Returning to the question:
Approximation of the convolution operator
And new discussion:
Convolution using the Laplace integral transform of certain functions
$f(t) = e^{-t}$
$g(t) = e^{-(e^{-t})^2}$
Using the convolution property, we can transform it into a product of folding functions, each of which is obtained using the direct Laplace transform of each of the original functions, i.e.:
$(f*g)(t) \rightarrow L^{-1}(L(f(t)) \cdot L(g(t)))$
But it is known that the original functions must meet certain requirements.
With these functions, the Laplace transform will look like:
$\frac{\Gamma \left(\frac{s}{2}\right)-\Gamma \left(\frac{s}{2},1\right)}{2 s+2}$
The inverse Laplace transform from this function is not searched. Mathematica and Maple do not produce adequate results.
How to use this approach to work with a wider class of functions? Use more advanced integral transforms? If so, which ones?
I will be glad to any advice and help.