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  "body": "\n## Definition\n\nLet $\\psi \\colon \\R \\to \\R$ be the [[step function|0AJXYH4DDQ]] with $\\psi = 1$ on $[0, \\tfrac12)$, $\\psi = -1$ on $[\\tfrac12, 1)$ and $\\psi = 0$ elsewhere.\nFor integers $n \\ge 0$ and $0 \\le k \u003c 2^n$, let\n$$h_{n,k}(t) = 2^{n/2} \\, \\psi(2^n t - k), \\qquad t \\in [0, 1].$$\nThe *Haar system* on $[0, 1]$ is the family consisting of the constant function $1$ and all the functions $h_{n,k}$.\n\n## Remarks\n\nThe function $h_{n,k}$ is supported on the dyadic interval $[k 2^{-n}, (k+1) 2^{-n}]$, where it equals $2^{n/2}$ on the left half and $-2^{n/2}$ on the right half.\nThe Haar system is an [[orthonormal basis|0S9GGTR0EX]] of $L^2([0, 1])$ (see [[Lp space|SASP14MA1Z]]).\nOn the whole line, the functions $2^{j/2}\\psi(2^j t - k)$ with $j, k \\in \\Z$ form an orthonormal basis of $L^2(\\R)$.\n\nExample: $h_{1,0}$ and $h_{1,1}$ are orthogonal, since their supports $[0, \\tfrac12]$ and $[\\tfrac12, 1]$ meet in one point.\nAlso $h_{0,0}$ and $h_{1,0}$ are orthogonal: on $[0, \\tfrac12]$ the function $h_{0,0}$ is constant $1$, and $h_{1,0}$ has integral $0$ there.\nAnd $\\norm{h_{1,0}}^2 = 2 \\cdot \\tfrac12 = 1$.\n\nNon-example: the family of all $h_{n,k}$ without the constant function $1$ is orthonormal but not a basis of $L^2([0, 1])$.\nIndeed, the constant $1$ is orthogonal to every $h_{n,k}$, since each $h_{n,k}$ has integral $0$.\n",
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