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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $S^{n-1}$ be the unit [[sphere|P9CM4GQ2T0]] in $\\R^n$, and let $d\\theta$ be its surface measure.\nLet $g \\colon S^{n-1} \\times \\R \\to \\C$ be continuous.\nThe *dual Radon transform* of $g$ is the function on $\\R^n$ given by\n$$(R^* g)(x) = \\int_{S^{n-1}} g(\\theta, x \\cdot \\theta)\\, d\\theta.$$\n\n## Remarks\n\nThe pair $(\\theta, x \\cdot \\theta)$ describes the hyperplane with normal $\\theta$ through $x$, so $R^* g(x)$ integrates $g$ over all hyperplanes through $x$.\nFor $f$ continuous with compact support and $g$ continuous, Fubini's theorem gives the duality\n$$\\int_{S^{n-1}} \\int_\\R (R f)(\\theta, s)\\, g(\\theta, s)\\, ds\\, d\\theta = \\int_{\\R^n} f(x)\\, (R^* g)(x)\\, dx,$$\nwhere $R$ is the [[Radon transform|VJNZJ352PC]].\n\nExample: $g(\\theta, s) = s^2$.\nThen $R^* g(x) = \\int_{S^{n-1}} (x \\cdot \\theta)^2\\, d\\theta = \\frac{\\omega_{n-1}}{n} \\abs{x}^2$, where $\\omega_{n-1}$ is the area of $S^{n-1}$, because the $n$ integrals $\\int \\theta_j^2\\, d\\theta$ are equal and add up to $\\omega_{n-1}$.\n\nNon-example: $R^*$ is not an inverse of $R$.\nFor $n = 2$ and $f$ continuous with compact support, polar coordinates give $R^* R f(x) = 2 \\int_{\\R^2} \\frac{f(x + z)}{\\abs{z}}\\, dz$, a smoothed version of $f$, not $f$ itself.\n\nOther notations are $R^\\#$ and $R^t$; some authors divide by the area of the sphere.\n",
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