Hello,

I am very new to the field of approximation theory, and since an extended search on the Internet did not provide answers for two rather basic questions, I decided to ask them here.

1) From my understanding upper bounds for

$$ \inf_{q} \int_{-1}^{1} |f(x) - q(x)|^{2p} dt $$

with $f$ continuous and $q$ a polynomial of degree $n$, are expressed in terms of the $L^p$ smoothness of $f$ and in terms of the degree $n$. Could somebody point me to a proof of such a result?

2) Heuristically, what kind of information do lower bounds for the above infinum contain ? (For example, suppose that I can give a lower bound of $p!$ for the above infinimum as $p \rightarrow \infty$).

My last question might not be well-posed, so if it doesn't make sense please ignore it.

Thank you.