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  "body": "\n## Statement\n\nLet $f \\colon \\R \\to \\C$ be a [[continuous|0X3WGX95FE]] $2\\pi$-periodic function with [[modulus of continuity|AXCJBSQZWF]]\n$$\\omega(\\delta) = \\sup \\set{\\abs{f(x) - f(y)} : x, y \\in \\R, \\ \\abs{x - y} \\le \\delta}.$$\nSuppose that\n$$\\lim_{\\delta \\to 0^+} \\omega(\\delta) \\log \\frac{1}{\\delta} = 0.$$\nThen the partial sums of the [[Fourier series|G12KXDHPPS]] of $f$ converge to $f$ [[uniformly|08VZ57PT26]] on $\\R$.\n\n## Remarks\n\nLet $E_N(f)$ be the uniform distance from $f$ to the trigonometric polynomials of degree at most $N$.\nSince $S_N T = T$ for such a polynomial $T$, one gets $\\sup \\abs{f - S_N f} \\le (1 + L_N) E_N(f)$, where $L_N$ is the [[Lebesgue constant|RWKQ5EP4YF]].\nJackson's theorem gives $E_N(f) \\le C \\omega(1/N)$ with an absolute constant $C$, and $L_N = O(\\log N)$, so the error is $O(\\omega(1/N) \\log N)$, which tends to $0$ by hypothesis.\nExample: every Hölder continuous $f$, with $\\omega(\\delta) \\le C \\delta^\\alpha$ for some $\\alpha \u003e 0$, satisfies the hypothesis; so does $f$ with $\\omega(\\delta) \\le C / (\\log(1/\\delta))^{2}$ for small $\\delta$.\n\n## Explanations\n\n### L1 human\n\nA continuous function can have a badly behaved Fourier series.\nThe criterion says that a little more regularity than continuity is enough to rule this out.\nThe required regularity just beats the logarithmic growth of the Lebesgue constants.\n\n### L2 human\n\nThe proof is a model of a general principle: if a linear approximation process has norms $L_N$, then it converges for all functions whose best approximation error is $o(1/L_N)$.\nHere the norms grow like $\\log N$, and Jackson's theorem converts best approximation error into modulus of continuity.\nThe criterion contains the uniform convergence of Fourier series of Hölder and Lipschitz continuous functions.\n",
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