1
$\begingroup$

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*}

  • 0
    Is there some Stieltjes-integral involved?2011-12-11

1 Answers 1

2

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 $t\rightarrow\pm\infty$.

  • 0
    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.2011-12-11