Definition of subspaces and of scaling functions A natural way to introduce wavelets is through the multiresolution analysis. Given a function f∈L2(R), a multiresolution of L2(R) will provide us with a sequence of spaces Vj,Vj+1,... such that the projections of f onto these spaces give finer and finer approximations (as j→∞) of the function f. (Multiresolution analysis (MRA) in the first generation) A multiresolution analysis of L2(R) is defined as a sequence of closed subspaces Vj⊂L2(R),j∈Z with the following properties: ... ⊂ V - 1 ⊂ V 0 ⊂ V 1 ⊂ ... The spaces Vj satisfy ⋃j∈ZVjisdenseinL2(R)and⋂j∈ZVj={0}. If f(x)∈V0,f(2jx)∈Vj, i.e. the spaces Vj are scaled versions of the central space V0. If f∈V0,f(.-k)∈V0,k∈Z, that is, V0 (and hence all the Vj) is invariant under translation. There exists ϕ∈V0 such that {ϕ(x-k);k∈Z} is a Riesz basis in V0. We will call `level' of a MRA one of the subspaces Vj. From , it follows that, for fixed j, the set {ϕjk(x)=2j/2ϕ(2jx-k);k∈Z} of scaled and translated versions of ϕ is a Riesz basis for Vj. Since ϕ∈V0⊂V1, we can express ϕ as a linear combination of {ϕ1,k}: ϕ ( x ) = ∑ k ∈ Z h k ϕ 1 , k ( x ) = 2 ∑ k ∈ Z h k ϕ ( 2 x - k ) . is called the two-scale equation or refinement equation. It is a fundamental equation in MRA since it tells us how to go from a fine levelV1 to a coarser levelV0. The function ϕ is called the scaling function. As said before, the spaces Vj will be used to approximate general functions. This will be done by defining appropriate projections onto these spaces. Since the union of all the Vj is dense in L2(R), we are guaranteed that any given function of L2(R) can be approximated arbitrarily close by such projections. As an example, define the space Vj as V j = { f ∈ L 2 ( R ) ; ∀ k ∈ Z , f | [ 2 - j k , 2 - j ( k + 1 ) [ = constant } Then the scaling function ϕ(x)=1[0,1)(x), called the Haar scaling function, generates by translation and dilatation a MRA for the sequence of spaces {Vj,j∈Z} defined in , see , .

The detail space and the wavelet function Rather than considering all the nested spaces Vj, it would be more efficient to code only the information needed to go from Vj to Vj+1. Hence we consider the space Wj which complements Vj in Vj+1 : V j + 1 = V j ⊕ W j . The space Wj is not necessarily orthogonal to Vj, but it always contains the detail information needed to go from an approximation at resolution j to an approximation at resolution j+1. Consequently, by using recursively the equation , we have for any j0∈Z, the decomposition L 2 ( R ) = V j 0 ⊕ ⊕ j = j 0 ∞ W j ¯ . With the notational convention that Wj0-1:=Vj0, we call the sequence {Wj}j≥j0-1 a multiscale decomposition (MSD). We call ψ a wavelet function whenever the set {ψ(x-k);k∈Z} is a Riesz basis of W0. Since W0⊂V1, there also exist a refinement equation for ψ, similarly to : ψ ( x ) = 2 ∑ k g k ϕ ( 2 x - k ) . The collection of wavelet functions {ψjk=2j/2ψ(2jx-k);k∈Z,j∈Z} is then a Riesz basis for L2(R). One of the main features of the wavelet functions is that they possess a certain number of vanishing moments. A wavelet function ψ(x) has Nvanishing moments if ∫ ψ ( x ) x p d x = 0 , p = 0 , ... , N - 1 . We now mention two interesting cases of wavelet bases.

