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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\C$ be a [[periodic function|0ATF6Q0633]] of period $2\\pi$ that is integrable on $[-\\pi, \\pi]$, let $x \\in \\R$ and let $s \\in \\C$.\nSuppose that\n$$\\int_0^{\\pi} \\frac{\\abs{f(x + t) + f(x - t) - 2s}}{t} \\, dt \u003c \\infty.$$\nThen the partial sums $S_N f(x)$ of the [[Fourier series|G12KXDHPPS]] of $f$ converge to $s$ as $N \\to \\infty$.\n\n## Remarks\n\nSince the [[Dirichlet kernel|3JY4BVE4DW]] is even with mean $1$, one has $S_N f(x) - s = \\frac{1}{2\\pi} \\int_0^{\\pi} \\big(f(x + t) + f(x - t) - 2s\\big) D_N(t) \\, dt$.\nWrite $D_N(t) = \\frac{\\sin((N + \\frac{1}{2}) t)}{\\sin(t/2)}$; the hypothesis says that $t \\mapsto \\frac{f(x + t) + f(x - t) - 2s}{\\sin(t/2)}$ is integrable on $[0, \\pi]$, because $\\sin(t/2) \\ge t/\\pi$ there.\nThe [[Riemann–Lebesgue lemma|WVKDC011KP]] then sends the integral to $0$.\nExample: if $f$ is differentiable at $x$, or only satisfies $\\abs{f(x + t) - f(x)} \\le C \\abs{t}^\\alpha$ for some $\\alpha \u003e 0$ and small $t$, then the hypothesis holds with $s = f(x)$, so the Fourier series converges to $f(x)$.\nContinuity of $f$ at $x$ alone does not suffice, since there are continuous functions whose Fourier series diverge at a point.\n\n## Explanations\n\n### L1 human\n\nThe Dini test checks convergence of a Fourier series at one point by looking only at how fast the function approaches a value near that point.\nIf the symmetric difference $f(x + t) + f(x - t) - 2s$ goes to zero fast enough that dividing by $t$ still leaves something integrable, the series converges to $s$.\nA small power of $t$ is enough, so all differentiable and all Hölder continuous functions pass.\n\n### L2 human\n\nThe test is local: it depends only on $f$ near $x$, in line with the Riemann localization principle.\nAt a jump with one-sided Hölder conditions, it applies with $s = \\frac{1}{2}(f(x^+) + f(x^-))$.\nIt is independent of the Dirichlet–Jordan test: some functions of bounded variation fail the Dini condition, and some functions satisfying it at every point are not of bounded variation.\n",
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