I'm trying to approximate a function by summing a series of orthogonal functions.
$f(x) \approx \sum_i a_i \phi_i(x)$
Since the set of functions $\phi_i(x)$ are orthogonal with respect to each other, I can directly solve for each coefficient $a_i$.
However, this involves solving an integral which I can't solve analytically: $\int f(x)\phi_i(x) dx$
I've opted to use simple Monte Carlo integration to approximately solve the integral.
My Question
For each coefficient I want to solve, I have to generate some set of randomly-chosen points for each Monte Carlo integration. Would it improve my approximation of $f(x)$ if I re-used the same set of random points for each of the integrations?
My thinking is that the solution of each individual term in $a_i$ is only an exact solution in the context of all other terms in the series. I'm worried that the random fluctuations in the estimates for each term will increase the overall error of the approximation a lot.