Scaling function spaces for the half-line

Wavelets on the interval 131 and kv − P j v k L p 2 − j s |v| B s pq , ∀v ∈ B s pq , ∀ j ∈ ✂ . 31 Thus, taking into account 1, we can apply the characterization Theorems 1 and 2 to the Besov space Z .

3. Scaling function spaces for the half-line

Starting from a biorthogonal decomposition on as described in Subsection 2.3, we aim to the construction of dual scaling function spaces V j + and e V j + which will form a multilevel decomposition of L p + and of L p ′ + , respectively. For simplicity, we will work on the scale j = 0 and again we will not explicitly describe the dual ˜ construction. Without loss of generality, we shall suppose L ≤ e L and ˜n ≤ n ≤ 0 ≤ n 1 ≤ ˜n 1 so that supp ϕ ⊆ supp e ϕ ; if this is not the case, it is enough to exchange the role of the primal and the dual spaces. From now on, we will append a suffix to all the functions defined on the real line. Note that if k ≥ −n , ϕ 0k have support contained in [0, +∞. More precisely supp ϕ 0k = [n + k, n 1 + k] . Let us fix a nonnegative integer δ and set k ∗ = −n + δ; observe that k ∗ = min n k ∈ ✂ : supp ϕ 0k ⊆ [δ, +∞ o . Let us define V + = span n ϕ 0k [0,+∞ : k ≥ k ∗ o ; 32 this space will be identified in a natural way with a subspace of V and will not be modified by the subsequent construction. To obtain the right scaling space V + for the half-line, we will add to the basis n ϕ 0k [0,+∞ : k ≥ k ∗ o of V + a finite number of new functions. These functions will be constructed so that the property of reproduction of polynomials is maintained. In fact we know that for any polynomial p ∈ ✁ L−1 and every fixed x ∈ , px = X k∈ ˘p 0k ϕ 0k x . So, if { p α : α = 0, . . . , L − 1} is a basis for ✁ L−1 , for every x ≥ 0, we have p α x = X k≥−n 1 +1 c α k ϕ 0k x = k ∗ −1 X k=−n 1 +1 c α k ϕ 0k x + X k≥k ∗ c α k ϕ 0k x , 33 where c α k : = ˘ p α 0k = Z p α y e ϕ y − k dy , α = 0, . . . , L − 1 . 34 132 L. Levaggi – A. Tabacco Since the second sum in 33 is a linear combination of elements of V + , in order to locally generate all polynomials of degree ≤ L − 1 on the half-line, we will add to this space all the linear combinations of the functions φ α x = k ∗ −1 X k=−n 1 +1 c α k ϕ 0k x , α = 0, . . . , L − 1 . 35 R EMARK 1. Let { p α |α = 0, . . . , L −1} and { p ⋆ α |α = 0, . . . , L −1} be two bases of ✁ L−1 . Let us denote by φ α and φ ⋆ α the functions defined by the previous argument and by M the matrix of the change of basis of ✁ L−1 , then one can prove that φ ⋆ α = L−1 X β= M αβ φ β for every α. P ROPOSITION 1. The functions φ α , α = 0, . . . , L − 1 are linearly independent. Proof. If δ is strictly positive, the linear independence of the boundary functions is obvious. Indeed, on [0, δ], they coincide with linearly independent polynomials. If δ = 0 let us observe that the functions ϕ 0k [0,+∞ involved in 35 have staggered support, i.e., supp ϕ 0k [0,+∞ = [0, n 1 +k], thus they are linearly independent. To obtain the linear independence of the functions φ α it is sufficient to prove that the matrix C = c α k k=−n 1 +1,... ,−n −1 α= 0,... ,L−1 induces an injective transformation. Thanks to Remark 1, we can choose any polynomial bases to prove the maximality of rank C; if p α x = x α for any α, one has c α k = α X β= α β k β e M α−β , where e M i = R x i e ϕ xd x is the i -th moment of e ϕ on . Let v be a vector in L such that C T v = 0, then the polynomial of degree L − 1 L−1 X β= L−1 X α=β α β v α e M α−β x β has n 1 − n − 1 distinct zeros; thus, recalling the relation 27, it is identically zero. This means Mv = 0 where M = M i j is an upper triangular matrix with M i j = j i e M j −i if j ≥ i. M is non-singular, in fact detM = R e ϕ xd x L 6= 0 and the proof is complete. The building blocks of our multiresolution analysis on 0, +∞ will be the border functions 35 and the basis elements of V + . Using Proposition 1 and the linear independence on of the functions ϕ 0k , one can easily check that P ROPOSITION 2. The functions φ α , α = 0, . . . , L−1 and ϕ 0k [0,+∞ , k ≥ k ∗ , are linearly independent. Wavelets on the interval 133 Thus, it is natural to define V + = span {φ α : α = 0, . . . , L − 1} ⊕ V + . 