© Hindawi Publishing Corp. BIORTHOGONAL MULTIRESOLUTION ANALYSES AND DECOMPOSITIONS OF SOBOLEV SPACES

The object of this paper is to construct extension operators in the Sobolev spaces Hk()−∞, 0)) and Hk((0, +∞() (k ≥ 0). Then we use these extensions to get biorthog- onal wavelet bases in H k (R). We also give a construction in L 2 ((−1, 1)) to see how to obtain boundaries functions. 2000 Mathematics Subject Classification. 41A58, 42C15.


Introduction.
The decomposition method was used by Ciesielski and Figiel [1,2] to construct spline bases of general Sobolev spaces W k p (M) (k ∈ Z and 1 ≤ p ≤ +∞) where M is a compact Riemannian manifold of dimension d.The extension operators constructed by Ciesielski and Figiel are based on the extension theorem of Stein [13] which do not permit to get multiresolution with compact support.
In 1993, we have constructed biorthogonal wavelet bases in Ω which is an interval and a bounded open set of R 2 (see [8,9]).These bases are adapted to study Sobolev spaces H 1 (Ω) and H 1 0 (Ω).Recently, in 1997, the decomposition was used by Cohen, Dahmen, and Schneider (see [3,4,5]) to construct biorthogonal wavelet bases (ψ λ , ∼ ψ λ ) λ∈∇ of L 2 (Ω) where Ω is a bounded domain of R d .These bases were those of Sobolev spaces H s (Ω) for s only in the interval ] − 3/2, 3/2[.There are related constructions given by Masson in [11].All these constructions are based on the decomposition method; there is a slight difficulty in their presentation, due to notational burden.Moreover, it is unclear how to obtain regularity Sobolev estimates for |s| ≥ 3/2 and also to get associated fast algorithms.
In this paper, we use a direct method based on the result described in [9] to define orthogonal and biorthogonal multiresolution analyses on the interval [0, 1] which are generated by a finite number of basis functions.These analyses are regular and have a compact support.Next we use a decomposition method to construct segmented biorthogonal multiresolution analyses in R. In this case, we get decompositions of the Sobolev space H k (R) (k ∈ Z) by using simple extension operators.These extensions permit to get fast algorithms for associated biorthogonal multiresolution analyses because all bases constructed in this paper satisfy the lifting scheme described in [11,14] in order to get wavelet bases with compact support and with the same regularity as for Daubechies bases [6].This analysis is adapted to the study of regular functions in H k ([0, +∞[), H k (]−∞, 0]), and H k (R), by using extensions.We also give a construction of segmented biorthogonal multiresolution analysis in L 2 ([−1, 1]) to see how to obtain boundary functions.Finally, recall that segmented multiresolution analyses are useful in many applications as numerical simulation for elliptic problems or image processing (see [11]).
The first object of this paper is studied in Section 2. In fact, we construct biorthogonal multiresolution (V j ,V * j ) (j ∈ Z) on the interval [0, 1].By a derivation on V j and an integration on V * j , we get biorthogonal multiscale analysis (V (1) j ,V (−1) j ) of the space L 2 ([0, 1]).Let P j be the projector on V j parallel to (V * j ) ⊥ and P (1) j be the projector in j ) ⊥ , then we have the following commutation property: If the multiscale function ϕ is regular, we develop a similar strategy for constructing biorthogonal multiresolution analysis (V Moreover, we have the commutation property between scale projectors and derivation The biorthogonal analysis (V ) are adapted to the study of Sobolev spaces ) and H k 0 ([0, 1]) for k ∈ Z. Section 3 is devoted to the construction of extension operators.We show that if we consider an extension operator E from H k (] − ∞, 0]) into H k (R) for k ∈ Z, we get decomposition of the Sobolev space H k (R) by using an isomorphism between the space H k (R) and the space E(H k (]−∞, 0]))+H k 0 ([0, +∞[).This isomorphism permits to get biorthogonal multiresolution analysis of H k (R) based on those of L 2 (] −∞, 0]) and L 2 ([0, +∞[).These multiscale analyses satisfy the commutation property between scale projectors and derivation.All wavelet bases constructed in this section have a compact support and are adapted to higher regularity analysis.
In conclusion, we describe "new" biorthogonal multiresolution analysis in L 2 ([−1, 1]) to show more clearly how to construct boundary functions.These analyses are adapted to the study of the Sobolev spaces

Multiresolution analyses on the interval and applications.
Recall that multiresolution analyses (denoted by MRA) on the interval are introduced by Meyer [12].For other related constructions see [7,8,9].In the first part of this section, we construct orthogonal multiresolution analyses V j on the interval [0, 1] and we show that there exists a new supplement X j of V j in V j+1 .In the second part, we introduce biorthogonal multiresolution analyses on the interval and we prove, by using a derivation, that we get other biorthogonal multiresolution analyses which are adapted to the study of Sobolev spaces H k ([0, 1]) and H k 0 ([0, 1]) for k ∈ Z.Moreover, we get the commutation property between scale projectors and derivation.These analyses have compact support and are adapted to higher regularity analysis.They will be used in the next section and, by using some natural extensions, we get segmented biorthogonal analyses.

