Bounding the coefficients of the approximating polynomial of a $1$-Lipschitz function

Question

Let $f$ be a $1$-Lipschitz function on the interval $[0,1]$. Then, by Jackson's theorem we get that there exists a polynomial $P_n$ of degree $n$ such that $$ |f(x)-P_n(x)|=O\!\left(\frac{1}{n}\right).$$ Let $\boldsymbol{a}$ denote that vector of $n+1$ coefficients $a_i$, $i=0,\ldots,n$, of $P_n$. I am interested in bounding $\Vert\boldsymbol{a}\Vert_1$. Are there any known results on this?

Answer 1

I have found this question and references therein which answers my above question

https://mathoverflow.net/questions/97769/approximation-theory-reference-for-a-bounded-polynomial-having-bounded-coefficie