# D Polynomial approximations of useful functions

Lemma D.1 (Polynomial approximations of logarithm [107]) 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 $$\text{deg}(\tilde{S})=O({\frac{1}{\beta}\log (\frac{1}{\eta} )})$$.