Orthogonal multiresolution analyses on the interval.
We recall that the orthogonal multiresolution analysis (denoted by OMRA) V j (R) of Daubechies [6] satisfies the following properties: • V 0 has an orthonormal base ϕ(x − k), k ∈ Z, where ϕ the scaling function with compact support.
, the sequence of real numbers (a k ) satisfies a 0 ≠ 0 and a 2N−1 ≠ 0.Moreover, we have and "∧" is the classical Fourier transform on R.
• The associated wavelet ψ is defined by (2.1) The multiresolution of Daubechies is orthogonal in L 2 (R), but if we take its restriction to [0, 1], we do not get an orthogonal multiresolution analysis in L 2 ([0, 1]).Moreover, if we consider the functions ϕ j,k (x) /[0,1] , we have an independent system but not orthogonal.However, if we consider the functions ψ j,k (x) /[0,1] we get a dependent system (see [12]).Then, the construction of orthogonal multiresolution analyses in [0, 1] (or biorthogonal) is technical specially near the boundaries 0 and 1.
In the following, we have to construct new orthogonal wavelet bases in [0, 1].For this purpose, we consider the OMRA V j (R) of Daubechies and we denote Example 2.2 (periodic wavelets).For j ≥ 0, we denote (2.3) Then V j is defined as the space generated by the functions . We then get periodic wavelet bases which are adapted to the study of periodic regular functions.
The following result of Meyer [12] gives another example of MRA of L 2 ([0, 1]) which is important to establish the first goal of this paper.Lemma 2.3.For j ≥ j 0 , the functions We describe a complement (not orthogonal) of V j in V j+1 .More precisely, we have the following important result from [9].Proposition 2.4.Let j 0 be the smallest integer satisfying 2 j 0 ≥ 4N − 4. For j ≥ j 0 , we denote (2.4) There exists an integer J such that for all j ≥ J, V j+1 = V j ⊕ X j .
Proof.First we remark that supp The matrix (m p,q ), with −N + 1 ≤ p, q ≤ 2 j − 1 of coefficients of x j,k relatively to ψ j,k/[0,1] (we decompose x j,k with respect to ϕ j,k/[0,1] and ψ j,k/[0,1] and we take only the coefficients corresponding to ψ j,k[0,1] ) is defined by where A = (m p,q ), with −N + 1 ≤ p, q ≤ 2 j−1 − 1, is a superior triangular matrix with all its diagonal coefficients are different from zero and B = (m p,q ), with 2 j − N + 1 ≤ p, q ≤ 2 j − N, is an inferior triangular matrix with all its diagonal coefficients are different from zero.The diagonal terms of A and B are defined by ) where we conclude that there exists an integer J such that for j ≥ J, V j+1 = V j ⊕ X j .Finally, by using Gram-Schmidt method, we get orthonormal wavelet basis on the interval [0, 1].

Biorthogonal multiresolution analyses on the interval.
First we give some definitions of biorthogonal multiresolution analysis (denoted by BMRA), then we describe constructions on the interval.
such that the multiscale functions g and g * have compact support, we can define a BMRA of (2.9) It is clear that ).This BMRA is adapted to the study of periodic functions.
Let (V j (R), V * j (R)) be a BMRA of L 2 (R) with associated multiscale functions (g, g * ).We assume that supp g = [N 1 ,N 2 ] and we denote (2.10) Our construction is based on the following result.
Theorem 2.7.We consider a BMRA (V j (R), V * j (R)) of L 2 (R), (g, g * ) are the multiscale functions with compact support and (V j ,V * j ) an associated BMRA of L 2 ([0, 1]).We assume that (i) g is differentiable and g (2.11) (2.12) Moreover, if we denote by P j (resp., ), then we have the following commutation property (2.13) ).Moreover, since V j contains the functions P β 0,j (x), we have In the same way, we have To see the duality between Then the derivation is an isomorphism from  (R) the MRA constructed by derivation and integration.Then the theorem described above proves that ).Moreover, if we denote by P (d) j the projector on (2.15) We define g and g * by (2.16) The functions g j,k/[0,1] form a basis of ) we take the functions g j,k with support in [0, 1] and the boundaries functions as follows: (2.17) The real constants α i,j,p satisfy (2.18) In this case the following results are proved in [9].
Theorem 2.8.We assume that ϕ is a C p+ε -function, p ∈ N, p ≥ d, and ε > 0. We denote by and j 0 an integer satisfying •d/dx allows to study the vector functions with divergence equal to zero (see [9]).

