Biorthogonal decomposition of the unit interval

152 L. Levaggi – A. Tabacco R EMARK 7. In Theorem 4, we have described how to characterize the family of Besov spaces B s pq,00 + using a scaling fuction ϕ ∈ B s pq + and removing S = [s ] boundary scaling functions. Of course, one can remove only S S boundary scaling functions and pro- ceed as before to construct a multiresolution analysis. In this case we characterize B s pq,00 + with 0 s ≤ S and B S pq,00 + ∩ B s pq + for every S s s .

8. Biorthogonal decomposition of the unit interval

In the previous sections, we have carried out the construction of a multiresolution analysis taking into account the presence of a boundary point. Now we want to exploit it to build a multilevel decomposition of the bounded interval 0, 1. Intuitively one easily sees that, provided the scale is finer enough, the presence of the left boundary point does not influence the construction at the right one. More precisely, we will choose a level j such that V j 0, 1 = span n ϕ j l : l ∈ L o ⊕ span{ϕ j k : k ∈ I } ⊕ span n ϕ 1 j r : r ∈ R o , ∀ j ≥ j , with I 6= ∅ and where the boundary functions ϕ j l and ϕ 1 j r are constructed independently. Here, and from now on, the suffix or 1 refers to the boundary point 0 or 1, respectively. The basic idea is to start from two multiresolution analyses on the half-lines I = 0, +∞ and I 1 = −∞, 1, and paste them together in a suitable way to get the spaces V j 0, 1. In turns, to obtain a decomposition on I 1 , we first consider a decomposition on − = −∞, 0 and then we translate it of a unit. Let us choose two bases for ✁ L−1 and ✁ e L−1 , say p 1 α : α = 0, . . . , L − 1 and ˜q 1 β : β = 0, . . . , e L − 1 , possibly different from the ones used to build V B and e V B for 0, +∞. Fixing two nonnegative integers δ 1 and ˜δ 1 , let us define the boundary functions as in 35 φ − α x = −n −1 X k=1−δ 1 −n 1 c 1 α k ϕ 0k x , x ≤ 0 , ∀α = 0, . . . , L − 1 . Matching the dimensions of V − and e V − , we obtain a relation similar to 40: ˜δ 1 − δ 1 = e L − L − ˜n 1 − n 1 . 93 Recalling the definition of the isometries T j see 49, we define V j − = span n φ − j α = T j φ − α : α = 0, . . . , L − 1 o ⊕ span n ϕ j k : k ≤ −δ 1 − n 1 o ; using the operator τ : x 7→ x − 1, we translate the origin into the right edge of our interval. It is easy to see that, calling φ 1 j α x = 2 j −n −1 X k=2 j +1−δ 1 −n 1 c 1 α, k−2 j ϕ j k x , x ≤ 1 , we have V j −∞, 1 = span n φ 1 j α : α = 0, . . . , L − 1 o ⊕ span n ϕ j k : k ≤ 2 j − δ 1 − n 1 o . Wavelets on the interval 153 As said before, we wish to maintain the situation at the two boundary points decoupled. This means we want to have at least one interior function in V j 0, 1. This requirement yields the condition −n + δ ≤ 2 j − δ 1 − n 1 . Keeping also into account the dual relation, we have to set a coarsest level j such that for all j ≥ j , 2 j ≥ max n 1 − n + δ + δ 1 , ˜n 1 − ˜n + ˜δ + ˜δ 1 . 94 By 40 and 93, we have ˜n 1 − ˜n + ˜δ + ˜δ 1 − n 1 − n + δ + δ 1 = 2 e L − L ≥ 0 , so we must fix j ≥ l log 2 ˜n 1 − ˜n + ˜δ + ˜δ 1 m . 95 Thus, we have, for j ≥ j , V j 0, 1 = span n φ j k : k = 0, . . . , L − 1 o ⊕ ⊕ span n ϕ j k : k = −n + δ , . . . , 2 j − δ 1 − n 1 o ⊕ ⊕ span n φ 1 j k : k = 0, . . . , L − 1 o , and similarly for the dual spaces e V j 0, 1. By construction, and thanks to the choice of j , all the biorthogonality properties are maintained. Finally, we observe that dim V j 0, 1 = 2 j + 2L + 1 − δ − δ 1 − ˜n 1 + ˜n , ∀ j ≥ j . Since V j +1 0, 1 = V j 0, 1 ⊕ W j 0, 1, this implies dim W j 0, 1 = 2 j +1 − 2 j = 2 j . Going through the construction in Section 5, one easily proves that, setting m : = δ 1 + n 1 − ˜n + 1 2 one has n ϕ 1k : k ≤ 2−δ 1 − n 1 + ˜n + 1 o ⊆ V − ⊕ span n ψ 0m : m ≤ −m o . As before, one out of two ϕ 1k , k = −δ 1 − n 1 , . . . , 2 −δ 1 − n 1 + ˜n + 2 is linearly dependent modulus V − on the previous ones; observe then that dim W B − = m − 1. Therefore, defining the projection operator P − on V − and ψ − 0,m −k : = ϕ 1,−2δ 1 +n 1 + n +2k −∞, 0] − P − ϕ 1,−2δ 1 +n 1 + n +2k −∞, 0] , k = 1, . . . , m − 1 154 L. Levaggi – A. Tabacco one gets W − = span n ψ 0m : m ≤ −m o ⊕ span n ψ − 0m : m = 1, . . . , m − 1 o . As in Section 5, we observe that D ψ 0m , e ψ 0n E = 0 , ∀n = 1, . . . , e m − 1 if m ≤ − 2 ˜n 1 + ˜δ 1 − n − ˜n 2 =: −m D ψ 0m , e ψ 0n E = 0 , ∀m = 1, . . . , m − 1 if n ≤ − 2n 1 + δ 1 − n − ˜n 2 =: −e m . Through the operators T j we can define W j − = T j W − and operating with the trans- lation τ : x 7→ x − 1 we finally have W j −∞, 1 = span n ψ j m : m ≤ −m + 2 j o ⊕ ⊕ span n ψ 1 j m : = τ T j ψ − 0m : m = 1, . . . , m − 1 o . As for the scaling spaces, we can paste the wavelet spaces at each level and we obtain W j 0, 1 = span n ψ j m : m = 0, . . . , m ∗ − 1 o ⊕ ⊕ span n ψ j m : m = m ∗ , . . . , 2 j − m o ⊕ ⊕ span n ψ 1 j m : m = 1, . . . , m − 1 o . The minimum level j must be taken in order to avoid intersection between the supports of the modified border wavelets corresponding to two different edges. It is not possible to have an easy and general formulation for it, but it will be computed later on for B-spline scaling functions. R EMARK 8. Obviously, we can state characterization theorems for the scales of Besov spaces B s pq 0, 1 and B s pq,00 0, 1 see Theorems 3, 4 and subsequent Remarks. Moreover, we can also characterize spaces of functions possibly satisfying different boundary conditions at the two edges 0 and 1.

9. The B-spline multiresolution