Jackson and Bernstein inequalities

Wavelets on the interval 137

4. Projection operators

Following the guiding lines of the abstract setting, we will define a sequence of continuous linear operators P j : L p + → V j for j ∈ ✁ , satisfying 7, 8 and 9. By 50, it is obvious that the definition of P gives naturally the complete sequence, by posing P j = T j ◦ P ◦ T −1 j , where T j is the isometry defined in 49. For v ∈ L p + , let us set P v = X k≥0 ˘v 0k ϕ 0k , with ˘v 0k = Z v x e ϕ 0k x d x . We will first prove that P is a well-defined and continuous operator. P ROPOSITION 5. We have kP v k p L p + ≍ X k≥0 | ˘v 0k | p kvk p L p + , ∀v ∈ L p + . Proof. Let us write P v = e L−1 X k=0 ˘v 0k ϕ 0k + X k≥e L ˘v 0k ϕ 0k . Observe that, by the H¨older inequality, e L−1 X k=0 | ˘v 0k | p ≤ kvk p L p +   e L−1 X k=0 ke ϕ 0k k p L p′ +   = C kvk p L p + . Thus, by the p-stability property on the line 26, X k≥0 | ˘v 0k | p kvk p L p + . This implies P v ∈ V + and the result follows by 47. Since T j is an isometry for any j , we immediately get 7. Equality 8, follows by the biorthogonality of the systems. Equality 9 is a consequence of the inclusion V j + ⊂ V j +1 + , proven in Proposition 4. Similarly, one can define a sequence of dual operators, e P j : L p ′ + → e V j + , satisfying the same properties of the primal sequence.

4.1. Jackson and Bernstein inequalities

The main property of the original decomposition on the real line we have inherited, is the way polynomials are reconstructed through basis functions. This is what we call the approximation property, and it is fundamental for the characterization of functional spaces. In this section we will exploit it to prove Bernstein- and Jackson-type inequalities on the half-line and then apply the general characterization results of the abstract setting Theorems 1 and 2. As in Section 2.2, we consider a Banach subspace Z of L p + and suppose that the scaling function ϕ belongs to Z . In the following, Z will be the Besov space B s pq + , with s 0 and 1 p, q +∞. 138 L. Levaggi – A. Tabacco P ROPOSITION 6. For any 0 ≤ s ≤ s , the Bernstein-type inequality |v| B s pq + 2 j s kvk L p + , ∀v ∈ V j + , ∀ j ∈ ✁ , 53 holds. Proof. Applying the definition of the operator T j , it is easy to see that |T j v | B s pq + = 2 j s |v| B s pq + , ∀v ∈ B s pq + , ∀ j ∈ ✁ ; 54 so, by 50, it is enough to prove the inequality for j = 0. Proceeding as in Proposition 3, we choose v ∈ V + and write v = v B + v I . Let us estimate separately the semi-norms of the two terms. We have |v B | p B s pq + = e L−1 X k=0 ˘v 0k ϕ 0k p B s pq + ≤   e L−1 X k=0 | ˘v 0k | |ϕ 0k | B s pq +   p e L−1 X k=0 | ˘v 0k | p , the constants depending on the semi-norms of the basis functions and the equivalence of norms in e L . Observe that v I is an element of V = V ; using the Bernstein inequality 30 and the p-stability on the line 26, we get |v I | B s pq + kv I k L p ≍   X k≥e L | ˘v 0k | p   1 p . Thus, by 47, |v| p B s pq + |v B | p B s pq + + | v I | p B s pq + k{ ˘v 0k } k∈ k p ℓ p kvk L p + . Next, we prove the generalized Jackson inequality, following the same ideas used in show- ing the analogous property 31 on the real line see [5]. P ROPOSITION 7. For each 0 ≤ s mins , L, we have kv − P j v k L p + 2 − j s |v| B s pq + , ∀v ∈ B s pq + , ∀ j ∈ ✁ . 55 Proof. As before see 54, it is enough to prove that kv − P v k L p + |v| B s pq + , ∀v ∈ B s pq + . Let us divide the half-line into unitary intervals, + = ∪ l≥0 I l where I l = [l, l +1], and estimate kv− P v k L p I l . Recalling that polynomials up to degree L −1 are locally reconstructed through the basis of V + , it is easy to see that for any q ∈ ✁ L−1 there exists v q in V + such that v q = q on I l . Then, using 8, we have kv − P v k L p I l = kv − q + P q − P v k L p I l ≤ kv − qk L p I l + kP v − qk L p I l . Wavelets on the interval 139 Moreover, from the compactness of the supports of the basis functions, setting R l = {k ∈ ✁ : supp ϕ 0k ∩ I l 6= ∅} and J l = ∪ k∈R l supp e ϕ 0k , for any f ∈ L p + , one gets kP f k p L p I l = Z I l X k∈R l ˘f 0k ϕ 0k x p d x ≤ max k∈R l k f k L p supp e ϕ 0k ke ϕ 0k k L p′ + p Z I l X k∈R l |ϕ 0k x | p d x k f k p L p J l . Thus kv − P v k L p I l kv − qk L p J l , ∀q ∈ ✁ L−1 . Taking the infimum over all q ∈ ✁ L−1 , we end up with kv − P v k L p + X l≥0 inf q∈ L−1 kv − qk L p J l . 56 A local version of Whitney’s Theorem see [21] for a given interval of the real line J , states that, for any v ∈ L p J , inf q∈ L−1 kv − qk L p J w L v, J , 57 where w L v, J = 1 2 |J | Z | J | −| J | dh Z J Lh |1 L h v | p d x 1 p and J s = {x ∈ J : x + s ∈ J }. Observing that h ∗ = |J l | independent of l and using 57, we have kv − P v k p L p + X l≥0 w L v, J l p = 1 2h ∗ Z h ∗ −h ∗ dh X l≥0 Z J l Lh |1 L h v | p d x sup |h|≤h ∗ Z + |1 L h v | p d x = ω L p v, h ∗ p , where ω L p is the modulus of smoothness see [21]. To conclude the proof, it is enough to observe that, for any 1 q ∞, ω L p v, h ∗   X j ∈ 2 j sq ω L p v, 2 − j q   1q ≍ |v| B s pq + . Since B s pq + is dense in L p + , the following property immediately follows. C OROLLARY 1. The union ∪ j ∈ V j + is dense in L p + . 140 L. Levaggi – A. Tabacco

5. Wavelet function spaces for the half-line