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Question. Suppose $f$ is periodic in $[0,2\pi]$. What conditions on the Fourier coefficients of $f$ would guarantee real analyticity of $f$? Please provide me with a reference.

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A function $f:\mathbb{R}\to\mathbb{C}$ periodic by $2\pi$ is real analytic if and only if it extends holomorphically to $\{z\in\mathbb{C}:\ |\Im z|<c\}$ for some $c>0$, because the interval $[0,2\pi]$ is compact. The latter condition is easily seen to be equivalent to the exponential decay of the Fourier coefficients: $a_n\ll e^{-c|n|}$ for come $c>0$. (Necessity of this condition follows from Cauchy's theorem about integrals of holomorphic functions. Sufficiency of the condition follows from Morera's theorem and Cauchy's theorem about the analyticity of holomorphic functions.)

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    $\begingroup$ For a reference (since OP asked for one): An Introduction to Harmonic Analysis by Y. Katznelson, 3rd edition, exercise 4 from section I.4 $\endgroup$ Apr 6, 2017 at 17:19
  • $\begingroup$ @WillieWong: Thank you. I did not know a reference from the top of my head, so I just thought it over. $\endgroup$
    – GH from MO
    Apr 6, 2017 at 17:20
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    $\begingroup$ Neither did I. After seeing your answer I had to go to my book case and look around a bit. I was convinced I've seen that condition before somewhere. $\endgroup$ Apr 6, 2017 at 17:22
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The condition is $|c_n|\le A e^{-B|n|}$ for some positive constants $A,B$ and all $n$. This is not difficult to prove directly (a reference might be exercise I.4.4 in Katznelson's Introduction to Harmonic Analysis, but it is given as an exercise...).

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    $\begingroup$ oh I was writing at the same time as GH but he was faster :) $\endgroup$ Apr 6, 2017 at 17:24

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