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  "body": "\n## Definition\n\nFor $x, \\omega \\in \\R$ and $g \\in L^2(\\R)$, let $T_x g(t) = g(t - x)$ and $M_\\omega g(t) = e^{2\\pi i \\omega t} g(t)$.\nLet $g \\in L^2(\\R)$ be nonzero and let $a, b \u003e 0$.\nThe *Gabor system* $\\mathcal{G}(g, a, b)$ is the family $(M_{bn} T_{ak} g)_{k, n \\in \\Z}$.\nIt is a *Gabor frame* if it is a frame of $L^2(\\R)$: there are constants $0 \u003c A \\le B \u003c \\infty$ with\n$$A \\norm{f}^2 \\le \\sum_{k, n \\in \\Z} \\abs{\\ip{f}{M_{bn} T_{ak} g}}^2 \\le B \\norm{f}^2 \\quad \\text{for all } f \\in L^2(\\R).$$\n\n## Remarks\n\nThe coefficients $\\ip{f}{M_{bn} T_{ak} g}$ are the values $V_g f(ak, bn)$ of the [[short-time Fourier transform|3DHNP5NEWH]] on the lattice $a\\Z \\times b\\Z$.\nIf $\\mathcal{G}(g, a, b)$ is a frame, then $ab \\le 1$; for $ab = 1$ it is a frame only if it is a Riesz basis.\nFor the Gaussian $g(t) = e^{-\\pi t^2}$, the system $\\mathcal{G}(g, a, b)$ is a frame exactly when $ab \u003c 1$, a result of Lyubarskii and of Seip.\n\nExample: let $g = 1_{[0, 1)}$ and $a = b = 1$.\nThe translates $T_k g$ are the indicators of the intervals $[k, k+1)$, which tile $\\R$.\nOn each such interval the functions $e^{2\\pi i n t}$, $n \\in \\Z$, form an orthonormal basis of $L^2([k, k+1))$.\nSo $\\mathcal{G}(g, 1, 1)$ is an orthonormal basis of $L^2(\\R)$, hence a Gabor frame with $A = B = 1$.\n\nNon-example: with $g = 1_{[0, 1)}$, $a = 2$ and $b = 1$, every function in the system vanishes on $[1, 2)$.\nSo a nonzero $f$ supported in $[1, 2)$ has all coefficients $0$, and the lower frame bound fails.\n",
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