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Show that if $ f \in L^1[(0,2\pi)]$ and $\sum^\infty_{-\infty} |\hat{f}(n)|^2 < \infty$ then $f \in L^2[(0,2\pi)]$

where $\hat{f}(n)$ is the Fourier transform

I started by by substituting the inverse transform into $||f||_2$, but it isnt going anywhere.

Will anyone give me some help?

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jake
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  • Do you know that the Fourier transform is a unitary isomorphism of $L^2$? – Neal Apr 17 '13 at 23:37
  • are you referring to Plancherel theorem or parsevels identity? – jake Apr 17 '13 at 23:42
  • The Plancherel theorem. – Neal Apr 17 '13 at 23:54
  • but doesnt Plancherel theorem requires $f$ to be in both $L^1$ and $L^2$ – jake Apr 18 '13 at 00:00
  • The sketch I have in mind is that since you know $g = \hat{f}\in L^2$, and you know by Plancherel that the Fourier transform can be extended to a unitary operator $L^2\to L^2$, the inverse Fourier transform of $g$ is also in $L^2$. Then I want to argue that $g = f$ and this implies $f\in L^2$. (I haven't checked details if this works, though, which is why I'm not posting it as an answer.) – Neal Apr 18 '13 at 00:04

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