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 Post subject: doesn't seem correctPosted: Thu, 2 Jun 2011 05:51:42 UTC
 S.O.S. Oldtimer

Joined: Mon, 27 Apr 2009 15:11:34 UTC
Posts: 156
I'm studying for analysis and I have the following question (given by a professor), which does not seem correct the way it is stated

Let be a function, so that
. Show that for any continuous function we have that the sequence
on the compact sets of IR.

First, the way it is written doesn't even seem to be make sense because it is integrating with respect to x so I believe that the integral should actually be with resepect to y. But that doesn't seem to be true, for instance if I define
and 0 otherwise. This integrates to 1. Moreoever, If I pick a function like f(x)= x^2 which is continuous on R, the function g_n doesn't even converge.
For instance,

I think the only way this problem could make sense is if the function is defined to have compact support.

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 Post subject: Re: doesn't seem correctPosted: Thu, 2 Jun 2011 06:29:11 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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TIMsetsFIRE wrote:
I'm studying for analysis and I have the following question (given by a professor), which does not seem correct the way it is stated

Let be a function, so that
. Show that for any continuous function we have that the sequence
on the compact sets of IR.

First, the way it is written doesn't even seem to be make sense because it is integrating with respect to x so I believe that the integral should actually be with resepect to y. But that doesn't seem to be true, for instance if I define
and 0 otherwise. This integrates to 1. Moreoever, If I pick a function like f(x)= x^2 which is continuous on R, the function g_n doesn't even converge.
For instance,

I think the only way this problem could make sense is if the function is defined to have compact support.

No! The question is essentially correct (and is how you construct approximations to Dirac delta).

For fixed and your

the function is supported on , so the first equality in the last equation should be

which then gives

pointwise, and uniformly on compacts.

The only typo in the question is integrate with respect to y, not x.

Edit: corrected shouldn't be in the support. Also, spoke too soon about the question being OK. I think there will be a problem if f grows way too fast, e.g. , , which means the convolution blows up dramatically for each fixed n. You want something like or rather than just . Of course, it is easiest if has compact support....

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Last edited by outermeasure on Thu, 2 Jun 2011 08:07:27 UTC, edited 2 times in total.

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 Post subject: Re: doesn't seem correctPosted: Thu, 2 Jun 2011 07:13:45 UTC
 S.O.S. Oldtimer

Joined: Mon, 27 Apr 2009 15:11:34 UTC
Posts: 156
outermeasure wrote:
the function is supported on , .

Of course!!! I completely missed this when I was looking at it. Thank you!

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