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  "body": "\n## Definition\n\nFor $k \\ge 1$, let $r_k$ be the $k$-th [[Rademacher function|51GRFQ4WY6]] on $[0, 1)$.\nThus $r_k(t) = (-1)^{t_k}$, where $t_k$ is the $k$-th binary digit of $t$, taken from the expansion that does not end in an infinite run of $1$s.\nLet $n \\ge 0$ be an integer with binary expansion $n = \\sum_{k \\ge 0} b_k 2^k$, $b_k \\in \\set{0, 1}$.\nThe *Walsh function* of index $n$ (in Paley order) is\n$$w_n = \\prod_{k \\ge 0} r_{k+1}^{\\,b_k},$$\na finite product, with $w_0 = 1$.\n\n## Remarks\n\nThe family $(w_n)_{n \\ge 0}$ is the *Walsh system*.\nIt is an [[orthonormal basis|0S9GGTR0EX]] of $L^2([0, 1))$ (see [[Lp space|SASP14MA1Z]]).\nOther orderings of the same functions are in use, such as the Walsh order by number of sign changes.\n\nExample: $n = 3 = 1 + 2$, so $w_3 = r_1 r_2$.\nHere $r_1 = 1$ on $[0, \\tfrac12)$ and $-1$ on $[\\tfrac12, 1)$, while $r_2 = 1$ on $[0, \\tfrac14) \\cup [\\tfrac12, \\tfrac34)$ and $-1$ elsewhere.\nSo $w_3$ is $1, -1, -1, 1$ on the four quarters of $[0, 1)$, and $\\int_0^1 w_3 = 0$, as orthogonality to $w_0$ requires.\n\nNon-example: the function $\\cos(2\\pi t)$ is orthogonal to many Walsh functions, but it is not a Walsh function, since it takes values other than $\\pm 1$.\n",
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