D Polynomial approximations of useful functions

Lemma D.1 (Polynomial approximations of logarithm (Gilyén and Li 2019)) 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 \(\text{deg}(\tilde{S})=O({\frac{1}{\beta}\log (\frac{1}{\eta} )})\).