36 We rename the functions in the following way ϕ 0k = φ k if k = 0, . . . , L − 1, ϕ 0,k ∗ +k−L if k ≥ L. 37 Observe that V + = span {ϕ 0k : k ≥ L} . 38 We study now the biorthogonality of the dual generators of V + and e V + . Setting k ∗ = max k ∗ , ˜ k ∗ = max −n + δ, − ˜n + ˜δ , 39 let us observe that {ϕ 0k : k ≥ k ∗ } and {e ϕ 0k : k ≥ k ∗ } are already biorthogonal. In order to get a pair of dual systems using our “blocks” we have therefore to match the dimensions of the spaces spanned by {φ α : α = 0, . . . , L − 1} ∪ n ϕ 0k : k = k ∗ , . . . , k ∗ − 1 o and by e φ β : β = 0, . . . , e L − 1 ∪ n e ϕ 0k : k = ˜k ∗ , . . . , k ∗ − 1 o . This requirement can be translated into an explicit relation between δ and ˜δ; indeed, we must have L − k ∗ = e L − ˜k ∗ , i.e., ˜δ − δ = e L − L + ˜n − n . 40 Since e L ≥ L, we get k ∗ = − ˜n + ˜δ. R EMARK 2. The two parameters δ and ˜δ have been introduced exactly because we want the equality of the cardinality of the sets previously indicated. On the other hand, we want to choose them as small as possible in order to minimize the perturbation due to the boundary. Thus, it will be natural to fix one, between δ and ˜δ, equal to zero and determine the other one from the relation 40. In particular, if e L − L + ˜n − n ≥ 0, we set δ = 0 and ˜δ = e L − L + ˜n − n ; in this case ˜δ e L; whereas if e L − L + ˜n − n 0, we choose ˜δ = 0 and δ = L − e L + n − ˜n . In analogy with the previous case, we will suppose ≤ δ L . 41 Let us define the spaces V B and e V B spanned by the so called boundary scaling functions as V B : = span ϕ 0k : k = 0, . . . , e L − 1 , e V B = span e ϕ 0k : k = 0, . . . , e L − 1 , 42 and the spaces V I and e V I spanned by the interior scaling functions as V I : = span ϕ 0k : k ≥ e L , e V I : = span e ϕ 0k : k ≥ e L . 43 134 L. Levaggi – A. Tabacco Thus we have V + = V B ⊕ V I , e V + = e V B ⊕ e V I . 44 Note that, if L e L, the subspace V I is strictly contained in V + see 38. In other words, some of the functions in V + which are scaling functions on supported in [0, +∞ are thought as boundary scaling functions, i.e., are included in V B . As we already observed, the basis functions of V I and e V I are biorthogonal. Recalling 35 and 37, if 0 ≤ k ≤ e L − 1 there exist coefficients α km such that ϕ 0k = P mk ∗ α km ϕ 0m ; thus, if l ≥ e L, due to the position of the supports, we have hϕ 0k , e ϕ 0l i = X mk ∗ α km Z ϕ 0m e ϕ 0,k ∗ +l−e L d x = 0 , i.e., V B ⊂ e V I ⊥ . The only functions that we have to modify in order to obtain biorthogonal systems are the border ones. The problem is to find a basis of V B , say η 0k : k = 0, . . . , e L −1 , and one of e V B , say ˜η 0l : l = 0, . . . , e L − 1 , such that hη 0k , ˜η 0l i = δ kl , k, l = 0, . . . , e L − 1 . Setting η 0k = P e L−1 m=0 d km ϕ 0m and ˜η 0k = P e L−1 m=0 ˜d km e ϕ 0m , and calling X the Gramian matrix of components X kl = hϕ 0k , e ϕ 0l i , k, l = 0, . . . , e L − 1 , 45 this is equivalent to the problem of finding two e L × e L real matrices, say D = d km and e D = ˜d km , satisfying D X e D T = I . 