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  "body": "\n## Summary\n\nA Gabor frame is a frame of $L^2(\\R)$ made of time shifts and frequency shifts of one window function $g$.\nThe shifts run over a lattice $a\\Z \\times b\\Z$ of the time–frequency plane.\nIt gives a stable, discrete version of the [[short-time Fourier transform|3DHNP5NEWH]].\nA function is determined by the samples of $V_g f$ on the lattice, and can be rebuilt from them.\nA Gabor frame can exist only when $ab \\le 1$.\nAt $ab = 1$, the [[Balian–Low theorem|XVVHKCYF2Z]] shows that the window cannot be well localized in both time and frequency.\n\n## Explanations\n\n### L1 human\n\nPick one short wave packet, say a bell curve.\nMake copies of it shifted in time by multiples of $a$ and shifted in pitch by multiples of $b$.\nIf these copies are dense enough in the time–frequency plane, every signal can be written as a stable combination of them.\nThis is how one turns a continuous time–frequency picture into a grid of numbers that still holds all the information.\n",
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