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I'm looking for a concrete example to understand the computation procedure for convolution:

Let $f, g \in C_{0}^{\infty}(\mathbb R)$ be defined as follows: $$f(x) := e^{-\frac{1}{1-x^2}}1_{(-1,1)} $$ $$g(x) := e^{-\frac{1}{1-(x-1)^2}}1_{(0,2)} $$ where $$1_{(a,b)} := \begin{cases} 1 & \text{if $x \in (a,b)$} \\ 0 & \text{otherwise} \end{cases}$$

By the above definitions, $f$ is supported on $(-1,1)$, and $g$ on $(0,2)$.

Then how would one go about computing $$(f*g)(x) = \int_{\mathbb R}f(y)g(x-y)dy$$ Also, how would you determine the support of $h := f*g$?

Thanks.

user3294195
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  • Oh man. I'm not sure how nice this will be. This is the prototypical bump function and it is quite an elegant example.. but it is far from a nice example to work with in practice. One of the standard tricks to doing a convolution is actually by doing a Fourier transform as that is often easier since there are deeper tools that can be used (residue theory for example) but I don't think that will lead anywhere good in this case either. – Cameron Williams Aug 09 '15 at 03:17
  • I guess brute force computation of the integral would be far too long to post here. If there are better examples to demonstrate the idea (computing and finding the support of $f*g$, for any compactly supported $f, g$), that would more than suffice. – user3294195 Aug 09 '15 at 03:30
  • Try just characteristic functions. Those are easy to convolve. – Cameron Williams Aug 09 '15 at 03:32

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