Let $P(x)$ be a real polynomial of degree at most $d$. Assume $|P(x)| \leq 1$ for $|x| \leq 1$. I would like a bound saying that each coefficient of $P(x)$ is at most $C^d$ in magnitude, for some absolute constant $C$.
This is surely a well-known, basic fact in approximation theory and I'm looking for a proper reference. I know one very recent paper which writes out a proof using the standard simple idea (Lagrange interpolation) -- Lemma 4.1 from a paper of Sherstov here:
Sherstov obtains $C = 4e$; I don't think either of particularly cares about getting the sharpest constant.
In any case, Sherstov and I agree that this must have appeared somewhere long ago. Could anyone provide a reference? Thanks!