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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 $[-\\pi, \\pi]$, with [[Fourier coefficients|5NN2YZZR3V]] $\\hat f(n)$, and let $k \\ge 0$ be an integer.\n\n1. If $f$ is $k$ times continuously differentiable, then $\\widehat{f^{(k)}}(n) = (in)^k \\hat f(n)$ for all $n$, and $\\abs{n}^k \\abs{\\hat f(n)} \\to 0$ as $\\abs{n} \\to \\infty$.\n2. If $\\sum_{n \\in \\Z} \\abs{n}^k \\abs{\\hat f(n)} \u003c \\infty$, then $f$ is equal almost everywhere to a $k$ times continuously differentiable function $g$, and $g^{(j)}(x) = \\sum_{n \\in \\Z} (in)^j \\hat f(n) e^{inx}$ for $0 \\le j \\le k$ and all $x$.\n3. $f$ is equal almost everywhere to a [[smooth function|0V33BNKZ9S]] if and only if, for every $m \\ge 0$, some constant $C_m$ satisfies $\\abs{\\hat f(n)} \\le C_m (1 + \\abs{n})^{-m}$ for all $n$.\n\n## Remarks\n\nFor part 1, integration by parts over a period gives $\\widehat{f'}(n) = in \\hat f(n)$, since the boundary terms cancel by periodicity; induction gives the formula for $f^{(k)}$.\nThe [[Riemann–Lebesgue lemma|WVKDC011KP]] applied to $f^{(k)}$ then gives $\\widehat{f^{(k)}}(n) \\to 0$.\nFor part 2, the series $\\sum (in)^j \\hat f(n) e^{inx}$ converge uniformly by the Weierstrass M-test, so their sums are continuous.\nThis justifies term-by-term differentiation.\nThe sum $g$ has the same Fourier coefficients as $f$, so $g = f$ almost everywhere, as in the [[uniform convergence of absolutely summable Fourier series|CSYZGA5YKS]].\nPart 3 follows from parts 1 and 2, since a bound $C_{m+2} (1 + \\abs{n})^{-m-2}$ makes $\\sum \\abs{n}^m \\abs{\\hat f(n)}$ finite.\nExample: the sawtooth $f(x) = x$ on $(-\\pi, \\pi)$ has a jump, and $\\abs{\\hat f(n)} = 1/\\abs{n}$ decays only like $1/\\abs{n}$.\nThe triangle wave $\\abs{x}$ is continuous with a piecewise continuous derivative, and its coefficients decay like $1/n^2$.\nPart 1 gives only $o(\\abs{n}^{-k})$ and part 2 needs slightly more than $O(\\abs{n}^{-k-1})$, so there is a gap between the two directions for a fixed $k$.\n\n## Explanations\n\n### L1 human\n\nA smooth function has small high-frequency content, and a rough function has a lot of it.\nEach derivative that $f$ possesses makes its Fourier coefficients decay faster by one power of $\\abs{n}$.\nA jump or a corner in the graph shows up as slow decay of the coefficients.\n\n### L2 human\n\nThis dictionary between regularity and decay is used constantly.\nIt explains why Fourier series of smooth functions converge very fast, and why the Fourier series of a function with a jump converges slowly and shows the Gibbs phenomenon.\nIt is the starting point of Sobolev spaces on the circle, where smoothness of order $s$ is measured by $\\sum (1 + n^2)^s \\abs{\\hat f(n)}^2 \u003c \\infty$.\nFor real analytic functions the coefficients decay exponentially, and conversely.\n",
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