Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and related applications

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Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and related

applications

Souleymane Kadri Harouna, Valérie Perrier

To cite this version:

Souleymane Kadri Harouna, Valérie Perrier. Homogeneous Dirichlet wavelets on the interval diago-

nalizing the derivative operator, and related applications. 2019. �hal-01568431v3�

Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and related

applications

Souleymane Kadri Harouna ^a,∗ , Val´ erie Perrier ^b

a Laboratoire de Math´ ematiques, Image et Applications, Avenue Michel Cr´ epeau, 17042 La Rochelle cedex 1 France.

b Univ. Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000 Grenoble, France.

Abstract

This paper presents a new construction of homogeneous Dirichlet wavelet basis on the unit interval, linked by a diagonal differentiation-integration relation to a standard biorthogonal wavelet basis. This new wavelet basis allows to compute the solution of the Poisson equation only by a wavelet coefficient renormaliza- tion - like in Fourier domain -, which yields a linear complexity O(N) for this problem. Another application concerns the construction of free-slip divergence- free wavelet bases of the hypercube, in general dimension, with an associated decomposition algorithm as simple as in the periodic case.

Keywords: Wavelets on the interval, Boundary condition, Poisson equation, Divergence-free wavelets

1. Introduction

1

Since the pioneering work of Lemari´ e-Rieusset [14], due to their important role in the construction of divergence-free or curl-free wavelets, biorthogonal multiresolution analyses linked by differentiation and integration have been widely studied [13, 19, 22, 23]. The main purpose was to construct two mul-

∗ Corresponding author

Email address: [email protected] (Souleymane Kadri Harouna )

tiresolution analyses of L ² (0, 1) provided by spaces V _j ¹ and V _j ⁰ such that

∀ j, d

dx V _j ¹ = V _j ⁰ . (1)

Relation (1) should be interpreted as: ∀f ∈ V _j ¹ , f ⁰ ∈ V _j ⁰ and ∀g ∈ V _j ⁰ , there

2

exists f ∈ V _j ¹ such that f ⁰ = g.

3

4

On the unit interval [0, 1], with non periodic boundary conditions, such a construction was firstly introduced by Jouini and Lemari´ e-Rieusset [13]. They started with V _j ¹ as a regular multiresolution analysis of L ² (0, 1) reproducing polynomial at boundaries [1, 5, 11], with the scaling function ϕ ¹ and wavelet ψ ¹ generators on R that satisfy [14]:

(ϕ ¹ ) ⁰ = ϕ ⁰ − ϕ ⁰ (· − 1) and (ψ ¹ ) ⁰ = 4ψ ⁰ . (2) Jouini and Lemari´ e-Rieusset [13] used the orthogonal construction of [5] for the space V _j ¹ . They show that, from relation (2) and properly setting the integer parameters in the construction of V _j ¹ , one can deduce the space V _j ⁰ that satisfies (1). In this case, the wavelet space W _j ⁰ is defined by differentiating the wavelet basis of W _j ¹ :

W _j ⁰ = span{ψ _j,k ⁰ := 2 ^−j (ψ ¹ _j,k ) ⁰ }.

The corresponding biorthogonal spaces ( ˜ V _j ¹ , V ˜ _j ⁰ ) are respectively constructed again using integration by part. However, the construction of [13] remains theoretical, for instance it is not obvious to compute numerically the wavelet filters of ψ _j,k ⁰ and ˜ ψ ⁰ _j,k :

ψ ⁰ _j,k = X

n

H _k,n ⁰ ϕ ⁰ _j+1,n and ˜ ψ _j,k ⁰ = X

n

H ˜ _k,n ⁰ ϕ ˜ ⁰ _j+1,n . (3)

where V _j ⁰ = span{ϕ ⁰ _j,k ; k} and ˜ V _j ⁰ = span{ ϕ ˜ ⁰ _j,k ; k}. This point has been raised

5

by Kadri-Harouna and Perrier in [19], they extended the construction of [13] to

6

any regular scaling function generator ϕ ¹ and provided a numerical algorithm

7

for the associated Fast Wavelet Transform.

8

9

One advantage of the construction [13, 19] is that the associated multiscale

10

projectors commute with the derivative operator in H ¹ (0, 1). This fundamental

11

property enables to construct divergence-free wavelet bases as it was done in

12

[19]. Another interest of this property is to make possible the Ladysenskaya-

13

Babuska-Brezzi (LBB) condition for a wavelet based method in the numerical

14

discretization of a mixed problem such as the Stokes problem [3, 6]. The key

15

ingredient is that, commutation with derivation allows to get easily the condition

16

of Fortin’s lemma [10], see [6].

17

18

Ensuring the commutation of multiscale projectors with differentiation im-

19

poses to the biorthogonal space ˜ V _j ⁰ to satisfy homogeneous Dirichlet boundary

20

condition [13]. In this case, ˜ V _j ⁰ ⊂ H ₀ ¹ (0, 1) and constitutes a multiresolution

21

analysis of this space (and not of L ² (0, 1)). Nethertheless, relation (2) remains

22

valid but only for internal scaling functions and wavelets (i.e. basis functions

23

having their support included into [0, 1]). The edge functions did not strictly

24

satisfy this diagonal relation, but a linear combination of them: a change of

25

basis is therefore introduced [20].

26

27

Recently, Stevenson [22] has proposed another construction which differs from the existing constructions by the choice of the boundary conditions for the dual spaces ˜ V _j ⁰ and ˜ V _j ¹ . Precisely, let us suppose that ψ ¹ _j,k and ˜ ψ _j,k ⁰ are the wavelets constructed from scaling function generators satisfying (2). Then, integration by part shows that:

ψ _j,k ¹ (1) ˜ ψ ⁰ _j,k (1) − ψ ¹ _j,k (0) ˜ ψ _j,k ⁰ (0) = hψ ⁰ _j,k , ψ ˜ _j,k ⁰ i − hψ _j,k ¹ , ψ ˜ ¹ _j,k i. (4) If the two systems (ψ _j,k ¹ , ψ ˜ _j,k ¹ ) and (ψ _j,k ⁰ , ψ ˜ _j,k ⁰ ) are biorthogonal, the boundary

28

terms of (4) should vanish. Instead of taking ˜ ψ ⁰ _j,k ∈ H ₀ ¹ (0, 1) like in [13], the

29

construction of [22] sets ψ ¹ _j,k (1) = 0 and ˜ ψ _j,k ⁰ (0) = 0. This choice of bound-

30

ary condition is more flexible and leads to ˜ V _j ⁰ as a multiresolution analysis of

31

L ² (0, 1): however the commutation of multiscale projectors with differentiation

32

is lost. Alternatively, to get (4) one can take (V _j ⁰ , V ˜ _j ⁰ ) as a multiresolution of

33

L 2,0 = {u ∈ L ² (0, 1) : R 1

0 u(t)dt = 0} [21]. In this case, V _j ¹ = R x

0 V _j ⁰ ⊂ H ₀ ¹ (0, 1),

34

thus only the spaces ˜ V _j ¹ = _dx ^d V ˜ _j ⁰ can provide a multiresolution analysis of

35

L ² (0, 1), see Proposition 3.1 of [21].