Segmented biorthogonal multiresolution analyses and extension operators.
We study in this section two constructions of biorthogonal wavelet bases.The first one is based on the OMRA of Daubechies and prove that we can analyze functions in H k (R) (k ∈ Z) with information in the past H k (] − ∞, 0]) (k ∈ Z), the relaxation of the past in the future near zero and information in the future The second construction is based on a symmetric multiresolution analyses with compact support and show how to obtain boundary functions in [−1 , 1].We obtain, by using extensions, biorthogonal multiresolution analyses of

General principles of extensions.
We begin with some notions of extensions which are important for decompositions of the Sobolev space H k (R).Next we give conditions on extensions to get simple algorithms of wavelet bases.Definition 3.1.We denote by Θ(R) the space of continuous operators on L 2 (R).
Example 3.2.We consider the interval I =] − ∞, 0] and f ∈ C k (I) (space of C kfunctions in I).We define the operator A by where β p are real constants such that β s < β s−1 < ••• < β 1 < 0. The real constants α p will be chosen such that Af ∈ C k (R).For this purpose, we must have s p=1 α p β m p = 1, m = 0, 1,...,k.The adjoint operator A * of A is defined by ..,k.We conclude that for s > 2k + 2, the constants α p exist and then A and A * are known.
The biorthogonality of bases depends on the operators A and A * .We remark that the operator defined above is not "good" because A or A * does not preserve the property of compact support, if we consider multiresolution analyses with compact support, and special properties near the boundaries [9].
Remark 3.4.The extension E relaxes the information of the past, and the function )) is supported by the future axis.We will show how Proposition 3.3 permits to get wavelet bases of L 2 (R) by using those of L 2 (] −∞, 0]) and L 2 ([0, +∞[).
The main problem is in the definition of extension operators which are adapted to scale, and permit to get regular wavelet bases.This problem is the object of the next section.

