Previously asked on Stats.SE
We have $N$ samples of the unknown function $f(x)$ on the finite interval $[a, b]$. The samples are subject to white noise of known variance. We want to approximate the function $f(x)$ by a polynomial $P_m(x)$ of order $m$. It is, however, known, that the original function $f(x)$ is smooth in some sence. For example, let's say we know that the original $f(x)$ is Lipschitz-continuous on the interval $[a, b]$ with some known constant $K$. Is there a standard procedure to select the best-fitting polynomial $P_m(x)$ while requiring that the polynomial satisfies the smoothness requirement.
I am not bound to Lipschitz continuity. I am happy to switch to another metric of smoothness if it is more convenient for this problem. However, the smoothness metric should be minimax. This means that the bound must apply to the smoothness of the worst point of the function, which is not necessarily the average such as L2.
Note: The question is about how to solve this problem in practice. Namely:
- Is there an analytical solution?
- Is there a known algotithm used in practice to solve this problem?
- Recommend an existing library that can perform this sort of fitting.