Absolute values of the coefficients of the polynomials in Weierstrass's approximation theorem

The following problem, bound up with Weierstrass's classical approximation theorem, is solved definitively: to determine the sequence of positive numbers $\{M_k\}$ such that, for any $f(z)\in C[0,1]$ and $\forall\,\varepsilon>0$ there exists the polynomial $P(z)=\sum_0^n\lambda_kz^k$ that $\|f-P\|<\varepsilon$ and $|\lambda_k|<\varepsilon M_k$, $k=1,\dots,n$.