46 A necessary and sufficient condition for 46 to have solutions is clearly the non-singularity of X , or equivalently V B ∩ e V B ⊥ = {0}. If this is the case, there exist infinitely many couples which satisfy equation 46; indeed if we choose e D non-singular then it is sufficient to set D = X e D T −1 . We know at present of no general result establishing the invertibility of X , although it can be proved, e.g., for orthogonal systems and for systems arising from B-spline functions see Section 9 and also [16]. From now on we will assume this condition is verified and we suppose renaming if necessary that {ϕ 0k } k≥0 and {e ϕ 0l } l≥0 are dual biorthogonal bases. Let us prove that the functions ϕ 0k , k ≥ 0, form a p-stable basis of V + . P ROPOSITION 3. We have V + =    v = X k≥0 α k ϕ 0k : {α k } k∈ ∈ ℓ p    with kvk L p + ≍ k{α k } k∈ k ℓ p , ∀v ∈ V + . 47 Wavelets on the interval 135 Proof. Let v be any function in V + ; by 44 it can be written as v = v B + v I with v B = P e L−1 k=0 α k ϕ 0k and v I = P k≥e L α k ϕ 0k . The sequence {α k } k≥e L is p-summable thanks to the “inclusion” of V I in V and by 26, so the first part of the Proposition is proved. To show 47, let us set K = max |supp ϕ 0k | : k = 0, . . . , e L − 1 and note that kv B k p L p 0,K ≍ e L−1 X k=0 |α k | p kvk p L p + . 48 Indeed, since for any N ∈ ✁ \ {0}, the application x = x , . . . , x N ∈ N +1 7−→ N X n=0 x n ϕ 0n L p 0,K , defines a norm on N +1 for a proof see, e.g., [5], Proposition 6.1, and every two norms on a finite dimensional space are equivalent, the first equivalence is proven. The second inequality follows from |α k | p = Z supp ˜ ϕ 0k v x ˜ϕ 0k x d x p ≤ Z supp ˜ ϕ 0k |vx| p d x · Z supp ˜ ϕ 0k | ˜ϕ 0k x | p ′ d x p p ′ kvk p L p + . Thus, by 48 and the p-stability of ϕ 0k on the line 26, we have kvk p L p + = k v B + v I k p L p + kv B k p L p 0,K + kv I k p L p + ≍ X k≥0 |α k | p . On the other hand, we have kv I k L p 0,K = kv − v B k L p 0,K ≤ kvk L p 0,K + kv B k L p 0,K kvk L p + and so kv I k p L p + kv I k p L p 0,K + kv I k p L p [K ,+∞ = kv I k p L p 0,K + kvk p L p [K ,+∞ kvk p L p + . Then X k≥0 |α k | p = e L−1 X k=0 |α k | p + X k≥e L |α k | p ≍ kv B k p L p 0,K + kv I k p L p + kvk p L p + , and the result is completely proven. Similarly, it is possible to show that the dual basis {e ϕ 0k } k≥0 of e V + is a p ′ -stable basis. Let us introduce the isometries T j : L p + → L p + and e T j : L p ′ + → L p ′ + defined as T j f x = 2 j p f 2 j x , e T j f x = 2 j p ′ f 2 j x . 49 136 L. Levaggi – A. Tabacco and set ϕ j k = T j ϕ 0k , e ϕ j k = e T j e ϕ 0k , ∀ j, k ≥ 0. We define the j-th level scaling function spaces as V j + : = T j V + , e V j + : = e T j e V + . 50 Let us now show that these are families of refinable spaces. P ROPOSITION 4. For any j ∈ ✁ , one has the inclusions V j + ⊂ V j +1 + and e V j + ⊂ e V j +1 + . Proof. As before, we will only prove the result for the primal setting. By 50, we can restrict ourselves to j = 0 and show that V + ⊂ V 1 + . For k ≥ L, rewriting 25, one has ϕ 0k x = ϕ 0,k ∗ +k−L x = 2 1 2 − 1 p X m h m−2k ∗ +k−L T 1 ϕ 0m x . As the filter {h n } is finite, we see that the first non vanishing term in the sum corresponds to ϕ 0,k ∗ +δ+2k−L , which belongs to V + for any k ≥ L, so that the function on the left hand side belongs to V 1 + . Suppose now k L; without loss of generality we can choose p α x = x α for any α, so that, by 33, 34 and 35, one has, for x ≥ 0, 2 1 p 2x k = 2 1 p   φ k 2x + X m≥k ∗ c km ϕ 0m 2x    = ϕ 1k x + X m≥k ∗ c km T 1 ϕ 0m x . Again, using 35, ϕ 0k = 2 −k+1 p   ϕ 1k + X m≥k ∗ c km ϕ 1m    − X m≥k ∗ c km ϕ 0m = 2 −k+1 p  ϕ 1k + X m≥L c k,k ∗ +m−L ϕ 1m   − X m≥L c k,k ∗ +m−L ϕ 0m , which completes the proof. R EMARK 3. Note that, choosing the basis of monomials for ✁ L−1 , the refinement equation for the modified border functions in V B takes the form φ α = 2 −α+1 p T 1 φ α + X k≥k ∗ H α k ϕ 1k 51 where H α k = c α k 2 −α+1 p − 2 1 2 − 1 p X l≥k ∗ c α l h k−2l , 52 involving only the respective modified border function in V B 1 . Wavelets on the interval 137

4. Projection operators