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  "body": "\n## Definition\n\nLet $L$ be a linear differential operator on an interval or a box $\\Omega$, and consider $L u = f$ with boundary or periodicity conditions.\nFix a family of smooth functions $\\varphi_k$ on all of $\\Omega$, each nonzero on most of $\\Omega$.\nTypical choices are $e^{ikx}$ for periodic problems, [[Chebyshev polynomials|M6EAR1JHPP]] or [[Legendre polynomials|QFYGJVA996]] otherwise.\nLet $I_N$ be a finite set of $n$ indices.\nA *spectral method* seeks $u_N = \\sum_{k \\in I_N} a_k \\varphi_k$ and fixes the $a_k$ by one of three conditions:\n\n1. Galerkin: $\\ip{L u_N - f}{\\varphi_j} = 0$ for all $j \\in I_N$, with the $\\varphi_k$ satisfying the boundary conditions;\n2. tau: the same equations for all but $m$ indices $j$, plus the $m$ boundary conditions written as equations on the $a_k$;\n3. collocation: $L u_N(x_j) = f(x_j)$ at chosen points $x_j$, together with the boundary conditions, with $n$ equations in all.\n\nHere $\\ip{\\cdot}{\\cdot}$ is the $L^2$ inner product, with the weight of the chosen polynomial family.\n\n## Remarks\n\nExample: on $[0, 2\\pi]$ with periodic conditions, solve $-u'' + u = f$ with the Fourier basis $e^{ikx}$, $\\abs{k} \\le N$.\nThe Galerkin condition gives $(k^2 + 1) a_k = \\hat f_k$, where $\\hat f_k$ are the [[Fourier coefficients|G12KXDHPPS]] of $f$.\nSo $u_N$ is the truncation of the exact Fourier series of $u$.\nIf $f$ is analytic, $\\hat f_k$ decays like $e^{-c \\abs{k}}$, and so does the error of $u_N$.\n\nNon-example: the finite element method with hat functions uses basis functions that vanish outside two mesh cells.\nThese are not global functions, so it is not a spectral method; its error for smooth $u$ decreases only like a fixed power of $1/N$.\n",
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