# How to prove or disprove that the series are Fourier transformation for some function.

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?

• By Fourier transform do you mean Fourier series? – LL 3.14 Jun 4 '20 at 16:06
• @LL3.14 Yes, Fourier series. – user Jun 4 '20 at 16:08
• Is your sum over $n$ or $k$? – James Arathoon Jun 4 '20 at 16:52
• What is exactly your question, you want to know if some given function $f$ is equal to this Fourier series ? – qdr Jun 4 '20 at 17:16
• 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. – paul garrett Jun 4 '20 at 17:23