36

37

The focal point of this work is the construction of a wavelet basis satisfy- ing homogeneous Dirichlet boundary conditions on the interval, associated to a biorthogonal multiresolution analyses of H ₀ ¹ (0, 1), and linked by a diagonal differentiation/integration relation to a standard wavelet bases of H ₀ ¹ (0, 1), as in[13, 19, 22]. As for the internal wavelets (2), emphasis is made on the con- struction of edge wavelets in order to get a diagonal differentiation relation:

ψ ¹ _j,k (x) = 2 ^j Z x

0

ψ ⁰ _j,k (t)dt and (ψ _j,k ¹ ) ⁰ (x) = 2 ^j ψ ⁰ _j,k (x). (5) Contrarily to our previous construction [19], which started with the wavelets ψ _j,k ¹ and ˜ ψ _j,k ¹ , in this work we begin with the knowledge of the wavelets ψ ⁰ _j,k and ˜ ψ _j,k ⁰ : for this step, we will use a standard orthogonal or biorthogonal wavelet basis on the interval [0, 1] allowing polynomial reproduction even at boundaries, see e.g; [1, 5]. Since R 1

0 ψ _j,k ⁰ (t)dt = 0, relation (5) leads to ψ _j,k ¹ ∈ H ₀ ¹ (0, 1) instead of ψ _j,k ¹ ∈ H ¹ (0, 1) as in [13, 19, 22, 23]. Denoting by W ¹ _j this new wavelet spaces, one obtains the multiscale decompositions:

V _j ¹ = V _j ¹

⊕ W ¹ _j

⊕ · · · ⊕ W ¹ _j−1 , (6) where incorporating homogeneous Dirichlet boundary condition in V _j ¹ is reduced to the treatment of this boundary condition only at the coarse scale j min :

V _j ¹ ∩ H ₀ ¹ (0, 1) = V _j ¹

min

∩ H ₀ ¹ (0, 1)

⊕ W ¹ _j

min

⊕ · · · ⊕ W ¹ _j−1 , (7) Notice that, due to the property of polynomial reproduction at boundaries, the

38

multiscale de composition (7) is stable in H ₀ ¹ (0, 1):

39

kuk ² _H

0

∼ X

k6=0,1

| < u, ϕ ˜ ¹ _j

_,k > | ² + X

j≥j

X

k

2 ^2j | < u, ψ ˜ ¹ _j,k > | ² , ∀u ∈ H ₀ ¹ (0, 1),

whereas (6) yields a non stable multiscale decomposition of L ² (0, 1).

40

41

Considering the particular case ψ ⁰ _j,k = ˜ ψ _j,k ⁰ , which corresponds to the or- thogonal setting, leads to:

< (ψ _j,k ¹ ) ⁰ , (ψ <sub>`,n</sub> <sup>1</sup> ) <sup>0</sup> >= 2 <sup>j+`</sup> < ψ ⁰ _j,k , ψ <sub>`,n</sub> <sup>0</sup> >= 2 <sup>j+`</sup> δ _j,` δ _k,n . (8) Then, from (8) we infer that the 1D Poisson equation with homogeneous Dirich-

42

let boundary can be solved with a linear numerical complexity in the multireso-

43

lution analysis provided by spaces V _j ¹ . Furthermore, this new construction still

44

maintains the properties of Fortin’s lemma [10] in the numerical discretization

45

of Stokes problem and allows to get a fast divergence-free wavelet transform

46

algorithm similar to this of the periodic case [9]. The main difficulty of such

47

a construction is the numerical implementation of the decomposition (6), this

48

point will be well documented in the present work.

49

50

In Section 2 we recall the construction of biorthogonal multiresolution anal-

51

ysis of L ² (0, 1) with polynomial reproduction and how to impose homogeneous

52

boundary conditions in such context to obtain a basis of H ₀ ¹ (0, 1). Section 3

53

reminds the principle of the construction of BMRA linked by differentiation

54

/ integration and its main properties needed for a numerical implementation.

55

The new construction of BMRA linked by differentiation / integration is de-

56

tailed in Section 4, while the associated fast wavelet transform algorithms are

57

provided in Section 5. Finally, Section 6 presents numerical examples showing

58

the potentiality of these new bases.

59

60

2. Biorthogonal multiresolution analyses of L ² (0, 1) reproducing poly-

61

nomial

62

The construction of biorthogonal multiresolution analyses (V _j , V ˜ _j ) of L ² (0, 1)

with polynomial reproduction (r, r) is classical [4, 7, 11]: the principle is to start ˜

with generators (ϕ, ϕ), that are biorthogonal scaling functions of a BMRA on ˜ R . We suppose that ϕ is compactly supported on [n min , n max ] and reproduces polynomials up to degree r − 1:

0 ≤ ` ≤ r − 1, x <sup>`</sup>

`! =

+∞

X

k=−∞

˜

p ` (k) ϕ(x − k), ∀ x ∈ R , (9) with ˜ p ` (k) = h ^x _`!

, ϕ(x ˜ − k)i. Similarly, ˜ ϕ reproduces polynomials up to degree

˜

r − 1 and we note p ` (k) = h <sup>x</sup> <sub>`!</sub>

, ϕ(x − k)i. For j sufficiently large, the spaces V j

on [0, 1] have the structure:

V j = V _j ^[ ⊕ V _j ^int ⊕ V _j ^] , (10) where V _j ^int = span{ϕ _j,k (x) = 2 ^j/2 ϕ(2 ^j x − k) ; k = k _[ , 2 ^j − k _] } is the space

63

generated by interior scaling functions whose supports are included into [ ^δ ₂

, 1 −

64

δ

2

] ⊂ [0, 1] (δ _[ , δ ] ∈ N be two fixed parameters), and k _[ = δ _[ − n min and k ] =

65

δ ] + n max . Moreover

66

V _j ^[ = span{Φ ^[ _{j,`</sub> (x) = 2 <sup>j/2</sup> Φ <sup>[</sup> <sub>`} (2 ^j x) ; ` = 0, · · · , r − 1},

V _j ^] = span{Φ ^] _{j,`</sub> (1 − x) = 2 <sup>j/2</sup> Φ <sup>]</sup> <sub>`} (2 ^j (1 − x)) ; ` = 0, · · · , r − 1}, are the edge spaces, the edge scaling functions at the edge 0 being defined in order to preserve the polynomial reproduction (9) on the interval [0, 1]:

0 ≤ ` ≤ r − 1, Φ <sup>[</sup> <sub>`</sub> (x) =

k

−1

X

k=1−n

˜

p _{`</sub> (k) ϕ(x − k) χ <sub>[0,+∞[</sub> . (11) At the edge 1, the edge scaling functions Φ <sup>]</sup> <sub>j,`} are constructed on ] − ∞, 1]

67

by symmetry, using the transform T f(x) = f (1 − x). In practice we have

68

to choose j ≥ j _min where j _min is the smallest integer which verifies j _min >

69

log ₂ [n _max −n _min + δ _] +δ _[ ] to ensure that the supports of edge scaling functions

70

at 0 do not intersect the supports of edge scaling functions at 1.

