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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\C$ be a [[continuous|0P27W9RW0G]] [[periodic function|089KWVYD69]] with period $2\\pi$.\nFor $n \\ge 0$ let $S_n f(x) = \\sum_{k=-n}^{n} c_k e^{ikx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]], where $c_k = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(t) e^{-ikt} \\, dt$.\nFor $N \\ge 1$ let\n$$\\sigma_N f(x) = \\frac{1}{N} \\sum_{n=0}^{N-1} S_n f(x)$$\nbe their [[Cesàro means|0P0DJHBGBB]].\nThen $\\sigma_N f \\to f$ [[uniformly|08FPABDQ4C]] on $\\R$ as $N \\to \\infty$.\n\n## Remarks\n\nThe proof writes $\\sigma_N f(x) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} F_N(t) f(x - t) \\, dt$ with the [[Fejér kernel|DCCA2T9GKH]] $F_N$.\nSince $F_N \\ge 0$ and $F_N$ has mean $1$, the error $\\sigma_N f(x) - f(x)$ is a weighted average of $f(x - t) - f(x)$.\nFor small $\\abs{t}$ this is small by uniform continuity of $f$, and for $\\abs{t} \\ge \\delta$ the weight $F_N(t)$ is uniformly small for large $N$.\n\nExample: the theorem shows that trigonometric polynomials are uniformly dense in the continuous $2\\pi$-periodic functions, since each $\\sigma_N f$ is one.\nIt also shows that two continuous periodic functions with the same Fourier coefficients are equal.\n\nContinuity is needed for uniform convergence: each $\\sigma_N f$ is continuous, so a uniform limit of them is continuous, and a function with a jump cannot be such a limit.\nThe theorem cannot be improved to the partial sums $S_n f$ themselves: there are continuous periodic functions whose Fourier series diverges at a point.\n\n## Explanations\n\n### L1 human\n\nThe Fourier series of a continuous periodic function may fail to converge.\nFejér's theorem says that averaging its partial sums always repairs this: the averages converge to the function, evenly over the whole line.\nSo a continuous periodic function is always recovered from its Fourier coefficients, by Cesàro summation of the series.\n\n### L2 human\n\nFejér's theorem is the model case of summability methods applied to Fourier series.\nIts proof uses only three properties of the Fejér kernel: positivity, unit mean and concentration at $0$.\nAny kernel with these properties gives a similar theorem, such as [[Abel–Poisson summation|RZZ78GXP5F]] with the Poisson kernel.\nVersions for integrable functions say that $\\sigma_N f \\to f$ in mean and at every point of continuity.\n",
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