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.
Existence of Fourier transform
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calculus
fourier-analysis
fourier-series
1 Answers
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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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0M$a$ybe the following wikipedia article helps http://en.wikipedia.org/wiki/Riesz%E2%80%93Fischer_theorem. – 2011-03-07