Let's say, there is a function $\Sigma^{\infty}_{k=1} \frac{\cos 2k x}{\sqrt{2k \ln 2k}}$, how do I prove or disprove that it is a Fourier transformation for some function?

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    $\begingroup$ By Fourier transform do you mean Fourier series? $\endgroup$ – LL 3.14 Jun 4 '20 at 16:06
  • $\begingroup$ @LL3.14 Yes, Fourier series. $\endgroup$ – user Jun 4 '20 at 16:08
  • $\begingroup$ Is your sum over $n$ or $k$? $\endgroup$ – James Arathoon Jun 4 '20 at 16:52
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    $\begingroup$ What is exactly your question, you want to know if some given function $f$ is equal to this Fourier series ? $\endgroup$ – qdr Jun 4 '20 at 17:16
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    $\begingroup$ If you know Parseval-Plancherel, you can see that it cannot be the Fourier series of an $L^2$ function... But what kind of "functions" do you allow? That series is certainly the Fourier series of a "generalized function" (distribution), lying in a (periodic) Sobolev space $H^{-1}$, for example. $\endgroup$ – paul garrett Jun 4 '20 at 17:23

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