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I was reading the proof of the Riemann-Lebesgue lemma on Wikipedia, and something confused me. It says the following:

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What does the author mean by "integration by parts in each variable"? If we integrate by parts with respect to $x$, then (filling in the limits, which I believe are $-\infty$ and $\infty$) I get $$\hat{f}(z) = \left[\frac{-1}{iz}f(x)e^{-izx}\right]_{-\infty}^{\infty} + \frac{1}{iz}\int_{-\infty}^{\infty}e^{-izx}f'(x)dx.$$

I think I am missing something here...it's not clear to me why the limit at $-\infty$ of the first term should exist. Can anyone clarify this for me?

Thanks very much.

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    $\begingroup$ $f$ is supposed to have compact support, so $f(x) = 0$ for $\lvert x\rvert \geqslant K$. $\endgroup$ – Daniel Fischer Jul 1 '13 at 20:30
  • $\begingroup$ Ah, I see. Any clue what they meant by "in each variable"? $\endgroup$ – Eric Auld Jul 1 '13 at 20:54
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    $\begingroup$ I guess it is just a leftover from when it was at least planned to write it down in higher dimensions. $\endgroup$ – Daniel Fischer Jul 1 '13 at 20:57
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This question was answered by @danielFischer in the comments.

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