lead by the observations in Section 3.1 to construct 26
j;1 of M0j;0 such that the (uniformly stable) rst some initial stable completions M complement bases j;1 ( j )T = (0j+1)T M (3.4.1) satisfy
j (0) = j (1) = 0: (3.4.2) 0 Here we denote at this point by Mj;0 the re nement matrices of the collections 0j prior to biorthogonalization (see (2.1.27)). We will consider rst a special case.
The piecewise linear case, d = 2: In the rst case of interest, d = 2, this is very
easy, namely the so called hierarchical bases do the job. Again for ` = 1; `~ = d~ one has j = f0; : : :; 2j g. Thus the collections
j := fj0 +1;2k?1 : k = 1; : : : ; 2j g (3.4.3)
correspond to taking the ne scale generators at the points in 2?j?1 j+1 n 2?1j . Obviously one has !j;k (0) = ! j;k (1) = 0; k = 1; : : : ; 2j : (3.4.4) Recall that in this case the basis 0j from (2.1.11) consists of the restrictions to the hat functions to [0; 1]. The corresponding re nement matrices are of the form 0
M0j;0 =
B B B B B B B B B B B B B B B B B B B B B @
p12 0 p1 p1 2 2 2 2 0 p12 0 p1 2
0 ...
0
2
0 ...
:::
0 0 0
p1 2 2
::: ...
::: ::: ::: 0 :::
p1 2 2
0
1
C C C C C C C C C C C C C C C C C C C 1 C p C 2 2 A
... 0
:
(3.4.5)
p1
2
The stable completion which corresponds to the hierarchical complement basis in (3.4.3) is given as 1 0 0 0 0 : : : 0 C B C B 1 0 0 C B C B C B 0 0 0 C B C B . . . . C B . . . . . . . C B . C: (3.4.6) Mj;1 = BB C 0 1 0 C B C B C B 0 0 0 C B C B C B 0 0 1 A @ 0 ::: 0 0 0 27
Since obviously, j0 +1;2k+1 = !j;k ; k = 0; : : : ; 2j ? 1; 0 2j;k ? 12 j0 +1;2k?1 + j0 +1 k + 1 p 0 1 ? 2 !j;k?1 ? 21 !j;k ; k = 1; : : : ; 2j ? 1; = 2j;k p p = 2j;0 0 ? 12 !j;0 0; j0 +1;2j+1 = 2j;0 2j ? 21 !j;0 2j ?1;
j0 +1;2k = j0 +1;0
p
(3.4.7)
?j 1 are readily identi ed as the corresponding blocks G j;0, G j;1 of the inverse G j = M 0 B B B B B B B B B @
G j;0 = and
p
2 0 0 0 p 0 0 2 0 ... p 2 0 ::: 0
0
? 12 1 ? 12 0 0 : : : B B B 0 0 ? 12 1 ? 21 B B
G j;1 = BBB ... B B @
Since obviously,
... 0 :::
::: ... ... 0 0
0 0
p0
2
1 C C C C C C C C C A
;
(3.4.8)
1
0 C ... C C C ... C C: C C ? 12 0 0 CCA ? 21 1 ? 12
(3.4.9)
(3.4.10) M j
;
G j
= O(1); j 2 IN 0; we infer form Proposition 3.3.1 that the bases 0j [ j are uniformly stable. When d > 2 the situation is a little more involved and we will apply the tools from the previous section. To this end, let j;`?d (f ) := 2?j=2f (0); j;2j ?`(d)?`+d (f ) := 2?j=2 f (1); (3.4.11) and set ~ j;0 := ~ j n f~j;`~?d~; ~j;2j ?`(d)?`~+d~g: (3.4.12) Lemma 3.4.1 The collections of functionals j := fj;`?d ; j;2j ?`(d)?`+d g [ ~ j;0 (3.4.13) satisfy (3.3.8) hj ; j i[0;1] = I; 28
as well as (3.3.10). De ning W (j ) as in (3.3.11) relative to the projectors Pj f := hf; j i[0;1]j onto S (j ), one has that g(0) = g(1) = 0; g 2 W (j ): (3.4.14) Proof: The biorthogonality of j and j is an immediate consequence of (2.1.32) and the boundary conditions (2.1.33), (2.1.34) combined with the de nition (3.4.13). The relation (3.4.14) follows from the de nition of the Pj and again (2.1.33). To identify stable bases j for the complements W (j ) of S (j ) in S (j+1), we will employ Theorem 3.3.1. As the initial stable completion we take here Mj;1 from (3.1.3) which corresponds to the compactly supported biorthogonal wavelet bases j con~ Tj;0; Gj;1 = structed in [DKU2]. Thus the corresponding inverses are given by Gj;0 = M M~ Tj;1. Since the Mj , Gj have a uniformly bounded number of non-zero entries per row and column which are also uniformly bounded, it is clear that (3.3.6) holds, so that the Mj;1 are indeed stable completions in the above sense. Lemma 3.4.2 Let j be de ned by (3.4.13). The matrices M j;1 := (I ? Mj;0hj+1 ; j iT[0;1])Mj;1 (3.4.15) are also uniformly stable completions of the Mj;0 and the inverses are given by G j;0 = Gj;0 + hj+1; j iT[0;1]Mj;1Gj;1; G j;1 = Gj;1: (3.4.16) Moreover, the collections j;1
Tj := Tj+1M (3.4.17) are uniformly stable bases of the spaces W (j ), W (j ) = S ( j ); (3.4.18) so that, by Lemma 3.4.1,
j (0) = j (1) = 0: (3.4.19) Proof: Since the elements of the bases j ; ~ j all have compact support the matrices hj+1 ; j iT have in each row and column only a uniformly bounded nite number of nonvanishing entries which are also uniformly bounded. In fact, de ning as in the proof of Proposition 2.1.3 0 det T (3.4.20) := det TL ; L we obtain by (2.1.34), 1=2 j;`?d (~j+1;k ) = 21=2 `~?d;k ~ ; j;2j ?`(d)?`+d (~j +1;k ) = 2 2j+1 ?`(d)?`~+d;k ~ ; while by (3.1.4) and (3.4.13), for k 2 j+1; k0 2 j n f` ? d; 2j ? `(d) ? ` + dg hj+1; j i[0;1] k;k0 = hj+1;k ; ~j;k0 i[0;1] = (M~ j;0)k;k0 :
