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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\C$ be a [[periodic function|0ATF6Q0633]] of period $2\\pi$ that has [[bounded variation|A8251BHCMP]] on $[-\\pi, \\pi]$.\nLet $S_N f$ be the partial sums of its [[Fourier series|G12KXDHPPS]].\n\n1. For every $x \\in \\R$ the one-sided limits $f(x^+)$ and $f(x^-)$ exist, and\n$$\\lim_{N \\to \\infty} S_N f(x) = \\frac{f(x^+) + f(x^-)}{2}.$$\n2. If moreover $f$ is continuous on an open interval $I$, then $S_N f \\to f$ uniformly on every closed bounded interval contained in $I$.\n\n## Remarks\n\nA function of bounded variation is a difference of two monotone functions, so it suffices to treat a monotone $f$.\nOne writes $S_N f(x) - s$ as an integral of $f(x + t) + f(x - t) - 2s$ against the [[Dirichlet kernel|3JY4BVE4DW]] over $[0, \\pi]$, with $s = \\frac{1}{2}(f(x^+) + f(x^-))$.\nThe second mean value theorem for integrals, together with the uniform boundedness of $\\int_0^a \\frac{\\sin((N + 1/2) t)}{t} \\, dt$ in $a$ and $N$, controls the part near $t = 0$, and the Riemann–Lebesgue lemma handles the rest.\nDirichlet's original conditions are that $f$ is bounded on a period, with finitely many maxima and minima and finitely many discontinuities there.\nSuch an $f$ has bounded variation, so Jordan's form above contains them.\nExample: for the square wave, equal to $\\operatorname{sgn} x$ on $(-\\pi, \\pi)$, the series converges to $0$ at the [[jumps|004FH91BN5]], the average of $-1$ and $1$, and to $f(x)$ elsewhere.\nBounded variation cannot be replaced by continuity: there are continuous functions whose Fourier series diverge at a point.\n\n## Explanations\n\n### L1 human\n\nIf a periodic function does not wiggle infinitely much, its Fourier series converges at every point.\nAt a point where the function jumps, the series picks the midpoint of the jump.\nWhere the function is continuous, it converges to the value of the function.\n\n### L2 human\n\nThis is the convergence theorem used in most applications, since piecewise monotone and piecewise smooth functions have bounded variation.\nIt is a pointwise test, like the Dini test; the two tests are independent, and neither implies the other.\nPart 2 explains why the convergence for the square wave is uniform away from the jumps but not near them, where the Gibbs phenomenon appears.\nCamille Jordan stated the test in this form, for functions of bounded variation.\n",
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