Convolutions, delta functions, etc.

[reklar]

Okay, these might be better off in two separate threads but...they are somewhat related I suppse.

Anyway, I would like to know how you go about computing the convolution of two functions on the unit circle. Let's say that f(x) = x and g(x) = 1 on the interval [0, Pi] and [0, Pi/2] respectively. I think I get the idea in the discrete case, but seem to have trouble with the continuous case for some reason...

Also, is there a good reference for reading about delta functions, approximate identities and the like? It seems like most texts I've run across barely touch on the subject, but I'd like to see a more thorough, understandable treatment.

Thanks!

 

[Tide]

Do you have an definition of "convolution?"

 

[reklar]

Do you have an definition of "convolution?"

Sure,

(f*g)(x) = integral f(x-y)g(y) dy

where the limits of integration are from -Pi to Pi...

 

[Tide]

Then all you have to do is apply that definition though the limits of integration will correspond to your particular problem. What, exactly, are you having trouble with?

 

[reklar]

Then all you have to do is apply that definition though the limits of integration will correspond to your particular problem. What, exactly, are you having trouble with?

It is a function of x, but what is going on with the y in there?
And the periodic thing throws me a bit I guess...

Anyway, if someone could just do one, perhaps that would make it clearer. I can't seem to find an example in any book I have where this is done...

 

[Tide]

"y" is a "dummy variable" that represent the variable of integration so, for example, one of your integrals will look like this:

$$\int_{0}^{\pi} (x - y) \times 1 dy$$

for f(x) = x and g(x) = 1. Notice how y no longer appears after performing the integration.