# Proof of Riemann-Lebesgue lemma: what does “integration by parts in each variable” mean?

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

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.

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