A question about the limit of a specific function, but not sure how to formulate it

**lajka** · August 24th 2011, 12:45 AM

Hi,

I was thinking of this problem for a couple of hours, but wasn't sure how to formulate it, hence I wasn't able to google it.

It's pretty simple to explain, though. Observe any function with a finite integral over the R line
$\int f(t)dt=A=const.$

And now look at the function
$G(\tau) = \int f(t - \tau)dt.$

Now, for any finite $\tau$ , $G(\tau)=A$ , obviously.
However, I asked myself what is the answer for $\lim_{\tau \to +\infty}G(\tau)$ ?

I'm not sure if I have or don't have the right to exchange the limit and the integral
$\lim_{\tau \to +\infty}\int f(t - \tau)dt =\int \lim_{\tau \to +\infty}f(t - \tau)dt$

mainly because I have no idea how to interpret this
$\lim_{\tau \to +\infty}f(t - \tau)$

So, what say you? I'm clueless.
Thanks!

**girdav** · August 24th 2011, 01:38 AM

Hi,
I understand you have no idea how to interpret $\lim_{\tau \to +\infty}f(t-\tau)$ since it's possible that this limit doesn't exist. For example, consider an even function f. We can choose for $n\geq 1$ $f$ affine on $\left[n-\frac 13,n\right]$ and $\left[n,n+\frac 13\right]$ , nonnegative and such that $\int_{n-\frac 13}^{n+\frac 13}f(t)dt=2^{-n}$ . Put $f=0$ on the complement of these intervals: we can choose $f$ continuous, integrable but which has no limit as $x\to \pm\infty$ .

**CaptainBlack** · August 24th 2011, 02:41 AM

Originally Posted by lajka

Hi,

I was thinking of this problem for a couple of hours, but wasn't sure how to formulate it, hence I wasn't able to google it.

It's pretty simple to explain, though. Observe any function with a finite integral over the R line
$\int f(t)dt=A=const.$

And now look at the function
$G(\tau) = \int f(t - \tau)dt.$

Now, for any finite $\tau$ , $G(\tau)=A$ , obviously.
However, I asked myself what is the answer for $\lim_{\tau \to +\infty}G(\tau)$ ?

I'm not sure if I have or don't have the right to exchange the limit and the integral
$\lim_{\tau \to +\infty}\int f(t - \tau)dt =\int \lim_{\tau \to +\infty}f(t - \tau)dt$

mainly because I have no idea how to interpret this
$\lim_{\tau \to +\infty}f(t - \tau)$

So, what say you? I'm clueless.
Thanks!

Because the limiting process only ever considers finite (but large) $\tau$ the limit is $A$

CB

**chisigma** · August 24th 2011, 04:19 AM

Originally Posted by lajka

Hi,

I was thinking of this problem for a couple of hours, but wasn't sure how to formulate it, hence I wasn't able to google it.

It's pretty simple to explain, though. Observe any function with a finite integral over the R line
$\int f(t)dt=A=const.$

And now look at the function
$G(\tau) = \int f(t - \tau)dt.$

Now, for any finite $\tau$ , $G(\tau)=A$ , obviously.
However, I asked myself what is the answer for $\lim_{\tau \to +\infty}G(\tau)$ ?

I'm not sure if I have or don't have the right to exchange the limit and the integral
$\lim_{\tau \to +\infty}\int f(t - \tau)dt =\int \lim_{\tau \to +\infty}f(t - \tau)dt$

mainly because I have no idea how to interpret this
$\lim_{\tau \to +\infty}f(t - \tau)$

So, what say you? I'm clueless.
Thanks!

In my opinion it is important to specify that You have to do with a definite integral over the R line, so that is...

$\int_{- \infty}^{+ \infty} f(t)\ dt = A= \text{cost}$ (1)

Now if You define a function of $\tau$ as...

$G(\tau)= \int_{- \infty}^{+\infty} f(t-\tau)\ dt$ (2)

... it is evident that is $G(\tau)=A= \text {cost}$ so that is $\lim_{\tau \rightarrow \infty} G(\tau)=A$ ...

Kind regards

$\chi$ $\sigma$

**lajka** · August 25th 2011, 10:09 PM

Hm, I think I get it now, hopefully. Thank you all!

@girdav Thanks for your reply bro, although to be honest - I didn't understand zip of it, I'm just a poor engineering student

But maybe someday I will.

edit:
The only thing that is still uncomfortable for me is that I can't seem to have an analytic form of the function I integrate, that is, $\lim_{\tau \to +\infty}f(t-\tau)$ as a function of t.
In other words, if I tell you the analytic form of the function $f(t)$ , for example $f(t)=e^{-t^2}$ , could you tell me the analytic form of the function $\lim_{\tau \to +\infty}f(t-\tau)$ , as a function of $t$ ?

Also, while we're on the subject, I haven't encountered interchange of limits and integrals like this before. It was usually the sequence of functions for me, and their interchange with the integral if the sequence converges uniformly etc.
What would be the conditions for the interchange in a case like this?
Thanks!

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