I have two finite sets of events $\{x_1, ..., x_N\}$ and $\{y_1, ..., y_N\}$ that are sampled from the PDFs $f(x)$ and $g(x)$, respectively, where $x \in [-1,+1]$. I want to estimate the Legendre expansion of $r(x) = f(x)/g(x)$ using these events. What is the best method? (Note that $g(x)\neq0$ over $[-1,1]$)
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$\begingroup$ Hi Sam, welcome to MO. As it is, the question is not clear. You need to evluate $r$'s Legendre expansion by a finite (different) number of samples from $f$ and $y$? I think it is better (for you and us) to devide the question in two: (1) How to evaluate $r$ if you know the expansions of $f$ and $g$. (2) How bad or good is this method if $f$ and $g$'s expansions are only determined using monte carlo $\endgroup$– Amir SagivJun 2, 2017 at 13:13
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$\begingroup$ Also, I think there's a good chance to get a good in stack.scicomp $\endgroup$– Amir SagivJun 2, 2017 at 13:16
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$\begingroup$ Hi Amir, thanks for your fast response! Yes that's correct - I have a finite number of samples from $f(x)$ and $g(x)$ and want to describe the Legendre expansion of $r(x) = f(x) / g(x)$. $\endgroup$– SamJun 2, 2017 at 13:20
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$\begingroup$ Good. Try and edit your questions so that'd be apparent. As it is, it's really hard to understand that. $\endgroup$– Amir SagivJun 2, 2017 at 13:20
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$\begingroup$ Erm... Legendre expansion makes sense only on a finite interval. So far the supports of your $f$ and $g$ are not restricted to anything and, moreover, the ratio $f/g$ is not obliged to be even finite a.e., forget the integrability. Some a priori assumptions have to be introduced to make the question meaningful, in which case it is a good idea to spell them out. $\endgroup$– fedjaJun 2, 2017 at 14:43
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