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Let $\{x_k\}$ be an infinite series. It is known that $\sum_{k=-\infty}^\infty \vert x_k\vert^2<\infty$ and $\sum_{k=-\infty}^\infty \vert x_k\vert$ does not converge. How do I prove that the Fourier transform of $\{x_k\}$, $\mathcal{F}(\omega)$ exists (or what other conditions need to be satisfied for it to exist)? I can solve this if $x_k$ is absolutely summable, but given that this is not satisfied, I don't know how.

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Whether it exists depends on which space you allow $\mathcal{F}$ to be. Using Parseval, you can immediately show from $\sum_k |x_k|^2$ that in your case $\mathcal{F}(\omega) \in L^2([0,2\pi])$. So if you allow for functions in $L^2$ the Fourier transform exist without further assumtions.

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But doesn't use of Parseval's/Plancheral's theorem require the existance of the Fourier transform in the first place? I thought that these theorems only allowed extensions of the Fourier transform to $L^2$. –  user7815 Mar 7 '11 at 22:15
    
The Fourier series exist due to the fact that $x \in \ell^2$ and Parseval tells you that the result is in $L^2$. –  Fabian Mar 7 '11 at 22:22
    
Maybe the following wikipedia article helps en.wikipedia.org/wiki/Riesz%E2%80%93Fischer_theorem. –  Fabian Mar 7 '11 at 22:37

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