Consider a function $f(x)$ and a Gaussian kernel function $g(x)$
The convolution $h(x)=(f\ast g)(x)$ results
and as observed, there is some smoothing around $x=5$. I want to prevent this smoothing and only have $f$ smoothed at it's boundaries ($x=-5,10$). This can be done in real space by just setting $h(x)=f(x)$ from (say) $x\in[-4,9]$ after the convolution is done.
However, I intend to perform this convolution in Fourier space using the convolution theorem $H(\kappa)=F(\kappa)G(\kappa)$. So my only option is to transform back $H(\kappa)$ to $h(x)$ and perform the above operation. If I want to avoid this last step, how can I do that in Fourier space? Maybe an adaptive kernel function can work, eg is Gaussian close to the boundaries and is a Dirac delta function in the interior part. Playing with the Gaussian standard deviation can provide something like this.
I think it is easy enough to implement this in real space in order to avoid that crude operation, but I can't think of a way to do it in Fourier space. How can I change the kernel shape when it's only a point-wise multiplication (and not a slide of one function on top of the other one)?