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  "body": "\n## Statement\n\nLet $n \\ge 2$, and let $R$ be the [[Radon transform|VJNZJ352PC]] on the [[Schwartz space|GE020MY43M]] $\\mathcal{S}(\\R^n)$.\nLet $\\mathcal{S}(S^{n-1} \\times \\R)$ be the space of smooth functions $g(\\theta, s)$ on $S^{n-1} \\times \\R$ that decay rapidly in $s$ with all derivatives.\nThat is, each derivative of $g$ in $\\theta$ along the [[sphere|P9CM4GQ2T0]] and in $s$, times any power of $\\abs{s}$, is bounded.\nA function $g \\in \\mathcal{S}(S^{n-1} \\times \\R)$ is of the form $g = R f$ for some $f \\in \\mathcal{S}(\\R^n)$ if and only if both conditions hold:\n\n1. $g(-\\theta, -s) = g(\\theta, s)$ for all $(\\theta, s)$.\n2. For every integer $k \\ge 0$, the function $\\theta \\mapsto \\int_\\R g(\\theta, s)\\, s^k\\, ds$ is the restriction to $S^{n-1}$ of a homogeneous polynomial of degree $k$ in $\\theta = (\\theta_1, \\dots, \\theta_n)$.\n\n## Remarks\n\nNecessity is a direct computation.\nCondition 1 holds because $(\\theta, s)$ and $(-\\theta, -s)$ give the same hyperplane.\nFor condition 2, Fubini's theorem gives $\\int_\\R R f(\\theta, s)\\, s^k\\, ds = \\int_{\\R^n} f(x)\\, (x \\cdot \\theta)^k\\, dx$, and expanding $(x \\cdot \\theta)^k$ shows a homogeneous polynomial of degree $k$ in $\\theta$.\nSufficiency is the deep part.\nFollowing the projection-slice theorem, one defines $\\hat f(\\sigma\\theta)$ as the Fourier transform of $g(\\theta, \\cdot)$.\nThe moment conditions are what make this function smooth at the origin $\\xi = 0$.\n\nExample: for $n = 2$, $g(\\theta, s) = \\sqrt{\\pi}\\, e^{-s^2}$ is the Radon transform of $e^{-\\abs{x}^2}$.\nIts moments vanish for odd $k$ and are constants $c_k$ for even $k$.\nOn the circle, $c_k = c_k (\\theta_1^2 + \\theta_2^2)^{k/2}$, a homogeneous polynomial of degree $k$.\n\nNon-example: for $n = 2$, $g(\\theta, s) = \\theta_1^2 e^{-s^2}$ satisfies condition 1.\nBut for $k = 0$ its integral in $s$ is $\\sqrt{\\pi}\\, \\theta_1^2$, which is not constant on the circle.\nSo condition 2 fails, and $g$ is not a Radon transform.\n\nThese are often called the Helgason–Ludwig consistency conditions.\n\n## Explanations\n\n### L1 human\n\nNot every function of lines is the Radon transform of something.\nThe theorem lists the conditions.\nThe data must not depend on how a line is labeled, and its moments in the offset must be polynomials of the right degree in the direction.\nFor example, the total integral of the data over each parallel family of lines must be the same number for every direction, namely the total mass of the function.\n\n### L2 human\n\nIn tomography, measured data are noisy and usually violate these conditions.\nProjecting the data onto the range, or checking the conditions, helps to detect errors, to estimate unknown calibration parameters and to fill in missing data.\nThe case $k = 0$ says that the mass is the same in every direction, and $k = 1$ says that the center of mass is a single point of $\\R^n$.\nThe theorem describes the range of $R$ on $\\mathcal{S}(\\R^n)$; for other function spaces the range is described by similar conditions.\n",
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