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If $\sum_{k=-\infty}^{\infty}|a_k|^2$ is not finite, does Parseval's theorem say that the Fourier transform of $a_k$ is also not finite?

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    Ah, I see, thanks Robert.2012-11-02

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What it says is there is no square-integrable function on $[0,2\pi]$ whose inverse Fourier transform is $a_k$.