# Numerical methods for solving the inverse Weierstrass transform

I have a measurement technique that results in a frequency distribution of the results. Let's say x is the parameter of interest, the method gives you a distribution F(x). I have the idea that the various physical processes in the measurement technique means that F(x) is really a convolution of the true distribution - let's call that f(x) - and the Gaussian function. Very similarly to a Gaussian blur - but the width of the Gaussian function actually varies with x (in a known way).

What I would like to do is be able to take the measurement F(x) and transform it back to f(x) - the true result. I've looked into this and my understanding is that effectively the measurement is making a generalised Weierstrass Transform of f(x).

My problem is that I can understand how to numerically do f(x) -> F(x), it's a simple averaging of f(x) with the Gaussian function, but I've no idea how to do the reverse numerically (the measurement doesn't give a result that can be written as some simple function). I have two ideas:

1. Do the transform directly using some appropriate method.
2. Solve the issue the other way - i.e. iterate over f(x)s until I find one that gives close to the measured F(x).

In both cases I'm at a bit of a loss as to the best approach - let alone which of the 2 options is most sensible.

For option 1, I've read the Wikipedia article and it gives three alternatives for the inverse transform, but all involve higher order differentiations - which I would guess lead to lots of numerical stability issues.

For option 2, I guess there are lots of ways to do this - I'm thinking I could try to modify some Monte-Carlo methods or something. But I'd really appreciate it if anyone has some pointers / knows of existing algorithms that can do this.