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  "body": "\n## Summary\n\nA generalized Fourier series expands a vector of an [[inner product space|0QVNCJXRY3]] along an orthogonal family, with coefficients computed by inner products.\nThe classical [[Fourier series|G12KXDHPPS]] is the case of the exponentials $e^{inx}$ in $L^2$ of the circle.\nOther cases are expansions in Legendre polynomials, Bessel functions or eigenfunctions of a Sturm–Liouville problem.\nThe partial sums are the best approximations from the span of the first terms, and the series recovers the vector when the family is complete.\n\n## Explanations\n\n### L1 human\n\nIn the plane, the coordinates of a vector along two perpendicular directions are found by dot products.\nA generalized Fourier series does the same in a space of functions, with infinitely many perpendicular directions.\nFor example, on $[-1, 1]$ the function $x^2$ equals $\\frac13 \\cdot 1 + \\frac23 \\cdot \\frac{3x^2 - 1}{2}$, a sum along the first and third Legendre polynomials.\nEach coefficient tells how much of one basic shape is present in the function.\n",
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