71

72

The polynomial reproduction in V j is then satisfied since, for 0 ≤ ` ≤ r − 1 and x ∈ [0, 1] we have:

2 ^j/2 (2 ^j x) ^`

`! = 2 <sup>j/2</sup> Φ <sup>[</sup> <sub>`</sub> (2 ^j x) +

2

−k

X

k=k

˜

p ` (k) ϕ j,k (x) + 2 <sup>j/2</sup> Φ <sup>]</sup> <sub>`</sub> (2 ^j (1 − x)). (12)

Similarly, the biorthogonal spaces ˜ V j are defined with the same structure, al- lowing the polynomial reproduction up to degree ˜ r − 1:

V ˜ _j = span{ Φ ˜ ^[ <sub>j,`</sub> } `=0,˜ r−1 ⊕ V ˜ _j ^int ⊕ span{ Φ ˜ ^] <sub>j,`</sub> } `=0,˜ r−1 , (13) where ˜ V _j ^int = span{ ϕ ˜ _j,k ; k = ˜ k _[ , 2 ^j − k ˜ _] } is the space generated by interior scaling functions ˜ ϕ _j,k (x) = 2 ^j/2 ϕ ˜ ¹ (2 ^j x − k), whose supports are included into [ ^δ ₂ ^˜

, 1 − ₂ ^{˜ δ}

] (˜ δ _[ , δ ˜ ] ∈ N be two parameters). The edge scaling functions at 0 are defined by:

0 ≤ ` ≤ r ˜ − 1, Φ ˜ <sup>[</sup> <sub>`</sub> (x) =

˜ k

−1

X

k=1−˜ n

p ` (k) ˜ ϕ(x − k) χ _[0,+∞[ .

The equality between dimensions of V j and ˜ V j is obtained by adjusting the

73

parameters ˜ δ _[ = ˜ k _[ − n ˜ max and ˜ δ ] = ˜ k ] + ˜ n min (with [˜ n min , n ˜ max ] = supp ϕ) ˜

74

such that: ∆ j = dim(V j ) = dim( ˜ V j ) = 2 ^j − (δ _[ + δ ] ) − (n max − n min ) + 2r + 1.

75

Remark that (δ _[ , δ ] ) remain ”free” parameters of the construction (often chosen

76

equal to 0 or 1). The last step of the construction lies in the biorthogonalization

77

process of the basis functions, since edge scaling functions of V j and ˜ V j are no

78

more biorthogonal [1, 7, 11, 16]. Finally, the spaces (V _j , V ˜ _j ) form a biorthogonal

79

MRA of L ² (0, 1).

80

Boundary conditions. A multiresolution analyses of

H ₀ ^m (0, 1) = {f ∈ H ^m (0, 1) : f ^(p) (0) = f ^(p) (1) = 0, 0 ≤ p ≤ m − 1}

can be defined from V j by taking V _j ^m,0 = V j ∩ H ₀ ^m (0, 1). For instance, if m = 1, as described in [15, 16], it suffices to remove the edge scaling functions Φ ^[ ₀ at edge 0 and Φ ^] ₀ at edge 1 which leads to:

V _j ^1,0 = span{Φ ^[ <sub>j,`</sub> ; ` = 1, r − 1} ⊕ V _j ^int ⊕ span{Φ ^] <sub>j,`</sub> ; ` = 1, r − 1}.

In such case, we also remove the edge functions ˜ Φ ^[ ₀ and ˜ Φ ^] ₀ from ˜ V j prior to

81

biorthogonalization, to adjust the dimension of the biorthogonal space. Then,

82

the spaces (V _j ^1,0 , V ˜ _j ^1,0 ) constitute a biorthogonal multiresolution analyses of

83

H ₀ ¹ (0, 1).

84

3. Existing construction of (V _j ⁰ , V ˜ _j ⁰ ) linked by differentiation / inte-

85

gration with (V _j ¹ , V ˜ _j ¹ )

86

In this section, we recall briefly the earlier construction of the multiresolution

87

analysis linked differentiation / integration. Then, we will mention their limita-

88

tions in some applications that we intend to take up with our new construction

89

below.

90

All the constructions of biorthogonal multiresolution analyses of L ² (0, 1)

91

linked by differentiation / integration are based on the following proposition

92

[14]:

93

Proposition 1. Let (V _j ¹ ( R ), V ˜ _j ¹ ( R )) be a biorthogonal MRA of L ² ( R ), with

94

associated scaling functions ϕ ¹ , ϕ ˜ ¹ and wavelets ψ ¹ , ψ ˜ ¹ . Assume that V _j ¹ ( R )

95

is regular, ϕ ¹ ∈ C ¹⁺ , > 0, and compactly supported. Then there exists a

96

biorthogonal MRA (V _j ⁰ ( R ), V ˜ _j ⁰ ( R )), with associated scaling functions ϕ ⁰ , ϕ ˜ ⁰ and

97

wavelets ψ ⁰ , ψ ˜ ⁰ , such that:

98

(ϕ ¹ ) ⁰ (x) = ϕ ⁰ (x) − ϕ ⁰ (x − 1) and (ψ ¹ ) ⁰ = 4 ψ ⁰ .

99

The biorthogonal functions verify: R x+1

x ϕ ˜ ¹ (t) dt = ˜ ϕ ⁰ (x) and ( ˜ ψ ⁰ ) ⁰ = −4 ˜ ψ ¹ .

