Haar Transform 0.1. Haar Wavelets on R. Definition 0.1. Let J, K ∈ Z. Define
Haar transform 0.1. Haar wavelets on R. Definition 0.1. Let j, k Z. Define the interval I to be ∈ j,k −j −j Ij,k = [2 k, 2 (k + 1)). We say that (j, k) =(j′, k′) if either j = j′ or k = k′. 6 6 6 Lemma 0.2. Let j, j′,k,k′ Z be such that (j, k) = (j′, k′). Then, one of the ∈ 6 following is true: (i) I I ′ ′ = , or j,k ∩ j ,k ∅ (ii) I I ′ ′ , or j,k ⊂ j ,k (iii) I ′ ′ I . j ,k ⊂ j,k Given the integers j, k Z and the interval I , we define Il (resp., Ir ) to be the ∈ j,k j,k j,k left half of Ij,k (resp., the right half of Ij,k). Thus, l −j −j −j−1 Ij,k = [2 k, 2 k + 2 ), r −j −j−1 −j Ij,k = [2 k + 2 , 2 (k + 1)). It is not difficult to see that: l r Ij,k = Ij+1,2k and Ij,k = Ij+1,2k+1. With this observation in mind, we have the following lemma. ′ ′ ′ ′ Lemma 0.3. Let j, j ,k,k Z be such that (j, k) =(j , k ). If I I ′ ′ then ∈ 6 j,k ⊂ j ,k l (i) either I I ′ ′ j,k ⊂ j ,k r (ii) or I I ′ ′ . j,k ⊂ j ,k Definition 0.4. The system of Haar scaling functions is defined as follows: Let χ[0,1) be the characteristic function of interval [0, 1), i.e., 1 x [0, 1), χ (x)= ∈ [0,1) 0 otherwise.
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