Re: question concerning integrals and convolution

From: G. A. Edgar (edgar_at_math.ohio-state.edu.invalid)
Date: 02/15/05 

Date: Tue, 15 Feb 2005 09:32:52 -0500

In article <4211faab$1_1@127.0.0.1>, bwelter
<bettina.welter@wmi.badw-dot-de.no-spam.invalid> wrote:

> Hello everybody,
>
> I have a question that keeps puzzling me and is really important for
> my work.
> I will try to put it down using the usual Latex-notation for the
> formulas.
> Let $f(x)$ and $g(x)$ be two functions in R.
>
> Suppose
> $\underset{x\rightarrow\infty}{\lim} \int_0^x \{f(t)-g(t)\} dt =0$
>
> Does that necessarily imply
>
> $\underset{x\rightarrow\infty}{\lim} \int_0^x \{ f(t)f(t-x)-g(t)g(t-x)
> \} dt =0$?
>

I tried a random example to see...
  f(x) = exp(-3*x)+exp(-6*x), g(x) = exp(-2*x) .
The numbers were chosen so that your first limit is zero.
But when I compute the second one, it goes to infinity.
If I change it to f(t)f(x-t)-g(t)g(x-t) then (at least
in this example) it does go to zero. 

-- 
G. A. Edgar                               http://www.math.ohio-state.edu/~edgar/