100

Proposition 1 provides biorthogonal multiresolution analysis of L ² ( R ) linked

101

by differentiation/ integration [14]. For the space L ² (0, 1), again based on

102

Proposition 1, the first construction was done by Jouini and Lemari´ e-Rieusset

103

[13]: they prove the existence of two biorthogonal multiresolution analyses of

104

L ² (0, 1), denoted (V _j ¹ ) and (V _j ⁰ ) linked by differentiation such that:

105

d

dx V _j ¹ = V _j ⁰ . (14)

The construction of [13] allows to study divergence-free vector functions on

106

hypercube [0, 1] ^d , thus to get commutation of multiscale projector with the

107

derivation operator, the biorthogonal spaces ˜ V _j ⁰ should satisfy:

108

V ˜ _j ⁰ = H ₀ ¹ (0, 1) ∩ Z x

0

V ˜ _j ¹ = n

f : f ⁰ ∈ V ˜ _j ¹ and f (0) = f (1) = 0 o . (15) Then, as mentioned before, ˜ V _j ⁰ can not be a multiresolution analysis of L ² ( R )

109

since ˜ V _j ⁰ ⊂ H ₀ ¹ (0, 1). Furthermore, if (P _j ¹ , P ˜ _j ¹ ) are the biorthogonal projectors

110

of (V _j ¹ , V ˜ _j ¹ ) and (P _j ⁰ , P ˜ _j ⁰ ) those of (V _j ⁰ , V ˜ _j ⁰ ) respectively, we have [13]:

111

Proposition 2.

112

(i) ∀ f ∈ H ¹ (0, 1), _dx ^d ◦ P _j ¹ f = P _j ⁰ ◦ _dx ^d f ,

113

(ii) ∀ f ∈ H ₀ ¹ (0, 1), _dx ^d ◦ P ˜ _j ⁰ f = ˜ P _j ¹ ◦ _dx ^d f .

114

115

Despite of satisfying Proposition 2, the construction of Jouini and Lemari´ e-

116

Rieusset [13] remains in a theoretical setting and inspired by the use of Daubechies

117

compactly supported orthogonal generators [8]. A construction that uses classi-

118

cal biorthogonal multiresolution analyses on the interval [4, 7, 11], with polyno-

119

mial reproduction at boundaries, was done and implemented by Kadri-Harouna

120

and Perrier [19]. In such a construction, the choice of integer parameters (δ [ , δ ] )

121

and (˜ δ _[ , δ ˜ _] ) is very important: they must be identical for the two multiresolu-

122

tion analyses to satisfy (14,15) and to provide the commutation of multiscale

123

projectors with the differentiation operator.

124

3.1. Wavelet spaces

125

For j ≥ j _min , the biorthogonal wavelet spaces associated to V _j ¹ are defined by W _j ¹ = V _j+1 ¹ ∩ ( ˜ V _j ¹ ) ^⊥ . As for the scaling function spaces, these spaces have the following structure:

W _j ¹ = W _j ^1,[ ⊕ W _j ^1,int ⊕ W _j ^1,] ,

where W _j ^1,[ is spanned by the edge wavelets at 0, W _j ^1,int is spanned by the

126

interior wavelets and W _j ^1,] is spanned by the edge wavelets at 1, see [1, 4, 7,

127

11, 16] and references therein. The biorthogonal spaces ˜ W _j ¹ = ˜ V _j+1 ¹ ∩ (V _j ¹ ) ^⊥ are

128

constructed in the same way, finally the wavelet bases of the two spaces must to

129

be biorthogonalized identically as the scaling functions. The resulting wavelet

130

bases are denoted by {ψ _j,k ¹ } k=1,2

and { ψ ˜ _j,k ¹ } k=1,2

without distinction.

131

132

The biorthogonal wavelets of W _j ⁰ and ˜ W _j ⁰ , linked to ψ ¹ _j,k and ˜ ψ _j,k ¹ by dif-

133

ferentiation/integration are defined by the following proposition, established in

134

the general framework by [13]:

135

Proposition 3. Let (V _j ¹ , V ˜ _j ¹ ) and (V _j ⁰ , V ˜ _j ⁰ ) BMRAs satisfying (14,15). The wavelet spaces W _j ⁰ = V _j+1 ⁰ ∩ ( ˜ V _j ⁰ ) ^⊥ and W ˜ _j ⁰ = ˜ V _j+1 ⁰ ∩ (V _j ⁰ ) ^⊥ are linked to the biorthogonal wavelet spaces associated to (V _j ¹ , V ˜ _j ¹ ) by:

W _j ⁰ = d

dx W _j ¹ and W ˜ _j ⁰ = Z x

0

W ˜ _j ¹ . (16) Moreover, let {ψ _j,k ¹ } _k=1,2

and { ψ ˜ ¹ _j,k } _k=1,2

be two biorthogonal wavelet bases of W _j ¹ and W ˜ _j ¹ . Biorthogonal wavelet bases of W _j ⁰ and W ˜ _j ⁰ are directly defined by:

ψ _j,k ⁰ = 2 ^−j (ψ ¹ _j,k ) ⁰ and ψ ˜ _j,k ⁰ = −2 ^j Z x

0

ψ ˜ _j,k ¹ . (17) This new edge wavelets preserve fast algorithms since they satisfy two-scale

136

equations [19]:

137

Proposition 4. Let {ψ _j,k ¹ } _k=1,2

and { ψ ˜ _j,k ¹ } _k=1,2

be two biorthogonal wavelet bases of W _j ¹ and W ˜ _j ¹ associated respectively to filters G ¹ _j and G ˜ ¹ _j :

ψ _j,k ¹ = X

n

(G ¹ _j ) k,n ϕ ¹ _j+1,n and ψ ˜ _j,k ¹ = X

n

( ˜ G ¹ _j ) k,n ϕ ˜ ¹ _j+1,n .

Then there exist sparse matrices G ⁰ _j and G ˜ ⁰ _j defined by:

G ⁰ _j = 2 ^−j G ¹ _j L ¹ _j+1 and G ˜ ⁰ _j = −2 ^j G ˜ ¹ _j L ^0T _j+1 , (18) such that the wavelets ψ _j,k ⁰ and ψ ˜ _j,k ⁰ satisfy:

ψ ⁰ _j,k = X

n

(G ⁰ _j ) _k,n ϕ ⁰ _j+1,n and ψ ˜ _j,k ⁰ = X

n

( ˜ G ⁰ _j ) _k,n ϕ ˜ ⁰ _j+1,n .

The matrices L ¹ _j and L ⁰ _j correspond to the change of basis between ( _dx ^d V _j ¹ , V _j ⁰ ) and ( R x

0 V _j ⁰ , V _j ¹ ), respectively:

d dx ϕ ¹ _j,k =

∆

−1

X

n=1

(L ¹ _j ) k,n ϕ ⁰ _j,n and − Z x

0

ϕ ⁰ _j,k =

∆

X

m=1

(L ⁰ _j ) k,m ϕ ¹ _j,m . (19)

138

Interior wavelets ψ _j,k ⁰ (x) = 2 ^j/2 ψ ⁰ (2 ^j x − k) in Proposition 3 correspond to

139

classical wavelets, ψ ⁰ being a wavelet on R associated to the scaling function ϕ ⁰

140

as in Proposition 1.

