Wavelet function spaces for the half-line

140 L. Levaggi – A. Tabacco

5. Wavelet function spaces for the half-line

We now have all the tools to build the detail spaces and the wavelets on the half-line. Recalling the abstract construction, we start from level j = 0 and look for a complement space W + such that V 1 + = V + ⊕ W + note that the sum is not, in general, orthogonal and W + ⊥ e V . To this end, let us consider the basis functions of V 1 + and let us write them as a sum of a function of V + and a function which will be an element of W + . Since we have based our construction on the existence of a multilevel decomposition on the real line, we report two equations that will be largely used in the sequel see, e.g., [6]: ϕ 1k = 2 1 p − 1 2   X ˜ n ≤k−2m≤ ˜n 1 ˜h k−2m ϕ 0m + X 1−n 1 ≤k−2m≤1−n ˜g k−2m ψ 0m   , 58 ψ 0m = 2 1 2 − 1 p 1+2m− ˜ n X l=1+2m− ˜ n 1 g l−2m ϕ 1l . 59 Interior wavelets. Since V + contains the subspace V + defined in 32, V 1 + contains the subspace T 1 V + = ϕ 1k [0,+∞ : k ≥ k ∗ . Considering equation 59, let us determine the integer m such that all the indices l in the sum are greater or equal to k ∗ . This is equivalent to 2m ≥ k ∗ + ˜n 1 − 1, so we set m ≥ k ∗ + ˜n 1 − 1 2 =: m ∗ . 60 Since, see 2.9 in [11], X n∈ ˜h n h n−2k = δ 0k , ∀k ∈ ✂ , 61 it is easy to see that ˜n 1 − n is always odd; thus, by 39, m ∗ = ˜n 1 − n − 1 2 + δ 2 . 62 Let us set W I : = span n ψ 0m [0,+∞ : m ≥ m ∗ o ; 63 we observe that W I can be identified with a subspace of W , thus it is orthogonal to e V and W I ⊆ W + . The functions ψ 0m : = ψ 0m [0,+∞ are called interior wavelets. Border wavelets. Let us now call W B a generic supplementary space of W I in W + and set V B 1 = T 1 V B . P ROPOSITION 8. The dimension of the space W B is m ∗ . Wavelets on the interval 141 Proof. Let K 0 be an integer such that on [K , +∞ all non-vanishing wavelets and scaling functions are interior ones. Then V B 1 ⊕ span ϕ 1k : k ∗ ≤ k −n + 2K = h V B ⊕ span ϕ 0k : k ∗ ≤ k −n + K i ⊕ h W B ⊕ span ψ 0m : m ∗ ≤ m ≤ K − 1 i , since, by 58, the first interior wavelet used to generate ϕ 1,−n +2K is ψ 0K . Then the result easily follows. To build W B we need some functions that, added to W I , will generate both V B 1 and the interior scaling functions that cannot be obtained in 58 using V + and W I . Thanks to 51, we only have to consider the problem of generating interior scaling functions. Let us now look for the functions ϕ 1k generated by ϕ 0m , for m ≥ k ∗ , and by ψ 0m , with m ≥ m ∗ . Let us work separately on the two sums of 58: a we must have m ≥ k − ˜n 1 2. Imposing m ≥ k ∗ and seeking for integer solutions, we get k − ˜n 1 2 ≥ k ∗ = −n + δ ; 64 b similarly, we obtain m ≥ n − 1 + k2. Again, we want m ≥ m ∗ , so we must have n − 1 + k 2 ≥ m ∗ . Using 62, this means n − 1 + k 2 ≥ −n + ˜n 1 − 1 2 + δ 2 . 65 Since 64 and 65 have to be both satisfied, we obtain the following condition k ≥ −2n + ˜n 1 + 2δ − 1 = 2k ∗ + ˜n 1 − 1 . 