Hence the hypotheses of Theorem 3.3.1 are satis ed, which con rms (3.4.18). The rest follows from Lemma 3.4.1. 29
Remark 3.4.1 Employing (3.1.9) and Remark 2.1.3 one easily con rms that the ma j ; G j all satisfy trices M M lj = M j ; G lj = G j :
3.5 Continuous Wavelets on Composite Curves
We wish to apply the above results to the construction of globally continuous biorthogonal wavelets on curves. Since it suces to consider just two adjacent arcs we return to the situation described in Section 2.2. For
rj := f1; : : :; 2j g let and set De ning
w(k) := 2?j (k ? 21 ); k 2 rj ;
rj; := (w(rj )); 2 f; g:
:= !j;k ; = (w(k)) 2 rj; ; (3.5.1) we immediately infer from (3.4.19) that these functions are continuous on ? and span ?j;1 is also a complement of S (j ) in S (j+1 ). The corresponding stable completion M j;1 from (3.4.17) satis es easily identi ed. In fact, we infer from Remark 3.2.1 that M j;1)`?d;k = (M j;1)2j+1 ?`(d)?`+d;k = 0; k 2 rj : (M j;
As a consequence there is no consistency problem when setting j;1)k0;k ; = (q(k0)) 2 2j+1; ; = (w(k)) 2 rj; ; 2 f; g: ?j;1); = (M (M (3.5.2) Proposition 3.5.1 For M?j;0; M~ ?0;j and M ?j;1 given by (2.2.9), (2.2.11) and (3.5.2) let
M?j;1 := (I ? M?j;0(M~ ?j;0)T )M ?j;1: Then the collections
(3.5.3)
Tj := Tj+1M?j;1 (3.5.4) S form biorthogonal wavelet bases, i.e., := j0 [ jj0 j form a Riesz basis for L2 (?). For the identi cation of the corresponding dual wavelets and their lters we refer to the discussion of corresponding construction for higher dimensional manifolds below.
30
3.6 Tensor Products
The next step is to take tensor products of the univariate constructions. This follows mostly canonical lines and one only has to x some notation. We will apply the following rules: Super- or subscripts 2 indicate quantities de ned on the unit n-cube 2 = [0; 1]n, usually obtained as tensor products. For instance, 2j = j j . Likewise, k is to be understood as a multiindex k = (k1; : : :; kn ) as soon as it is associated with a multivariate quantity. The wavelets require now a further index e 2 f0; 1gn , namely 2 (x) = ! !j;e;k j;e1 ;k1 (x1) !j;en ;kn (xn ); where ( 0; 2 !j;e;k = !j;k ;;kk 22 rj ;; ifif ee = = 1; and
j;k
j
rj := f1; : : : ; 2j g:
2 , i.e., not to use It will thererfore sometimes be convenient to write !j;20;k instead of j;k an extra notation for the scaling functions. The matrices M2j ; G2j are now naturally blocked into components M2j;e ; G2j;e, e 2 f0; 1gn , where (3.6.1) (M2j;e)k;k0 = (Mj;e1 )k1 ;k10 (Mj;en )kn ;kn0 ; and M2j;1 is comprised of all the components M2j;e ; e 2 f0; 1gn n f0g, while M2j;0 is the re nement matrix of 2j . Tensor products of functionals de ned on univariate functions are canonically de ned by their action on the respective variables. Interpreting the collections 2j in this sense, we will make crucial use of the projectors
Pj2f := hf; 2j i2 2j
(3.6.2)
onto the spaces S (2j ). To describe their relevant properties, let us denote by
@ 2j := fk 2 2j : ki 2 f` ? d; 2j ? `(d) ? ` + dg; for some 1 i ng and analogously by @ ~ j the set of indices associated with the boundary of 2.
Lemma 3.6.1 The projectors Pj2 have the following properties. (i) Whenever k 2 2j belongs to @ 2j , then the quantities hf; 2j;k i2 depend only on
values of f restricted to the intersection of all faces of 2 which k is associated with. In particular, one has that Pj2 interpolates at the vertices of 2, i.e.,
(Pj2 f )(e) = f (e); e 2 f0; 1gn : (ii) When d > n=2 the projectors Pj2 have the following approximation properties
kf ? Pj2 f kL2 (2) n=2, and any p 2 j;2, one has kf ? Pj2f k2L2 (j;k ) < kf ? pk2L2 (j;k ) + kPj2(f ? p)kL2 (j;k ) 2 2 (3.6.5) kf ? pkL2 (j;k ) + kmax 0 2 jj;k0 (f ? p)j ; j;k
2k where we have used that kj;k 1. Recall that the functionals 2j;k are tensor L2 (2) < products of L2-inner products and (scaled) point evaluations. Taking the respective scalings into account, it is easy to see that (3.6.6) jj;k0 (f )j n=2 the Sobolev imbedding theorem ensures that kf ? pkL1 (2)
