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  "body": "\n## Definition\n\nLet $T$ be an [[integral transform|YCK709QB06]], defined on a class $A$ of functions, and assume that $T$ is injective on $A$.\nThe *inverse transform* of $T$ is the map $T^{-1} \\colon T(A) \\to A$ that sends $F$ to the unique $f \\in A$ with $Tf = F$.\nIt is the [[inverse function|0W766JAPSN]] of $T$ viewed as a bijection from $A$ onto $T(A)$.\n\n## Remarks\n\nExample: the Fourier transform $\\hat f(\\xi) = \\int_\\R f(x) e^{-2\\pi i x \\xi}\\, dx$ is a bijection of the Schwartz space onto itself, and its inverse is $f(x) = \\int_\\R \\hat f(\\xi) e^{2\\pi i x \\xi}\\, d\\xi$.\nSo here the inverse transform is again an integral transform, with kernel $e^{2\\pi i x \\xi}$.\nNon-example: the transform $Tf(u) = \\int_0^1 f(t)\\, dt$ with kernel $K(t, u) = 1$ on $[0,1] \\times \\R$ has no inverse transform on the continuous functions, since $f(t) = 0$ and $f(t) = 2t - 1$ both have $Tf = 0$.\nWhen the inverse is an integral transform with kernel $H$, one writes $f(t) = \\int F(u)\\, H(u, t)\\, du$, and $(f, F)$ is a transform pair.\n",
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