Orthogonal bases In an orthogonal multiresolution analysis, the spaces Wj are defined as the orthogonal complement of Vj in Vj+1. The following theorem tells us one of the main advantages of such a MRA. (, Theorem 5.1.1) If a sequence of closed subspaces (Vj)j∈Z in L2(R) satisfies , and if, in addition, {ϕ(x-k),k∈Z} is an orthogonal basis for V0, then there exists one function ψ(x) such that {ψ(x-k);k∈Z} forms an orthogonal basis for the orthogonal complement W0 of V0 in V1. An immediate consequence of is that {ψjk,k∈Z} constitutes an orthogonal basis for the orthogonal complement Wj of Vj in Vj+1. In this section, let Pj (resp. Qj) be the orthogonal projection operator onto Vj (resp. Wj). The orthogonal expansion f = P j 0 f + ∑ j = j 0 ∞ Q j f = ∑ k f , ϕ j 0 , k ϕ j 0 , k + ∑ j = j 0 ∞ ∑ k f , ψ j k ψ j k tells us that a first, coarse approximation of f in Vj0 is further refined with the projection of f onto the detail spaces Wj. shows two examples of orthogonal wavelet functions. The first is the Haar wavelet, associated to the Haar scaling function defined in "Definition of subspaces V j and of scaling functions". ψ ( x ) Haar = 2 - 1 / 2 ϕ Haar ( 2 x - 1 ) - ϕ Haar ( 2 x ) = 1 [ 1 2 , 1 ) ( x ) - 1 [ 0 , 1 2 ) ( x ) . The Haar wavelet has only one vanishing moment and consequently is optimal only to represent functions having a low degree of regularity, like, for example, β-Hölder functions with 0<β<1. Daubechies constructed in , compactly supported wavelets which have more than one vanishing moment. Compactly supported wavelets are desirable from a numerical point of view, while having more than one vanishing moment allows to reconstruct exactly polynomials of higher order. These wavelets cannot, in general, be written in a closed analytic form. However, their graph can be computed with arbitrarily high precision using a subdivision scheme algorithm. (b) represents the Daubechies Least Asymmetric wavelet with N=4 vanishing moments.

(a) N=1 (b)N=4 Some orthogonal basis functions: (a) the Haar wavelet function bases with N=1 vanishing moments, (b) the Least Asymmetric wavelet function of Daubechies , , with N=4 vanishing moments. This figure also illustrates the reason behind the name `wavelet': since wavelets are functions with a certain number of vanishing moments, they have the shape of a `little wave' or `wavelet'.

Biorthogonal bases Having an orthogonal MRA puts strong constraints on the construction of a wavelet basis. For example, the Haar wavelet is the only real-valued function which is compactly supported and symmetric. However, if we relax orthogonality for biorthogonality, then it becomes possible to have real-valued wavelet bases of fixed but arbitrary high order (see Definition 1 from Approximation of Functions) which are symmetric and compactly supported . In a biorthogonal setting, a dual scaling function ϕ˜ and a dual wavelet function ψ˜ exist. They generate a dual MRA with subspaces V˜j and complement spaces W˜j such that V ˜ j ⊥ W j and V j ⊥ W ˜ j . In other words, ϕ ˜ , ψ ( · - k ) = 0 and ϕ , ψ ˜ ( · - k ) = 0 Moreover, the dual functions also have to satisfy ϕ ˜ , ϕ ( · - k ) = δ k , 0 and ψ ˜ , ψ ( · - k ) = δ k , 0 , where δk,0 is the Kronecker symbol. By construction, the dual scaling and wavelet functions satisfy a refinement equation, similarly to the equations and . In this work, we use the following convention: the dual MSD will be used to decompose a function (or a signal), while the original, or primal MSD reconstructs the function. This yields the following representation of a function f∈L2(R) f ( x ) = ∑ k f , ϕ ˜ j 0 , k ϕ j 0 , k ( x ) + ∑ j = j 0 ∞ ∑ k f , ψ ˜ j k ψ j k ( x ) . shows an example of a biorthogonal wavelet basis built by Cohen, Daubechies and Feauveau in , (called CDF-wavelets hereafter).

Primal and dual scaling and wavelet functions for the (3,1)-Cohen-Daubechies-Feauveau (CDF) biorthogonal basis. The primal wavelet function ψ has one vanishing moment while the dual wavelet ψ˜ has three vanishing moments.