Construction of segmented BMRA of L 2 (R).
The object of this section is to construct a BMRA of L 2 (R).For this purpose, we consider the OMRA V j (R) of Daubechies with associated scaling function ϕ and wavelet ψ, and we construct a BMRA of L 2 (] − ∞, 0]) and L 2 ([0 + ∞[).Next, we use some extension operators to get a BMRA of L 2 (R).
Recall that the functions ϕ j,k (x) = 2 j/2 ϕ(2 j x − k), k ∈ Z, form a Riesz basis of V j (R) and ψ j,k (x) = 2 j/2 ψ(2 j x − k), k ∈ Z, form a Riesz basis of W j (R) (orthogonal complement of V j (R) in V j+1 (R)).We define the extension operators E j and E j by and the two extension operators E * 0 and We consider V j = Vect{ϕ j,k , k ∈ Z} (OMRA of Daubechies).To define the dual space, we need to construct the new functions ϕ * j,q given by where a j,q,l satisfy the following conditions: a j,q,l 2 j/2 ϕ 2 j x − l ϕ 2 j x − k dx = δ q,k . (3.6) The precedent system of (2N −2) equations and (2N −2) unknowns has one solution because the functions ϕ j,k/]−∞,0] , 2 − 2N ≤ k ≤ −1, are independent (see [5]).As the conditions on a j,q,l do not depend on j, then we have We define now the dual space We remark that To give a basis of W j , we need to construct the new wavelets α j,k as follows: for 2 − 2N ≤ k ≤ −N, we put (3.9) We have (N − 1) functions which can be orthonormalized (for the scalar product of L 2 (]−∞, 0])) to get the functions α j,k , 2−2N ≤ k ≤ −N, indicated above.Then a basis of W j is given by the following functions: • β j+1,2k = ϕ j+1,2k − q≥0 ϕ j+1,2k ,ϕ j,q ϕ j,q − q≥0 ϕ j+1,2k ,ψ j,q ψ j,q , 0 ≤ k ≤ N − 2.
We construct now a basis of the space Obviously, we have α j,k ,ψ * j,q L 2 (R) = δ k,q .We denote by β * j+1,2k , 0 ≤ k ≤ N − 2, the dual system of β j+1,2k , 0 ≤ k ≤ N − 2, for the scalar product of L 2 (R), and we define ,ψ j,q ψ j,q . (3.11) We conclude that a basis of W * j is given by the following functions: The functions ϕ j,k , k ∈ Z, form a basis of the space V j and the functions ϕ * j,k , k ∈ Z, form a basis of the space V * j such that (3.12) The projector P j can be written as and we have the following property: The following theorem proves that the MRA (V j ,V * j ) described above is a segmented BMRA of L 2 (R).
(ii) We consider the spaces We define in the same way the spaces We denote Then W j and W * j are in duality for the scalar product of L 2 (R) and we have (iii) If we define V (1) j and V (−1) j in the same way as (3.15) by replacing ) is a BMRA of L 2 (R).Moreover, if P j is the projector from is the projector from L 2 (R) into V Proof.We consider the BMRA (V j ,V * j ) of L 2 (R) described above.In the following we prove the properties (i) and (ii).By taking the restrictions, respectively, on ]−∞, 0] and [0, +∞[ of functions of V j and V * j .We get the spaces (3.20) In the same way, the spaces defined in (3.16) are completely described.In fact, we have These multiscales define a BMRA of L 2 (R).We denote by U j (R) the closed linear hull of ϕ j,k , k ∈ Z, where ϕ j,k (x) = 2 j/2 ϕ(2 j x − k), and in the same way U * j (R) the closed linear hull of ϕ * j,k , k ∈ Z, where ϕ * j,k (x) = 2 j/2 ϕ * (2 j x − k).The important properties of ϕ and ϕ * are There exists ε > 0 such that ϕ ∈ H 1+ε (R), (3.37) The last point allows that the functions ϕ j,k , k ∈ Z, form a Riesz base of ⊥ can be written in the form and satisfies P j+1 • P j = P j • P j+1 = P j , such that Q j = P j+1 − P j is a projector from A base of W j (R) is given by the functions ψ j,k (x) = 2 j/2 ψ(2 j x − k), k ∈ Z, where ψ(2ξ) = e −iξ m(ξ + π) ϕ(ξ).In the same way, a base of W * j (R) is given by the functions ψ * j,k (x) = 2 j/2 ψ * (2 j x − k), k ∈ Z, where ψ * (2ξ) = e −iξ m(ξ + π) ϕ * (ξ).Moreover, the two bases are in duality for the scalar product of L 2 (R): The fundamental properties of ψ and ψ * are (3.41) We construct now a segmented BMRA of L 2 ([−1, 1]) by using those in L 2 ([−1, 0]) and L 2 ([0, 1]) and the extension operators E j and E j described above.
We define the following spaces: We have the same result for V j ([0, 1]) and V * j ([0, 1]).We apply extensions as follows: • The function ϕ β j is extended by ), which is symmetric and with compact support where (3.43) In the same way, we get the associated wavelet spaces where Q j f is given by The functions 1 and x belong to V j ([−1, 1]) (see (3.36)), then we get 1 −1 x α η * (x)dx = 0, for 0 ≤ α ≤ 1 (see [8]).We have and for a sequence (λ j,k ) j,k ∈ Z 2 , we have where C is a positive constant.Then we get j≥j 0 4 j Q j f 2 2 ≤ C f 2 , where C is a positive constant.To prove the other inequalities, we write The properties (3.45) and (3.52) give the result.

Conclusion.
We have constructed in this paper two multiresolution analyses (OMRA and BMRA) of L 2 ([0, 1]) which are generated by a finite number of basis functions.For the first one, we used a direct method based on the result described in [9] to define an orthonormal multiresolution analysis on [0, 1] which is regular and has compact support.For the second one, we used the idea of "derivation and integration" to get new biorthogonal multiresolution analyses on the interval.In this case, we get the commutation property (2.13) between scale projectors and differentiability.Next, we use the decomposition method to construct two segmented biorthogonal multiresolution analyses.For the first one, we show that if we consider an extension operator E from H k (] − ∞, 0]) into H k (R), we get decomposition of the Sobolev space H k (R) (k ∈ Z) by using an isomorphism between the space H k (R) (k ∈ Z) and the space Recall that all bases constructed in this work satisfy the lifting scheme [10,14]; thus, we get wavelet bases with compact support and with the same regularity as for Daubechies bases.
x 0 β i (t)dt then by integration, we conclude that the bases ( ∼ α i ) and ( ∼ β i ) are biorthogonal and we have a duality between ∼ V j and ∼ V * j .Finally the commutation property is satisfied.In fact, we have

(2. 14 )
Fundamental example.Let V j (R) be the OMRA of Daubechies where the scale function ϕ is of class C d .We denote by V (d) j (R) and V (−d) j