Explicit polynomial approximation of the reciprocal on the left semicircle
= Explicit polynomial approximation of the reciprocal on the left semicircle
On $K=\{z:|z|=1,\operatorname{Re}z\leq0\}$, the polynomials
$$
p_n(z)=-\frac14\sum_{j=0}^n\sum_{k=0}^{n^2}\binom{j+k}{k}\left(\frac z4\right)^k
$$
converge uniformly to $1/z$. Expand $1/z$ first in powers of $4/(z-4)$, whose modulus is at most $4/\sqrt{17}$ on $K$, and then expand each $(z-4)^{-j-1}$ in powers of $z/4$. The estimate $\binom{j+k}{k}\leq2^{j+k}$ controls the diagonal truncation.