D Polynomial approximations of useful functions

{lemma, poly-approx-ln, name="Polynomial approximations of logarithm [@distributional]"} Let $\beta\in(0,1]$, $\eta\in(0,\frac{1}{2}]$ and $t\geq 1$. There exists a polynomial $\tilde{S}$ such that $\forall x\in [\beta,1]$, $|\tilde{S}(x)-\frac{\ln(1/x)}{2\ln(2/\beta)}|\leq\eta$, and $\,\forall x\in[-1,1]\colon -1\leq\tilde{S}(x)=\tilde{S}(-x)\leq 1$. Moreover $\text{deg}(\tilde{S})=O({\frac{1}{\beta}\log (\frac{1}{\eta} )})$.