# Convolutions, delta functions, etc.

1. Mar 3, 2006

### 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!

2. Mar 3, 2006

### Tide

Do you have an definition of "convolution?"

3. Mar 4, 2006

### reklar

Sure,

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

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

4. Mar 4, 2006

### 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?

5. Mar 4, 2006

### reklar

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 somone 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...

6. Mar 5, 2006

### 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.