66 Indeed, this can be seen considering all possible situations. For instance, if n and δ are even, then ˜n 1 is odd and l δ 2 m = δ 2 . If k satisfies both 64 and 65, so does k − 1; thus we can look for the least k as an even integer. In this case l k− ˜ n 1 2 m = k− ˜ n 1 2 + 1 2 and l n −1+k 2 m = n −1+k 2 + 1 2 , and 66 easily follows. The other cases are dealt with similarly. Let us set k = 2k ∗ + ˜n 1 − 1 , 67 so that span n ϕ 1k [0,+∞ : k ≥ k o ⊆ V + ⊕ W I . We are left with the problem of generating some functions of V 1 + , precisely ϕ 1k with k ∗ ≤ k k. Observe that we have to generate k − k ∗ functions using a space of dimension m ∗ = k−k ∗ 2 . In fact one can show that one out of two ϕ 1k , for k = k ∗ , . . . , k − 1, depends on the previous ones through elements of level zero. 142 L. Levaggi – A. Tabacco P ROPOSITION 9. Let e = 0 if δ is even, e = 1 if δ is odd. For all 1 ≤ m ≤ m ∗ − e, one has ϕ 1,k−2m [0,+∞ ∈ S m ⊕ V I ⊕ W I , with S m : = span n ϕ 1,k−2l+1 [0,+∞ : 1 ≤ l ≤ m o . Proof. Let us set ˜n 1 − n = 2r + 1 with r 0 recall that ˜n 1 − n is odd. Indeed, it is not difficult to see that for r = 0 there is nothing to prove. Observe that, by 58, for any l ∈ ✂ , we have ϕ 1,k−2l =2 1 p − 1 2     X n≥k ∗ −l ˜h k−2l−2n ϕ 0n + X n≥m ∗ + j δ 2 k −l ˜g k−2l−2n ψ 0n     =2 1 p − 1 2 ˜h ˜ n 1 −1 ϕ 0,k ∗ −l + ˜h ˜ n 1 −3 ϕ 0,k ∗ −l+1 + . . . + ˜g −n ψ 0,m ∗ + j δ 2 k −l + ˜g −n −2 ψ 0,m ∗ + j δ 2 k −l+1 + . . . 68 and ϕ 1,k−2l+1 =2 1 p − 1 2     X n≥k ∗ −l ˜h k−2l+1−2n ϕ 0n + X n≥m ∗ + j δ 2 k −l ˜g k−2l+1−2n ψ 0n     =2 1 p − 1 2 ˜h ˜ n 1 ϕ 0,k ∗ −l + ˜h ˜ n 1 −2 ϕ 0,k ∗ −l+1 + . . . + ˜g −n +1 ψ 0,m ∗ + j δ 2 k −l + ˜g −n −1 ψ 0,m ∗ + j δ 2 k −l+1 + . . . . 69 Let us prove the stated result by induction on m. For m = 1 we consider the linear combination h n ϕ 1,k−2 + h n +1 ϕ 1,k−1 = 2 1 p − 1 2    h n ˜h ˜ n 1 −1 + h n +1 ˜h ˜ n 1 ϕ 0,k ∗ −1 + X n≥k ∗ c 1n ϕ 0n + h n ˜g −n + h n +1 ˜g −n +1 ψ 0,m ∗ + j δ 2 k −1 + X n≥m ∗ + j δ 2 k d 1n ψ 0n     , for some coefficients c 1n and d 1n . Writing 61 with k = n − ˜ n 1 +1 2 6= 0 and X n∈ ˜g n h n−2k = 0 see 3.29 in [11] with k = −n , we have h n ˜h ˜ n 1 −1 + h n +1 ˜h ˜ n 1 = 0 , h n ˜g −n + h n +1 ˜g −n +1 = 0 . Wavelets on the interval 143 Thus we have h n ϕ 1,k−2 [0,+∞ + h n +1 ϕ 1,k−1 [0,+∞ ∈ V I ⊕ W I , and the result follows because h n 6= 0. Set now 1 m ≤ m ∗ − e. As before, we choose a certain linear combination of the scaling functions ϕ 1,k−2l and ϕ 1,k−2l+1 with 1 ≤ l ≤ m. Then we use 68, 69 to represent them through functions of level 0. More precisely h n ϕ 1,k−2m + h n +1 ϕ 1,k−2m+1 + . . . + h n +2m−2 ϕ 1,k−2 + h n +2m−1 ϕ 1,k−1 = h n ˜h ˜ n 1 −1 + h n +1 ˜h ˜ n 1 ϕ 0,k ∗ −m + h n ˜h ˜ n 1 −3 + h n +1 ˜h ˜ n 1 −2 + h n +2 ˜h ˜ n 1 −1 + h n +3 ˜h ˜ n 1 ϕ 0,k ∗ −m+1 + . . . + h n ˜h ˜ n 1 −2m+1 + h n +1 ˜h ˜ n 1 −2m+2 + . . . + h n +2m−1 ˜h ˜ n 1 ϕ 0,k ∗ −1 + X n≥k ∗ c mn ϕ 0n + h n ˜g −n + h n +1 ˜g −n +1 ψ 0,m ∗ + δ 