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  "body": "\n## Definition\n\nLet $f$ be a function in the [[Schwartz space|CG9JEATW3Q]] $\\mathcal{S}(\\R)$.\nThe *Zak transform* of $f$ is the function $Zf \\colon \\R^2 \\to \\C$,\n$$Zf(x, \\omega) = \\sum_{k \\in \\Z} f(x + k) \\, e^{-2\\pi i k \\omega}.$$\nFor general $f \\in L^2(\\R)$ (see [[Lp space|SASP14MA1Z]]), $Zf$ is defined by continuous extension, with the series converging in $L^2([0, 1)^2)$.\n\n## Remarks\n\nFor fixed $x$, $Zf(x, \\cdot)$ is a [[Fourier series|G12KXDHPPS]] with coefficients $f(x + k)$.\nThe map $Z$ extends to a unitary operator from $L^2(\\R)$ onto $L^2([0, 1)^2)$.\nThe values are quasi-periodic: $Zf(x, \\omega + 1) = Zf(x, \\omega)$ and $Zf(x + 1, \\omega) = e^{2\\pi i \\omega} Zf(x, \\omega)$.\nThe second rule follows by putting $j = k + 1$ in the series.\nSome authors include a lattice parameter $a \u003e 0$ and use $f(x + ak)$.\n\nExample: let $f = 1_{[0, 1)}$.\nFor $x \\in [0, 1)$ only the term $k = 0$ is nonzero, so $Zf(x, \\omega) = 1$ on $[0, 1)^2$.\nThen $\\norm{Zf}^2_{L^2([0,1)^2)} = 1 = \\norm{f}_2^2$, as unitarity requires.\n\nNon-example: the map $f \\mapsto \\sum_{k \\in \\Z} f(x + k)$ without phase factors is the periodization of $f$.\nIt loses information, for instance it sends $1_{[0, 1/2)} - 1_{[1, 3/2)}$ to $0$.\n",
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