141

142

In the previous works [19, 22], to construct divergence-free wavelet satisfy-

143

ing free slip boundary condition, one needs to differentiate wavelets of V _j+1 ¹ ∩

144

H ₀ ¹ (0, 1) that satisfy homogeneous Dirichlet boundary condition, which deriva-

145

tives differ from the wavelets defined in (17). In this case, the numerical com-

146

putation of the Helmholtz-Hodge decomposition or the numerical simulation of

147

the incompressible Navier-Stokes equations should required the use of four dif-

148

ferent kind of edge wavelet filters. Precisely, in two space dimension using the

149

multiresolution analysis (V _j ¹ ⊗V _j ⁰ )×(V _j ⁰ ⊗V _j ¹ ), one should use the wavelet filters

150

of (ψ ¹ _j,k ; ψ _j,k ⁰ = 2 ^−j (ψ _j,k ¹ ) ⁰ ) for the usual decomposition and the wavelet filters

151

of (ψ ^1,0 _j,k ∈ V _j+1 ¹ ∩ H ₀ ¹ (0, 1); 2 ^−j (ψ _j,k ^1,0 ) ⁰ 6= ψ _j,k ⁰ ), due to the free slip boundary

152

condition, see [18, 20] for details.

153

154

The new construction detailed in the next section will lead to edge wavelets

155

that satisfy relation (17) even if with homogeneous Dirichlet boundary condi-

156

tion.

157

4. New construction of (V _j ¹ , V ˜ _j ¹ ) linked to (V _j ⁰ , V ˜ _j ⁰ ) by differentiation

158

/ integration to handle boundary conditions in V _j ¹ .

159

In this section we present our new construction of biorthogonal multiresolu-

160

tion analyses linked by differentiation and integration. The construction of the

161

primal spaces (V _j ¹ , V _j ⁰ ) remains the same as in the classical construction [13, 19].

162

However, the construction of the biorthogonal spaces ( ˜ V _j ¹ , V ˜ _j ⁰ ) will be different.

163

Indeed, to handle Dirichlet boundary conditions in V _j ¹ , we will construct new

164

wavelet bases (ψ ¹ _j,k ) which will constitue a Riesz basis for the homogeneous space

165

H ₀ ¹ (0, 1). This is an issue of major benefit in the construction of divergence-free

166

wavelet satisfying physical boundary condition [19].

167

168

The construction starts with (V _j ⁰ , V ˜ _j ⁰ ) as a standard biorthogonal multireso-

169

lution analyses of L ² (0, 1) [1, 4, 5] (which can be orthogonal), where the scaling

170

function generators (ϕ ⁰ , ϕ ˜ ⁰ ) satisfy Proposition 1, with at least two vanishing

171

moments for the wavelet ψ ⁰ : ˜ r ⁰ ≥ 2. We denote by (δ _[ , δ _] ) and (˜ δ _[ , ˜ δ _] ) the

172

integer parameters used in the construction of (V _j ⁰ , V ˜ _j ⁰ ). Following [13, 19],

173

the classical multiresolution spaces V _j ¹ is constructed from the scaling function

174

generator ϕ ¹ with the same integer parameters (δ _[ , δ ] ) and satisfy:

175

d

dx V _j ¹ = V _j ⁰ and ∆ ¹ _j − 1 = ∆ ⁰ _j .

In this case, for the biorthogonal space ˜ V _j ⁰ , since ∆ ⁰ _j = dim( ˜ V _j ⁰ ), we see that:

176

∆ ¹ _j − 2 = dim( d dx

V ˜ _j ⁰ ).

The construction of ˜ V _j ¹ follows similar approach with the generator ˜ ϕ ¹ . To get

177

equality between dimensions of spaces V _j ¹ and ˜ V _j ¹ one needs:

178

∆ ¹ _j = ˜ ∆ ¹ _j ,

which imposes to the integer parameters to be used for the construction of ˜ V _j ¹

179

to be fixed to (˜ δ _[ − 1, δ ˜ ] − 1). It follows therefore that:

180

d dx

V ˜ _j ⁰ 6⊂ V ˜ _j ¹ .

This is a major difference compared to the existing construction.

181

4.1. A new wavelet space for V _j ¹

182

The construction of the wavelet basis associated to V _j ¹ is the major contri- bution of the present work. In the classical construction, on defines the wavelet space as:

V _j+1 ¹ = V _j ¹ ⊕ W _j ¹ , where W _j ¹ = V _j+1 ¹ ∩ ( ˜ V _j ¹ ) ^⊥ .

Then, the space W _j ¹ does not necessarily satisfy homogeneous Dirichlet bound-

ary condition. To compensate for that, in this work the wavelet space is defined

as:

W ¹ _j = Z x

0

W _j ⁰ ,

where W _j ⁰ is the wavelet space associated to V _j ⁰ [1, 4, 7, 11, 16]:

W _j ⁰ = V _j+1 ⁰ ∩ ( ˜ V _j ⁰ ) ^⊥ . Remark 1.

From the zero mean value property of the wavelet ψ _j,k ⁰ , by construction the space W ¹ _j satisfies:

W ¹ _j ⊂ H ₀ ¹ (0, 1).

In the previous section, the wavelet space W _j ⁰ was defined as W _j ⁰ = _dx ^d W _j ¹ and this choice of W _j ⁰ led in general to:

W _j ¹ 6=

Z x 0

W _j ⁰ .

For all j ≥ j _min , the spaces W ¹ _j verify the following proposition.

183

Proposition 5.

Let W _j ⁰ be the wavelet space associated to V _j ⁰ , where V _j ⁰ = _dx ^d V _j ¹ . Then the space V _j+1 ¹ can be decomposed as follows:

V _j+1 ¹ = V _j ¹ ⊕ W ¹ _j , with W ¹ _j = Z x

0

W _j ⁰ , (20)

and

V _j+1 ¹ = V _j ¹

⊕ W ¹ _j

⊕ · · · ⊕ W ¹ _j . (21) Proof. As _dx ^d V _j ¹ = V _j ⁰ , we get:

W ¹ _j = Z x

0

W _j ⁰ ⊂ Z x

0

d

dt V _j+1 ¹ ⊂ V _j+1 ¹ and Z x

0

W _j ⁰ ⊂ H ₀ ¹ (0, 1).

Moreover, let u _j be a function of V _j ¹ ∩ W ¹ _j : u j = X

k

c k ϕ ¹ _j,k = X

n

d n

Z x 0

ψ ⁰ _j,n ,

we deduce that:

d

dx u j ∈ V _j ⁰ ∩ W _j ⁰ ⇒ d

dx u j = 0 ⇒ u j = C ∈ R .

