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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $S^{n-1}$ be the unit [[sphere|P9CM4GQ2T0]] in $\\R^n$.\nEach line in $\\R^n$ is $\\set{x + t\\theta : t \\in \\R}$ for a direction $\\theta \\in S^{n-1}$ and a point $x \\in \\theta^\\perp = \\set{y : y \\cdot \\theta = 0}$.\nLet $f \\colon \\R^n \\to \\C$ be continuous and [[integrable|X7GTF3D98F]] on every line.\nThe *X-ray transform* of $f$ is the function\n$$(P f)(\\theta, x) = \\int_\\R f(x + t\\theta)\\, dt, \\qquad \\theta \\in S^{n-1},\\ x \\in \\theta^\\perp.$$\n\n## Remarks\n\nThe same line is given by $(\\theta, x)$ and $(-\\theta, x)$, and $P f(-\\theta, x) = P f(\\theta, x)$.\nFor $f \\in L^1(\\R^n)$ the transform is defined for almost every $x \\in \\theta^\\perp$, for each $\\theta$.\nBy Fubini's theorem, $\\int_{\\theta^\\perp} P f(\\theta, x)\\, e^{-i x \\cdot \\xi}\\, dx = \\hat f(\\xi)$ for $\\xi \\in \\theta^\\perp$, where $\\hat f(\\xi) = \\int f(y)\\, e^{-i y \\cdot \\xi}\\, dy$; this is the projection-slice theorem for lines.\n\nExample: $f(y) = e^{-\\abs{y}^2}$.\nFor $x \\perp \\theta$, $\\abs{x + t\\theta}^2 = \\abs{x}^2 + t^2$, so $P f(\\theta, x) = \\sqrt{\\pi}\\, e^{-\\abs{x}^2}$.\n\nNon-example: $f(y) = (1 + \\abs{y})^{-1}$ is continuous and bounded but has no X-ray transform, since $\\int_\\R (1 + \\abs{x + t\\theta})^{-1}\\, dt$ diverges like $\\int dt / \\abs{t}$.\n\nFor $n = 2$, let $\\theta'$ be $\\theta$ rotated by a right angle.\nThen $P f(\\theta, s\\theta')$ equals the [[Radon transform|VJNZJ352PC]] $R f(\\theta', s)$, since both integrate $f$ over the line $y \\cdot \\theta' = s$.\n",
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