Gaussian integral as a convolution integral

[Prometheus]

For part of a homework question I need to write:

 

[latex]

f(x,T)=\int_a^b G_{w^2}(x-s)ds

[/latex]

 

as a convolution integral, where:

 

[latex] G_{\sigma^2}(x) = (2{\pi}{\sigma^2})^\frac{-1}{2}e^\frac{-x^2}{2{\sigma^2}} [/latex]

 

So my understanding is that I need to express the above integral in the form:

 

[latex] \int_{-\infty}^{\infty} f(x-y)g(y)dy[/latex]

 

Is this the correct understanding? Also, is anyone able to give a little insight into these convolutions integrals - I find them very strange and do not understand them in the least.

 

Thanks to any and all help.

 

 

[Prometheus]

I managed to make some sense of it by letting [latex] g(s) = 1[/latex] in the first integral and playing around with the limits a bit.

 

Still, any intuition behind the convolution integral would be awesome.

[TCHAPAIEV]

You need to find an "f" corresponding to Gw2*f; clearly the only candidate is just NOT the function I ( equal to 1 everywhere ) BUT the step function which is the caracateristic function of [a,b] that means the function equal to 1 on [a,b] and 0 elsewhere !

Intuition for convolution is easy by practicing it is a remarquale tool which allow to FOURIER TRANSFORM in one click ; * becomes just ordinary product of functions ( the Fourier transform of ........ ) in case of the gaussian the forier transform is almost identical to itself ; so what remains is to compute the Fourier transfrorm of the indicatrice of the intreval [a, b]; What do you expect preicely by "more intuition" there are a lot of books for students ;all which are relied to convolution are related to Fourier transforms ans series ; please have some look to those

[Prometheus]

Thanks. I've handed this one in as I described so I'll just see how it goes for now.

 

In terms of the intuition, at the moment I just imagine the transform as a mapping onto another space because it is easier to perform certain operations in the space. Kind of like why we sometimes work in log space. I've seen a few books but they are all engineering books so they talk about quite specific mappings I'm not familiar with. Can you recommend any good non-engineering orientated books on the subject?