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  "body": "\n## Summary\n\nLet $S_N f$ be the partial sums of the [[Fourier series|G12KXDHPPS]] of a periodic function $f$.\nThe convergence of Fourier series is the question whether, and in which sense, $S_N f$ tends to $f$.\nThe answer depends on the mode of convergence: at a point, uniformly, almost everywhere, or in the norm of $L^p$.\nSmoothness or bounded variation of $f$ gives pointwise convergence, while mere continuity does not.\nIn $L^2$ the partial sums always converge, and by Carleson's theorem they also converge almost everywhere for $f \\in L^2$.\n\n## Explanations\n\n### L1 human\n\nAdding more and more waves of the Fourier series should give a better and better picture of the function, but this can fail.\nFor a smooth function the picture improves everywhere at once.\nAt a jump the sums settle at the midpoint of the jump and overshoot nearby.\nThere even exist continuous functions whose Fourier series diverge at some point.\nMeasured by the average squared error instead of point by point, the picture always improves, for every square integrable function.\n",
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