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  "body": "\n## Summary\n\nA Chebyshev series expands a function on $[-1, 1]$ as a sum $\\sum a_k T_k(x)$ of [[Chebyshev polynomials|M6EAR1JHPP]].\nUnder the substitution $x = \\cos \\theta$ it becomes the Fourier cosine series of $f(\\cos \\theta)$.\nTruncated Chebyshev series are cheap to compute and are close to minimax approximations, within a factor that grows like $\\log n$.\nSo they are central in numerical analysis.\n\n## Explanations\n\n### L1 human\n\nA Fourier series writes a periodic function as a sum of cosines and sines.\nA Chebyshev series does the same for a function on $[-1, 1]$, using the polynomials $T_k$, which are cosines in disguise.\nFor example, $x^2 = \\frac{1}{2} T_0(x) + \\frac{1}{2} T_2(x)$, since $T_0 = 1$ and $T_2 = 2x^2 - 1$.\nFor smooth functions the coefficients shrink very fast, so a few terms already give a very accurate polynomial.\n",
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