In my work I often encounter functions with somewhat exponential decay, usually they are non-negative and non-zero everywhere and of even symmetry (so let's just assume $x\ge0$ for simplicity). I would be nice to have a robust way to find the best or near-best approximation $$ f(x) \approx e^{-ax}\sum_{i=0}^nb_ix^i . $$ Best or near-best in the sense that $n$ is small. I can think of some ways to get an approximation, but no systematic robust idea.
I'm very much familiar with polynomial approximations in every shape or form (e.g. Chebyshev polynomials), as well as approximations by sums of exponentials (e.g. Prony's method). I couldn't find anything when looking for related keywords (e.g. approximation by gamma pdfs, exponential polynomials) when a "combination" of both is used.
Is there something like this and/or what could I look for?
EDIT: I found a modified Prony's method in http://num.math.uni-goettingen.de/plonka/pdfs/gamm-preprint.pdf and currently checking if this would work. For simple cases this seems to already work fairly well, and now I'm working to get it robust for more complicated cases.
EDIT 2: Seems like the same as the normal Prony's method the modified one is not really robust, even with only one exponential factor.