Since

h1, Z x

0

ψ _j,n ⁰ i = −hx, ψ _j,n ⁰ i = 0,

we get u j = 0, which implies V _j ¹ ∩ W ⁰ _j = {0}. Let f j+1 ∈ V _j+1 ¹ , then:

d

dx f j+1 ∈ V _j+1 ⁰ = V _j ⁰ ⊕ W _j ⁰ .

Since f _j+1 (0) ∈ V _j ¹ (the constants are in V _j ¹ ), integration gives:

f _j+1 (x) = f _j+1 (0) + Z x

0

P _j ⁰ ( d

dx f _j+1 ) + Z x

0

Q ⁰ _j ( d

dx f _j+1 ) ∈ V _j ¹ + W ¹ _j , and this ends the proof.

184

4.2. A new multiscale decomposition of H ₀ ¹ (0, 1), and relation with the derivative

185

operator

186

We recall that from the results of Section 2, incorporating homogeneous boundary conditions in V _j ¹ consists on removing the two scaling functions that do not satisfy the desired boundary conditions. In addition to that, one interest of this new wavelet space construction is that the treatment of homogeneous Dirichlet boundary conditions in V _j ¹ is done only at the coarse scale j min . In- deed, by construction the space W ¹ _j = R x

0 W _j ⁰ ⊂ H ₀ ¹ (0, 1) and relation (21) allow to get the following decomposition:

V _j+1 ^1,0 = V _j+1 ¹ ∩ H ₀ ¹ (0, 1) = V _j ¹

min

∩ H ₀ ¹ (0, 1)

⊕ W ¹ _j

min

⊕ · · · ⊕ W ¹ _j . (22) Moreover, as a matter of fact, the wavelet space W ¹ _j is the classical wavelet

187

space of H ₀ ¹ (0, 1) associated to the multiresolution analysis constituted by V _j ^1,0

188

(but the wavelet basis is different), as proved in the following proposition.

189

Proposition 6.

Let (V _j ¹ , V ˜ _j ¹ ) and (V _j ⁰ , V ˜ _j ⁰ ) be two BMRAs of L ² (0, 1) linked by differentiation and integration constructed using the parameters (δ [ , δ ] ) and (˜ δ [ , δ ˜ ] ). Defining the biorthogonal spaces (V _j ^1,0 , V ˜ _j ^1,0 ) by:

V _j ^1,0 = V _j ¹ ∩ H ₀ ¹ (0, 1) and d dx

V ˜ _j ⁰ = ˜ V _j ^1,0 , (23) then we have:

190

• (i) The spaces V _j ^1,0 provide a multiresolution analysis of H ₀ ¹ (0, 1).

191

• (ii) The space W ¹ _j = R x

0 W _j ⁰ is the classical wavelet space associated to V _j ^1,0 :

V _j+1 ^1,0 = V _j+1 ¹ ∩ H ₀ ¹ (0, 1) = V _j ^1,0 ⊕ Z x

0

W _j ⁰ and Z x

0

W _j ⁰ = V _j+1 ^1,0 ∩ ( ˜ V _j ^1,0 ) ^⊥ . (24) Proof.

The first point (i) is evident. Since V _j ¹ is a multiresolution analysis of L ² (0, 1) and V _j ^1,0

min

= V _j ¹

∩ H ₀ ¹ (0, 1), we have:

∪ j≥j

V _j ^1,0 = H ₀ ¹ (0, 1). (25) For the second point (ii), from the vanishing moment condition of the wavelet basis of W _j ⁰ we get:

Z 1 0

W _j ⁰ = 0 ⇒ Z x

0

W _j ⁰ ⊂ H ₀ ¹ (0, 1).

The differentiation relation gives:

d

dx V _j+1 ¹ = V _j+1 ⁰ ⇒ Z x

0

V _j+1 ⁰ ⊂ V _j+1 ¹ ,

thus

Z x 0

W _j ⁰ ⊂ V _j+1 ¹ ∩ H ₀ ¹ (0, 1) = V _j+1 ^1,0 . Moreover, the differentiation relation:

d dx

V ˜ _j ⁰ = ˜ V _j ^1,0

states that for any ˜ f _j ^1,0 ∈ V ˜ _j ^1,0 , there exists ˜ f _j ⁰ ∈ V ˜ _j ⁰ such that ( ˜ f _j ⁰ ) ⁰ = ˜ f _j ^1,0 , then:

h Z x

0

ψ _j,k ⁰ , f ˜ _j ^1,0 i = h Z x

0

ψ _j,k ⁰ , d dx

f ˜ _j ⁰ i = −hψ ⁰ _j,k , f ˜ _j ⁰ i = 0 ⇒ Z x

0

W _j ⁰ ⊂ ( ˜ V _j ^1,0 ) ^⊥ . Then we deduce that

Z x 0

W _j ⁰ ⊂ V _j+1 ^1,0 ∩ ( ˜ V _j ^1,0 ) ^⊥

and since the two spaces have the same dimension, we get:

Z x 0

W _j ⁰ = V _j+1 ^1,0 ∩ ( ˜ V _j ^1,0 ) ^⊥ .

192

We remind that the integer parameters used in the construction of ˜ V _j ¹ and

193

V ˜ _j ⁰ are not the same. Then one can not expect to get commutation between

194

multiscale projectors and derivation as in Proposition 2, but for the oblique

195

multiscale projectors of (V _j ^1,0 , V _j ⁰ ) and ( ˜ V _j ⁰ , V ˜ _j ^1,0 ) we can prove the following

196

proposition:

197

Proposition 7.

198

Let (P _j ^1,0 , P _j ⁰ ) be the biorthogonal projectors associated with (V _j ^1,0 , V _j ⁰ ) and ( P e _j ⁰ , P e _j ^1,0 )

199

the biorthogonal projectors associated with ( ˜ V _j ⁰ , V ˜ _j ^1,0 ). Then, we have:

200

(i) ∀ f ∈ H ₀ ¹ (0, 1), _dx ^d ◦ P _j ^1,0 f = P _j ⁰ ◦ _dx ^d f.

201

(ii) ∀ f ∈ H ¹ (0, 1), _dx ^d ◦ P ˜ _j ⁰ f = ˜ P _j ^1,0 ◦ _dx ^d f.

202

Proof.

