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  "body": "\n## Summary\n\nThe Haar system is an orthonormal basis of $L^2([0, 1])$.\nIt consists of the constant function $1$ and the rescaled, shifted copies of one step function.\nThis step function is $1$ on the left half and $-1$ on the right half of $[0, 1)$.\nAlfréd Haar introduced it in 1910, as an orthonormal system in which every continuous function has a uniformly convergent expansion.\nIt is the simplest example of a [[wavelet|4PZ35C0GV3]] basis, and it is a [[Schauder basis|528G11SDWR]] of $L^p([0, 1])$ for $1 \\le p \u003c \\infty$.\n\n## Explanations\n\n### L1 human\n\nStart with the average of a function over $[0, 1]$.\nThen record how the average over the left half differs from the average over the right half.\nThen do the same inside each half, inside each quarter, and so on.\nEach step uses one function that is $+1$ on one half of a dyadic interval and $-1$ on the other half, scaled to have length $1$ in $L^2$.\nThese numbers rebuild the function step by step, with finer and finer detail.\n",
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