2 −m + h n ˜g −n −2 + h n +1 ˜g −n −1 + h n +2 ˜g −n + h n +3 ˜g −n +1 ψ 0,m ∗ + δ 2 −m+1 + . . . + h n ˜g −n −2m+2 + h n +1 ˜g −n −2m+1 + . . . + h n +2m−2 ˜g −n +1 ψ 0,m ∗ + δ 2 −1 + X n≥m ∗ + δ 2 d mn ψ 0n , 70 for some c mn and d mn . The coefficients of the functions ϕ 0,k ∗ −m , . . . , ϕ 0,k ∗ −1 can be written as X n∈ h n ˜h n−2k = δ 0k , 71 with k = n − ˜ n 1 +1 2 , . . . , n − ˜ n 1 +1 2 + m − 1 = −r, · · · , −r + m − 1, respectively. Similarly, the coefficients of the functions ψ 0,m ∗ + δ 2 −m , . . . , ψ 0,m ∗ + δ 2 −1 can be written as X n∈ h n−2k ˜g n = 0 , with k = −n , · · · , −n − m + 1, respectively. Observe now that m ≤ m ∗ − e = r + δ 2 . If m ≤ r, all indices k in 71 are negative, so h n ϕ 1,k−2m + h n +1 ϕ 1,k−2m+1 [0,+∞ ∈ − h n +2 ϕ 1,k−2m+2 + h n +3 ϕ 1,k−2m+3 + . . . + h n +2m−2 ϕ 1,k−2 + h n +2m−1 ϕ 1,k−1 [0,+∞ + V I ⊕ W I 144 L. Levaggi – A. Tabacco and the result is proven by induction since h n 6= 0. If m r i.e., δ ≥ 2, we get h n ϕ 1,k−2m + h n +1 ϕ 1,k−2m+1 [0,+∞ ∈ − h n +2 ϕ 1,k−2m+2 + h n +3 ϕ 1,k−2m+3 + . . . + h n +2m−2 ϕ 1,k−2 + h n +2m−1 ϕ 1,k−1 [0,+∞ +2 1 p − 1 2 ϕ 0,k ∗ −m+r [0,+∞ + V I ⊕ W I . Therefore, by the induction hypothesis, we only have to prove that ϕ 0,k ∗ −m+r [0,+∞ ∈ S m ⊕ V I ⊕ W I . We immediately get m − r ≤ m ∗ − e − r = δ 2 , so we show that ϕ 0,k ∗ −l [0,+∞ ∈ S l , 1 ≤ l ≤ δ 2 , 72 by induction on l. If l = 1, from 69 we have ϕ 0,k ∗ −1 [0,+∞ = 2 1 2 − 1 p ˜h ˜ n 1 ϕ 1,k−1 [0,+∞ + X n≥k ∗ c 1n ϕ 0n [0,+∞ + X n≥m ∗ + δ 2 −1 d 1n ψ 0n [0,+∞ ∈ S 1 ⊕ V I ⊕ W I for some c 1n and d 1n . If l 1, using induction, we similarly get ϕ 0,k ∗ −l [0,+∞ = 2 1 2 − 1 p ˜h ˜ n 1 ϕ 1,k−2l+1 [0,+∞ + X n≥k ∗ −l+1 c l,n ϕ 0n [0,+∞ + X n≥m ∗ + δ 2 −l d l,n ψ 0n [0,+∞ ∈ S l ⊕ V I ⊕ W I again c ln and d ln are fixed coefficients. Thus we have proven 72, and this completes the proof. Using this result, setting ψ 0,m ∗ −l : = ϕ 1,k−2l+1 [0,+∞ − P ϕ 1,k−2l+1 [0,+∞ , l = 1, . . . , m ∗ , 73 and W B = {ψ 0m | m = 0, . . . , m ∗ − 1} , 74 we get W + = W B ⊕ W I , 75 Wavelets on the interval 145 with W I as defined in 63. With the same process we can build e W + = e W B ⊕ e W I . As for the scaling functions, we must find a couple of biorthogonal bases for the spaces W + and e W + . To fix notations, we suppose that e m ∗ ≤ m ∗ otherwise we only have to exchange e m ∗ and m ∗ in what follows. From the biorthogonality properties on the real line we have hψ 0m , e ψ 0n i = δ mn , ∀m, n ≥ m ∗ . Note, however, that the modified wavelets we have defined are no longer orthogonal to the inte- rior ones. In fact, from definition 73, it follows that hψ 0m , e ψ 0n i = ϕ 1,k−2m ∗ −m+1 , e ψ 0n , m = 0, . . . , m ∗ − 1 , n ≥ m ∗ . Using the refinement equation for wavelets on the real line, we easily show that hψ 0m , e ψ 0n i = 0, n = 0, . . . , e m ∗ − 1 if m ≥ ˜k + ˜n 1 − 1 2 =: m ∗ hψ 0m , e ψ 0n i = 0, m = 0, . . . , m ∗ − 1 ifn ≥ k + n 1 − 1 2 =: e m ∗ . 