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  "body": "\n## Definition\n\nLet $S^2 = \\set{x \\in \\R^3 : \\abs{x} = 1}$ be the unit [[sphere|P9CM4GQ2T0]].\nFor $\\theta \\in S^2$, let $C_\\theta = \\set{x \\in S^2 : x \\cdot \\theta = 0}$ be the great circle with pole $\\theta$, with its arc length measure $ds$.\nLet $f \\colon S^2 \\to \\C$ be continuous.\nThe *Funk transform* of $f$ is the function on $S^2$ given by\n$$(F f)(\\theta) = \\int_{C_\\theta} f(x)\\, ds(x).$$\n\n## Remarks\n\nSince $C_{-\\theta} = C_\\theta$, $F f$ is even.\nIf $f$ is odd, $f(-x) = -f(x)$, then $F f = 0$, because $C_\\theta$ is symmetric under $x \\mapsto -x$.\nOn a [[spherical harmonic|3F32Y21HP4]] $Y_k$ of degree $k$, $F Y_k = 2\\pi P_k(0)\\, Y_k$, where $P_k$ is the Legendre polynomial; $P_k(0) = 0$ for odd $k$ and $P_k(0) \\ne 0$ for even $k$.\nHence $F$ is injective on even continuous functions.\n\nExample: $f = 1$ gives $F f = 2\\pi$, the length of a great circle; this is the case $k = 0$, with $P_0(0) = 1$.\n\nExample: $f(x) = x_3$ is odd, so $F f = 0$, although $f \\ne 0$.\n\nNon-example: the integral of $f$ over a small circle $\\set{x \\in S^2 : x \\cdot \\theta = c}$ with $0 \u003c \\abs{c} \u003c 1$ is not a value of the Funk transform, which uses only circles through the center of the sphere.\n\nThe same definition works on $S^{n-1}$ with the great subspheres $S^{n-1} \\cap \\theta^\\perp$.\n",
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