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I hope you don't mind that rather than typing this question up I took a screenshot and uploaded it:

http://www.math.ualberta.ca/~schlitt/stackexchangeproblems/tempered-distributions-calc.png

The step that I can't follow is the one clearly identified in red.

It's an integration change of variable I think that is really confusing me.

Update:

I tried doing integration by parts using $u = F(t)$ and $dv = \phi'(t)dt$.

I get

$$ \begin{eqnarray*} \int F(t)\;d\phi(t) &=& \int F(t)\phi'(t)\;dt\\ &=& F(t)\phi(t) - \int \phi(t)F'(t)\;dt\\ &=& F(t)\phi(t) - \int \phi(t)\;dF(t) \end{eqnarray*} $$

Michael Hardy
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roo
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1 Answers1

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Ok, I post it as an answer. It is just an integration by parts that uses the fact that "tempered distributions" imply 0 in the limit $t\rightarrow\pm\infty$.

Jon
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  • Thanks! Now I realize that the answer was exactly what you said the first time. I completely forgot I was dealing with a improper definite integral. – roo Dec 11 '11 at 19:35