76 Observing that m ∗ ≥ e m ∗ , it is sufficient to find two m ∗ ×m ∗ matrices E = e mr and e E = ˜e ns such that m ∗ −1 X r=0 e mr ψ 0r , m ∗ −1 X s=0 ˜e ns e ψ 0s + = δ mn , ∀m, n = 0, . . . , m ∗ − 1 . Calling Y the m ∗ × m ∗ matrix of components Y mn = hψ 0m , e ψ 0n i, this condition is equivalent to E Y e E T = I . Again, it is enough to prove that the matrix Y is non-singular. In fact this follows from the assumed invertibility of the matrix X defined in 45; since det Y 6= 0 iff W + ∩ e W + ⊥ = {0} , we immediately get the result observing that W + ∩ e W + ⊥ ⊂ V 1 + ∩ e V ⊥ 1 = {0} . Moreover W + ⊂ e V ⊥ ; indeed, for any v ∈ L p + , we have hv − P v, e ϕ 0k i = ˘v 0k − X l≥0 ˘v 0l hϕ 0l , e ϕ 0k i = 0 . Finally, for any j ∈ ✁ , we set W j + = T j W + and e W j + = e T j e W + ; 77 146 L. Levaggi – A. Tabacco setting ψ j m = T j ψ 0m , e ψ j m = e T j e ψ 0m for every j, m ∈ ✁ , it is easy to check that the biorthog- onality relations hψ j m , e ψ j ′ n i = δ j j ′ δ mn , ∀ j, j ′ , m, n ≥ 0 , hold. Moreover, with a proof similar to the one of Proposition 3, one has P ROPOSITION 10. The bases 9 j = {ψ j m : m ∈ ✁ } of W j + , are uniformly p-stable bases and the bases e 9 j = e ψ j m : m ∈ ✁ of e W j + , are uniformly p ′ -stable bases for all j ≥ 0. Moreover, let us state a characterization theorem for Besov spaces based on the biorthogonal multilevel decomposition V j + , e V j j ≥0 as described in Section 3. With the notation of the Introduction, for 1 p, q +∞, let us set X s pq : = B s pq + if s ≥ 0 , B −s p ′ q ′ + ′ if s 0 , and denote by + the space of distributions. T HEOREM 3. Let ϕ ∈ B s pq + for some s 0, 1 p, q +∞. For all s ∈ S := − mins , L, mins , L \ {0}, the following characterization holds: X s pq =              n v ∈ L p + : P j ≥0 2 sq j P k≥0 |v j k | p q p +∞ o if s 0 , n v ∈ + : v ∈ X s pq , for some s ∈ S and P j ≥0 2 sq j P k≥0 |v j k | p q p +∞ o if s 0 , where v j k = ˆv j k = hv, e ψ j k i if s 0 , ˆev j k = hv, ψ j k i if s 0 . In addition, for all s ∈ S and v ∈ X s pq , we have |v| X s pq ≍    X j ≥0 2 sq j   X k≥0 |v j k | p   q p    1q . 78 Finally, if p = q = 2, the characterization and the norm equivalence hold for all index s ∈ − mins , L, mins , L. Proof. It is sufficient to apply Theorems 1 and 2 since the Bernstein- and Jackson- type in- equalities have been proven in Proposition 6 and 7, respectively and remember the interpolation result 1. R EMARK 4. It is possible to obtain a characterization result using the same representation of a function v ∈ X s pq for both positive and negative s. Indeed, if ϕ ∈ B s pq , e ϕ ∈ B ˜s pq , given any s ∈ − min˜s , e L, mins , L \ {0} and v = P j,k≥0 ˆv j k ψ j k ∈ X s pq , we have the same norm equivalence as in 78 with ˆv j k instead of v j k see, e.g., [14]. We also observe that, in general, ˜s s and thus Theorem 3 gives characterization for a larger interval. Wavelets on the interval 147

6. Boundary values of scaling functions and wavelets