From proposition 6, there exist two matrices denoted L ^1,0 _j and L e ^1,0 _j of size (∆ ¹ _j − 2) × (∆ ¹ _j − 1), such that:

ϕ ^1,0 _j,k =

∆

−1

X

n=1

(L ^1,0 _j ) k,n

Z x 0

ϕ ⁰ _j,n and

Z x 0

˜ ϕ ^1,0 _j,k =

∆

−1

X

n=1

(e L ^1,0 _j ) k,n ϕ ˜ ⁰ _j,n . (26) Then, the biorthogonality of the basis functions ϕ ^1,0 _j,k and ˜ ϕ ^1,0 _j,k , with an integra-

203

tion by part give:

204

δ k,m = hϕ ^1,0 _j,k , ϕ ˜ ^1,0 _j,m i = X

`

(e L ^1,0 _j ) m,` hϕ ^1,0 _j,k , d

dx ϕ ˜ ⁰ _j,` i = X

`

(e L ^1,0 _j ) m,` h− d

dx ϕ ^1,0 _j,k , ϕ ˜ ⁰ _j,` i

= − X

n

X

`

(L ^1,0 _j ) k,n ( L e ^1,0 _j ) m,` hϕ <sup>0</sup> <sub>j,n</sub> , ϕ ˜ <sup>0</sup> <sub>j,`</sub> i = − X

n

(L ^1,0 _j ) k,n (e L ^1,0 _j ) m,n , which means that:

I _∆

₋₂ = −L ^1,0 _j ^t L e ^1,0 _j .

Thus, the proof of the point (i) becomes a change of basis. Indeed, for f ∈

205

H ₀ ¹ (0, 1), we have:

206

d

dx P _j ^1,0 (f ) = X

k

hf, ϕ ˜ ^1,0 _j,k i d

dx ϕ ^1,0 _j,k = X

k

X

n

(L ^1,0 _j ) k,n hf, ϕ ˜ ^1,0 _j,k i ϕ ⁰ _j,n

= X

n

hf, X

k

(L ^1,0 _j ) k,n ϕ ˜ ^1,0 _j,k i ϕ ⁰ _j,n = X

n

hf, X

k

X

m

(L ^1,0 _j ) k,n (e L ^1,0 _j ) k,m

d

dx ϕ ˜ ⁰ _j,m i ϕ ⁰ _j,n

= X

n

hf, − d

dx ϕ ˜ ⁰ _j,n i ϕ ⁰ _j,n = X

n

h d

dx f, ϕ ˜ ⁰ _j,n i ϕ ⁰ _j,n = P _j ⁰ ( d dx f ).

For the second point (ii), let us consider the matrix L e ⁰ _j defined by:

d dx ϕ ˜ ⁰ _j,k =

∆

−1

X

n=1

( L e ⁰ _j ) k,n ϕ ˜ ^1,0 _j,n . Again, the duality of the basis and integration by part give:

h d

dx ϕ ^1,0 _j,k , ϕ ˜ ⁰ _j,m i = (L ^1,0 _j ) _k,m = hϕ ^1,0 _j,k , − d

dx ϕ ˜ ⁰ _j,m i = −( ˜ L ⁰ _j ) _m,k , then

207

d dx

P ˜ _j ⁰ (f ) = X

k

hf, ϕ ⁰ _j,k i d

dx ϕ ˜ ⁰ _j,k = X

k

X

n

( ˜ L ⁰ _j ) k,n hf, ϕ ⁰ _j,k i ϕ ˜ ^1,0 _j,n

= X

n

hf, X

k

( ˜ L ⁰ _j ) k,n ϕ ⁰ _j,k i ϕ ˜ ^1,0 _j,n = X

n

hf, − X

k

( ˜ L ^1,0 _j ) n,k ϕ ⁰ _j,k i ϕ ˜ ^1,0 _j,n

= X

n

hf, − d

dx ϕ ^1,0 _j,n i ϕ ˜ ^1,0 _j,n = X

n

h d

dx f, ϕ ^1,0 _j,n i ϕ ˜ ^1,0 _j,n = ˜ P _j ^1,0 ( d dx f ).

208

4.3. Fast decomposition algorithm

209

In this section we provide the decomposition of a given function f ∈ H ₀ ¹ (0, 1) in the MRA (V _j ^1,0 ) using (25, 22). As usual in Fast Wavelet Transforms, the complete decomposition uses a binary tree whose elementary step is given by the decomposition:

V _j+1 ^1,0 = V _j ^1,0 ⊕ W ¹ _j (27) Remembering that the space V _j ^1,0 is obtained from V _j ¹ only removing one bound- ary scaling function at each boundary 0 and 1:

V _j ^1,0 = span{ϕ ^1,0 _j ; j = 2, · · · , ∆ ¹ _j − 1}

and that (22) holds, we first study the elementary step:

V _j+1 ¹ = V _j ¹ ⊕ W ¹ _j .

Therefore we will study the computation of the projection of f _j+1 ∈ V _j+1 ¹ onto V _j ¹ and W ¹ _j respectively. Precisely, starting with:

f _j+1 =

∆

X

k=1

c _j+1,k ϕ ¹ _j+1,k ,

we want to compute the coefficients c j,k and d j,k from c j+1,k such that:

f j+1 =

∆

X

k=1

c j,k ϕ ¹ _j,k +

2

X

m=1

d j,m

Z x 0

ψ _j,m ⁰ .

Firstly, we notice that f _j+1 can be split as

f j+1 =

∆

X

k=1

c j+1,k ϕ ¹ _j+1,k = f _j+1 ⁰ + f _j+1 ^1,0 + f _j+1 ¹ , with

210

f _j+1 ⁰ = c j+1,1 ϕ ¹ _j+1,1 ⇒ f _j+1 ⁰ (0) 6= 0 and f _j+1 ⁰ (1) = 0,

f _j+1 ¹ = c _j+1,∆

j+1

ϕ ¹ _j+1,∆

j+1

⇒ f _j+1 ¹ (0) = 0 and f _j+1 ¹ (1) 6= 0, and

f _j+1 ^1,0 =

∆

−1

X

k=2

c _j+1,k ϕ ¹ _j+1,k ∈ V _j+1 ^1,0 ⊂ H ₀ ¹ (0, 1).

Thus, the two scale decomposition of f _j+1 ^1,0 is a classical decomposition in the multiresolution analysis of H ₀ ¹ (0, 1) provided by the scaling function filter of V _j ^1,0 and wavelet filter of W ¹ _j . To compute the projection of f _j+1 ⁰ , one way to proceed is to use the two scale relations satisfied by ϕ ¹ _j+1,1 :

ϕ ¹ _j+1,1 =

∆ ˜

X

n=1

H ˜ _n,1 ¹ ϕ ¹ _j,n +

2

X

m=1

G ˜ ¹ _m,1 ψ ¹ _j,m =

∆ ˜

X

n=1

H ˜

1

n,1 ϕ ¹ _j,n +

2

X

m=1

G ˜

1 m,1

Z x 0

ψ _j,m ⁰ ,

where the first decomposition corresponds to V _j+1 ¹ = V _j ¹ ⊕ W _j ¹ , with W _j ¹ a

211

chosen classical wavelet space associated to V _j ¹ [4, 7, 11, 16] and the second one

212

corresponds to V _j+1 ¹ = V _j ¹ ⊕ W ¹ _j .

