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!
Best Answer
Dear Ryan, I hope the following references will be useful for you:
V.A. Markov has solved your posed problem back in 1892, see pages 80-81 in
http://www.math.technion.ac.il/hat/fpapers/vmar.pdf
Compare also the book
I.P. Natanson: Constructive Function Theory, Vol. I. Uniform Approximation, F. Ungar Publishing, New York, 1964, page 56.
You will find more detailed information in the papers
H.-J. Rack: On V.A. Markov´s and G. Szegö´s inequality for the coefficients of polynomials in one and several variables, East Journal on Approximations 14 (2008), pages 319 - 352
H.-J. Rack: On the length and height of Chebyshev polynomials in one and two variables, East Journal on Approximations, 16 (2010), pages 35 - 91.