I have a function $f(x), x\in[0,1],$ which I can sample at points $x = 1/k$ for integer $k$, $k<K$ where $K$ is some sufficiently large integer that can be chosen depending on the error. I want to learn a degree $K$ polynomial $p_K(x)$ such that $error = sup_{x\in[-1,1]}|f(x) - p_K(x)|\leq \epsilon$ with as few samples as possible.
Are there any results on how efficiently this can be done in terms of $K, \epsilon$? I am aware of results in the case where the samples are drawn from the Chebyshev or Normal distribution, but unaware of any others.