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The Stone Weierstrass theorem for $C([0,1])$ claims that for any continuous function $f:[0,1]\to\mathbb{R}$ and each $n\in\mathbb{N}$, there is a polynomial $p_{n,f}(x)=\sum_ia_{f,n,i}x^i$ such that $\left\|f-p_{n,f}\right\|_\infty<\frac{1}{n}$. I would like to know results about bounds of $\sum_i|a_{f,n,i}|$ in terms of $f$ and $n$.

To be more specific, are there known bounds $U(N,L,n)$ such that for any $L$-Lipschitz function $f:[0,1]\to\mathbb{R}$ with $\|f\|_\infty<N$ we can find a polynomial $p_{f,n}(x)$ as above with $\sum_i|a_{f,n,i}|<U(N,L,n)$?

My motivation to ask is this MO question, maybe it has a positive answer if the bounds $U(N,L,n)$ exist and don't grow very fast.

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    $\begingroup$ There is constructive proof of the Weierstrass approximation theorem due S. N. Bernstein. Keyword: Bernstein polynomials. It gives a bound on coefficients and the degree in terms of the modulus of continuity. $\endgroup$
    – Oleg Eroshkin
    Commented Mar 12, 2023 at 2:12
  • $\begingroup$ Thanks! After searching "Bernstein polynomials" I found this answer which contains a few references $\endgroup$
    – Saúl RM
    Commented Mar 12, 2023 at 3:23
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    $\begingroup$ Basically you are asking how to bound the coefficient of a polynomial with its sup-norm. A way is to use iteratively Markoff inequality saying that $\|P'\|_\infty \leq 2n^2 \|P\|_\infty$ if $n$ is the degree. $\endgroup$
    – Giorgio Metafune
    Commented Mar 12, 2023 at 10:34

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