TITLE: Approximation theory reference for a bounded polynomial having bounded coefficients
QUESTION [7 upvotes]: 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: 
http://eccc.hpi-web.de/report/2012/037/download 
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!

REPLY [2 votes]: I think that an interesting historical point to mention is that this problem was first posed and solved for quadratic polynomials by Medeleev (a chemist). There is a nice little article about the problem, including its origins, in the American Mathematical Monthly:

R.P. Boas, Extremal problems for polynomials,  Amer. Math. Monthly 85 (1978), No. 6, 473--475.

You can also find Markov's theorem written up (including generalizations to polynomials of several variables) in the following textbooks:

P.B. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, GTM 161, Springer, 1995.
P. Borwein, Computational Excursions in Analysis and Number Theory, CMS Books in Mathematics, Springer, 2007.