213

214

Then, to get the new filters H ˜

1

n,1 and G ˜

1

n,1 , according to the biorthogonal- ization procedure that we adopted, where only the dual basis are modified, we have:

G ˜ ¹ _k,1 =

2

X

m=1

G ˜

1 m,1 h

Z x 0

ψ _j,m ⁰ , ψ ˜ ¹ _j,k i, (28) and

H ˜ _k,1 ¹ = H ˜

1 k,1 +

2

X

m=1

G ˜

1 m,1 h

Z x 0

ψ _j,m ⁰ , ϕ ˜ ¹ _j,k i. (29) Equations (28) and (29) define two linear systems with respect to the edge

215

scaling function and wavelet filters G ˜

1

n,1 and H ˜

1

n,1 . From [2, 20], the computation

216

of coefficients h R x

0 ψ ⁰ _j,m , ψ ˜ ¹ _j,k i and h R x

0 ψ _j,m ⁰ , ϕ ˜ ¹ _j,k i is straightforward and this is

217

done only for functions whose support intersect the edge function support, due

218

to the compactly support properties of the generators. Then, solving these linear

219

systems, and similar relations at the boundary 1, allows to get the new edge

220

filters. Finally, the main steps of the decomposition algorithm are summarized

221

as:

222

c j,1 = c j+1,1 H ˜

1

1,1 , (30)

c _j,k =

∆

−1

X

n=2

c _j+1,n H ˜ _n,k ^1,0 + c _j+1,1 H ˜

1

k,1 + c _j+1,∆

H ˜

1

k,∆

, 2 ≤ k ≤ ∆ _j − (31) 1, c j,∆

= c j+1,∆

H ˜

1

∆

,∆

, (32)

and

223

d _j,k =

∆

−1

X

n=2

c _j+1,n G ˜ ^1,0 _n,k + c _j+1,1 G ˜

1

k,1 + c _j+1,∆

G ˜

1

k,∆

, 1 ≤ k ≤ 2 ^j (33) Remark 2. Working in V _j ^1,0 ⊂ H ₀ ¹ (0, 1) amounts to work directly with f _j+1 ^1,0 . In

224

this case the elementary decomposition step (27) is reduced to the computations

225

of coefficients (31, 33).

226

Fast reconstruction algorithm. For f j+1 ∈ V _j+1 ¹ , let us suppose that we know its projection onto V _j ¹ ⊕ W ¹ _j in terms of:

f j+1 =

∆

X

k=1

c j,k ϕ ¹ _j,k +

2

X

m=1

d j,m

Z x 0

ψ _j,m ⁰ ,

and we want to compute its projection onto V _j+1 ¹ in terms of:

f j+1 =

∆

X

k=1

c j+1,k ϕ ¹ _j+1,k .

Setting

∆

X

k=1

c j,k ϕ ¹ _j,k = f _j ⁰ + f _j ^1,0 + f _j ¹ , with f _j ⁰ = c j,1 ϕ ¹ _j,1 and f _j ¹ = c _j,∆

ϕ ¹ _j,∆

j

, it is easy to see that

f _j ^1,0 +

2

X

m=1

d j,m

Z x 0

ψ _j,m ⁰ ∈ V _j+1 ^1,0 ,

thus we get:

c j+1,k =

∆

X

n=1

c j,n H _n,k ¹ +

2

X

m=1

d j,m G ¹ _m,k , 2 ≤ k ≤ ∆ ¹ _j+1 − 1, and

c j+1,1 =

∆

X

n=1

c j,n H _n,1 ¹ , c _j+1,∆

=

∆

X

n=1

c j,n H _n,∆ ¹

.

Remark 3. Again working in V _j ^1,0 ⊂ H ₀ ¹ (0, 1) amounts to work directly with

227

f _j ^1,0 and f _j+1 ^1,0 , assuming that f _j ⁰ = 0 and f _j ¹ = 0. Using (30,32), we obtain

228

c j+1,1 = 0 and c _j+1,∆

= 0.

229

5. Numerical examples

230

We present in this section numerical examples to illustrate the effective-

231

ness and the potential application of this new construction of multiresolution

232

analyses linked by differentiation and integration. We first show the shape of

233

0 0.5 1 1.5 2 2.5 3 -0.02

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06

0 0.5 1 1.5 2 2.5 3

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06

Figure 1: Plot of the internal scaling function ϕ ⁰ (left) and the internal wavelet ψ ⁰ (right). Daubechies orthogonal generator with r = 3.

generating functions and study the approximation errors provided by the MRA

234

(V _j ¹ ). Second we apply the new bases to the resolution of a Dirichlet-Laplace

235

problem, only using Fast Wavelet Transforms, leading to a linear complexity for

236

the resolution of the problem.

237

5.1. Basis functions and approximation errors

238

For the different examples, the scaling function and wavelet generators (ϕ ⁰ , ψ ⁰ ) considered are Daubechies orthogonal generators, with three vanishing moments for the wavelet [8]. The integer parameters of the construction of V _j ⁰ thus are:

r = 3, δ _[ = δ ] = 1, n min = −r + 1 and n max = r.

On Figure 1, we show the plot of the internal scaling function ϕ ⁰ and the wavelet ψ ⁰ . The edge orthogonal scaling functions and wavelets are plotted on Figure 2 and Figure 3 respectively. The generators (ϕ ¹ , ψ ¹ ) are computed from (ϕ ⁰ , ψ ⁰ ) using the formula:

ϕ ¹ (x) = Z x

x−1

ϕ ⁰ (t)dt and ψ ¹ (x) = 4 Z x

−∞

ψ ⁰ (t)dt. (34) The graphs of ϕ ¹ and ψ ¹ are plotted on Figure 4, Figure 5 and Figure 6 show the

239

plot of the edge scaling function graphs. The edge wavelet graphs are plotted on

240

Figure 7 and Figure 8. We notice that these edge wavelets satisfy homogeneous

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Dirichlet boundary condition while this boundary condition is not required for

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0 1 2 3 4 5 6 -0.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

(a) Φ ^0,[ _j,0

0 1 2 3 4 5 6

-4 -3 -2 -1 0 1 2 3

(b) Φ ^0,[ _j,1

0 1 2 3 4 5 6

-2 -1 0 1 2 3 4

(c) Φ ^0,[ _j,2

-5 -4 -3 -2 -1 0 1

-0.5 0 0.5 1 1.5 2 2.5 3 3.5

(d) Φ ^0,] _j,0

-5 -4 -3 -2 -1 0 1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(e) Φ ^0,] _j,1

-5 -4 -3 -2 -1 0 1

-0.5 0 0.5 1 1.5 2 2.5 3 3.5

(f) Φ ^0,] _j,2

Figure 2: Plot of the edge orthogonal scaling functions of V _j ⁰ . Daubechies orthogonal

generator with r = 3.