# 1-D local interpolation of sampled function that is polynomial in 1/x

I have a number of evenly-spaced samples of functions that are polynomial in $1/x$ where $x$ is a continuous variable on $x\in\left[a,b\right]$, i.e. $$f(x_i)=\sum_{n=0}^{N} f_n x_i^{-n}.$$ I want to use some sort of local interpolation function like a hat function (linear interpolation), but it is pretty inaccurate (about 10% relative error in the $L_2$ norm) for values of $x$ near $a$. Near $b$, the interpolation is much better. I haven't yet implemented higher-order local Lagrange interpolation polynomials. I was wondering if anyone knows of a local interpolation method better suited to this problem. For reference, $N$ may be anywhere from about 5 up to 1000, and the number of samples reflects this. I'm not completely wedded to evenly-spaced samples, either, if a good interpolation method requires a different spacing. My thinking is that perhaps concentrating sampling points closer to $a$ and interpolating on a uniformly-spaced transform grid might work.

• Have you considered using splines? Jul 22 '17 at 5:28
• Rational B-Splines should capture your function nicely, since the basis functions are of similar form.
– Bort
Jul 22 '17 at 10:33