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  "body": "\n## Summary\n\nThe Zak transform turns a function on $\\R$ into a function of two variables by periodizing it with phase factors.\nIt is a unitary map from $L^2(\\R)$ onto $L^2([0, 1)^2)$, and its values are quasi-periodic.\nIt diagonalizes Gabor systems at critical density, which makes it the main tool behind the [[Balian–Low theorem|XVVHKCYF2Z]].\nIt has been discovered several times; Joshua Zak introduced it in solid-state physics.\n\n## Explanations\n\n### L1 human\n\nTake a function on the line and cut it into pieces of length $1$.\nStack the pieces on top of each other, and add them up with phase factors $e^{-2\\pi i k \\omega}$ that depend on a second variable $\\omega$.\nThe result depends on a position $x$ within one piece and on $\\omega$.\nIt is a Fourier series in the index $k$ of the pieces, for each fixed $x$.\nFor the box function $1_{[0, 1)}$ there is just one piece, and the Zak transform is constant $1$ on the unit square.\n",
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