What is the Limit of a Function with Finite Integral?

[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!

 

[Hurkyl]

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

Er, what's wrong with this? f denotes an ordinary function, right? (or are you doing something weird with, e.g., distributions?) So for each value of t you're taking the limit of a single-valued function of one variable. (and so the result is a function of t)

(And no, in general yo do not have the right to do the swap. It's easy to come up with examples that disprove it)


But anyways, you're making the problem way too hard:

$G(\tau)=A$
$\lim_{\tau \to +\infty}G(\tau)$

 

[Lajka]

Hm, you got to point there. I guess I just have a problem visualising it:

It never stops moving to the right, right? So I wasn't able to visualise this function of t, which is to be integrated. Still can't.

So, if I understood correctly, $\lim_{\tau \to +\infty}G(\tau)=A$?

 

[Hurkyl]

Hm, you got to point there. I guess I just have a problem visualising it:

Try visualizing it for a single t, rather than all t at once.

So, if I understood correctly, $\lim_{\tau \to +\infty}G(\tau)=A$?

Right.

 

[Lajka]

Hm, okay.

The only thing that is still uncomfortable for me is that I can't seem have an analytic form of the function which I integrate, e.g. if I tell you the analytic form of the function $f(t)$ is $f(t)=e^{-t^2}$, 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 here, 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!

 

[Hurkyl]

I tell you the analytic form of the function $f(t)$ is $f(t)=e^{-t^2}$, could you tell me the analytic form of the function $\lim_{\tau \to +\infty}f(t-\tau)$, as a function of $t$?

$$\lim_{\tau \to +\infty}f(t-\tau) = 0$$
(In fact, the limit is zero no matter what f is, so long as $\int_{-\infty}^{+\infty} f$ is finite and this limit exists)
A buzzword is "pointwise convergence".

Also, while we're here, 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?

I honestly don't remember; I would have to consult a reference book. I find it plausible that uniform convergence of the limit would still suffice. (Notice the above limit does not converge uniformly in t)


Edit: I suppose I should comment that one can define other limits; e.g. a limit computed in some space of functions. But without any other comment, a limit with a free variable should be assumed to be computed pointwise.

Edit2: Unless you're reading physics, when it's just as plausible they meant the limit to be computed in some space of distributions.