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  "body": "\n## Definition\n\nLet $k \\ge 1$ be an integer.\nThe *$k$-th Rademacher function* $r_k \\colon [0, 1) \\to \\set{1, -1}$ is given by\n$$r_k(t) = \\sgn\\big(\\sin(2^k \\pi t)\\big)$$\nwhere $\\sin(2^k \\pi t) \\ne 0$.\nAt the finitely many other points of $[0,1)$, its value is fixed so that $r_k$ is right-continuous.\nEquivalently, $r_k(t) = (-1)^{t_k}$, where $t = \\sum_{i \\ge 1} t_i 2^{-i}$ is the binary expansion of $t$ that does not end in an infinite run of $1$s.\nThe family $(r_k)_{k \\ge 1}$ is the *Rademacher system*.\n\n## Remarks\n\nThe Rademacher system is orthonormal in $L^2([0, 1))$ (see [[Lp space|SASP14MA1Z]]).\nIt is not complete: the product $r_1 r_2$ has norm $1$ and is orthogonal to every $r_k$.\nUnder [[Lebesgue measure|2Z2QVWZ262]] on $[0, 1)$, the $r_k$ are independent random variables, each equal to $1$ or $-1$ with probability $\\tfrac12$.\n\nExample: $r_2$ equals $1$ on $[0, \\tfrac14)$ and $[\\tfrac12, \\tfrac34)$ and $-1$ on $[\\tfrac14, \\tfrac12)$ and $[\\tfrac34, 1)$.\nCheck: $\\sin(4 \\pi t) \u003e 0$ exactly for $t \\in (0, \\tfrac14) \\cup (\\tfrac12, \\tfrac34)$.\n\nNon-example: the function $\\sgn(\\sin(3 \\pi t))$ is a $\\pm 1$ square wave on $[0, 1)$.\nIts sign changes are at $\\tfrac13$ and $\\tfrac23$, not at dyadic points, so it is not a Rademacher function.\n",
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