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  "body": "\n## Example\n\nThe *Haar wavelet* is the function $\\psi \\colon \\R \\to \\R$ with\n$$\\psi(x) = \\begin{cases} 1 \u0026 0 \\le x \u003c \\tfrac12, \\\\ -1 \u0026 \\tfrac12 \\le x \u003c 1, \\\\ 0 \u0026 \\text{otherwise.} \\end{cases}$$\nIt is an orthonormal [[wavelet|4PZ35C0GV3]]: the functions $\\psi_{j,k}(x) = 2^{j/2} \\psi(2^j x - k)$, $j, k \\in \\Z$, form an orthonormal basis of $L^2(\\R)$.\n\nCheck of orthonormality.\n$\\psi_{j,k}$ is supported on the dyadic interval $I_{j,k} = [k 2^{-j}, (k+1) 2^{-j})$ and $\\norm{\\psi_{j,k}}_2^2 = 2^j \\cdot 2^{-j} = 1$.\nTwo dyadic intervals are either disjoint or nested.\nIf they are disjoint, the product $\\psi_{j,k} \\psi_{j',k'}$ is $0$.\nIf $I_{j',k'}$ lies strictly inside $I_{j,k}$, then it lies inside one half of $I_{j,k}$, where $\\psi_{j,k}$ is constant, and $\\psi_{j',k'}$ has integral $0$.\n\nCompleteness comes from the Haar [[multiresolution analysis|1GHQMBR7DE]], whose space $V_j$ consists of the functions in $L^2(\\R)$ that are constant on the intervals $I_{j,k}$.\nThe [[scaling function|YH4BERB952]] is $\\varphi = 1_{[0, 1)}$, with $\\varphi(x) = \\varphi(2x) + \\varphi(2x - 1)$ and $\\psi(x) = \\varphi(2x) - \\varphi(2x - 1)$.\nOn $[0, 1]$, the constant $1$ together with the $\\psi_{j,k}$ with $j \\ge 0$ and $0 \\le k \u003c 2^j$ is the [[Haar system|N0B51FAA9S]].\n\nThe Haar wavelet has exactly one [[vanishing moment|7QERXM0C52]]: $\\int \\psi = 0$, while $\\int_\\R x \\psi(x) \\, dx = \\tfrac18 - \\tfrac38 = -\\tfrac14$.\nIt is discontinuous, so for smooth functions its expansions converge slowly compared with smoother wavelets.\n\n## Explanations\n\n### L1 human\n\nThe Haar wavelet is a single up-step followed by a down-step.\nIts squeezed and shifted copies work at every scale and place.\nEach one measures how the average of a function on the left half of a small interval differs from its average on the right half.\nAlfréd Haar used these functions in 1910, long before the word wavelet existed.\nIt is the first member of the family of Daubechies wavelets.\n",
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