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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $g$ be a nonzero function in the [[Schwartz space|CG9JEATW3Q]] $\\mathcal{S}(\\R^d)$, called the window.\nFor a [[tempered distribution|AES2T5JPFP]] $f \\in \\mathcal{S}'(\\R^d)$, let $V_g f$ be its [[short-time Fourier transform|3DHNP5NEWH]]:\n$$V_g f(x, \\omega) = \\int_{\\R^d} f(t) \\, \\overline{g(t - x)} \\, e^{-2\\pi i t \\cdot \\omega} \\, dt, \\quad x, \\omega \\in \\R^d,$$\nunderstood as the action of $f$ on the Schwartz function $t \\mapsto \\overline{g(t - x)} e^{-2\\pi i t \\cdot \\omega}$.\nLet $1 \\le p, q \u003c \\infty$.\nThe *modulation space* $M^{p,q}(\\R^d)$ is the set of $f \\in \\mathcal{S}'(\\R^d)$ with\n$$\\norm{f}_{M^{p,q}} = \\left( \\int_{\\R^d} \\left( \\int_{\\R^d} \\abs{V_g f(x, \\omega)}^p \\, dx \\right)^{q/p} d\\omega \\right)^{1/q} \u003c \\infty.$$\nIf $p = \\infty$ or $q = \\infty$, the corresponding integral is replaced by an essential supremum.\n\n## Remarks\n\nThe space does not depend on the choice of the window $g$, and different windows give equivalent norms.\nWith this norm $M^{p,q}(\\R^d)$ is a Banach space.\nOne writes $M^p = M^{p,p}$.\nSpecial cases are $M^{2} = L^2(\\R^d)$ with an equivalent norm, and $M^{1}$, the Feichtinger algebra.\nWeighted versions $M^{p,q}_m$ insert a weight $m(x, \\omega)$ into the integrand.\n\nExample: let $d = 1$ and $\\varphi(t) = e^{-\\pi t^2}$, and take $g = \\varphi$.\nCompleting the square in $t$ gives\n$$\\abs{V_\\varphi \\varphi(x, \\omega)} = 2^{-1/2} e^{-\\pi (x^2 + \\omega^2)/2}.$$\nThis is integrable to every power in $x$ and in $\\omega$, so $\\varphi \\in M^{p,q}(\\R)$ for all $p, q$.\n\nNon-example: the constant function $f = 1$ is not in $M^{p,q}(\\R)$ for $p \u003c \\infty$.\nHere $\\abs{V_g 1(x, \\omega)} = \\abs{\\hat{\\overline{g}}(\\omega)}$ does not depend on $x$, so the inner integral is infinite whenever $\\hat{\\overline{g}}(\\omega) \\ne 0$.\nIt does lie in $M^{\\infty, 1}(\\R)$.\n",
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