I have a formular for the convolution theorem, and read several chapters in several scripts about it. This is the formula:
$(f*g)(x)=\int_{\mathbb{R}^d}f(x-y)g(y)dy$
However much I read, I cannot figure out where exactly the y comes from.
So, if I had an excercise where I have to convolute to functions, or one with itself, where do I put the y? I have found examples, but mostly they just fill in the formula and in the end the y just miraculously vanishes through some clever crossing out.
But I want to convolute to conditional functions (like, though not quite so simple, the Kronecker $\delta$. And I am just confused.
What exactly is the y, and is there anything its value depends on? if so, what?
Thanks
To the comments:
Say I have a function which gives f(x) = 1 if x<0 and f(x) = 0 in every other case.
I now want to convolute f with itself, so f*f.
I know there are different cases but which?
I just do not know what to put into the formula.
Say for instance the case x<0.
Then I put:
$ =\int_{\mathbb{R}^d}f(x-y)f(y)dy$ =
$ =\int_{\mathbb{R}^d}(f(x)-f(y))f(y)dy$
$ =\int_{\mathbb{R}^d}(1-f(y)f(y))dy$ and that is it?
Is there really no way to figure out whether x-y is greater zero, because the y does not really exist?