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  "body": "\n## Summary\n\nThe Legendre integral transform sends a function $f$ on $[-1, 1]$ to the sequence of numbers $\\int_{-1}^{1} P_n(x) f(x)\\, dx$, where $P_n$ is the $n$-th [[Legendre polynomial|QFYGJVA996]].\nThese numbers are, up to the factors $\\frac{2n+1}{2}$, the coefficients of the [[Fourier–Legendre series|RK2G9HN63E]] of $f$, so the series is the inverse transform.\nIt plays for the Legendre differential operator the role that Fourier coefficients play for $d^2/dx^2$.\nIt turns Legendre's differential equation into an algebraic equation.\nIt should not be confused with the Legendre transformation of convex analysis.\n\n## Explanations\n\n### L1 human\n\nThe Legendre polynomials $1, x, \\frac{3x^2 - 1}{2}, \\dots$ are like a set of perpendicular directions for functions on $[-1, 1]$.\nThe transform measures how much of each direction a function contains.\nFor example, the function $f(x) = x$ is exactly $P_1$, so its transform is $\\frac{2}{3}$ at $n = 1$ and $0$ at every other $n$.\n",
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