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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asked Apr 6, 2017 at 16:53
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2 Answers
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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.)
answered Apr 6, 2017 at 17:16
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5$\begingroup$ For a reference (since OP asked for one): An Introduction to Harmonic Analysis by Y. Katznelson, 3rd edition, exercise 4 from section I.4 $\endgroup$ Commented Apr 6, 2017 at 17:19
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$\begingroup$ @WillieWong: Thank you. I did not know a reference from the top of my head, so I just thought it over. $\endgroup$ Commented Apr 6, 2017 at 17:20
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2$\begingroup$ 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. $\endgroup$ Commented 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...).
answered Apr 6, 2017 at 17:24
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2$\begingroup$ oh I was writing at the same time as GH but he was faster :) $\endgroup$ Commented Apr 6, 2017 at 17:24