Let ${\mathbb C}_+$$\mathbb {C} _ + $ denote the right halfplane and $A$ the algebra $$ A = \{ f \in H^\infty({\mathbb C}_+) \cap C(\overline{{\mathbb C}_+}): \; |f(z)| \le M (1+|z|)^{-\eps} \text{ for some } \eps>0, M>0\} $$$$ A = \{ f \in H^\infty({\mathbb C} _ +) \cap C(\overline{{\mathbb C} _ +}): \; |f(z)| \le M (1+|z|)^{-\epsilon} \text{ for some } \epsilon > 0, M > 0 \} $$ Assume $f \in A$ and $(f_n) \subset A$ an approximating sequence with respect to sup norm, i.e. $\sum_{\Re(z)>0} | f_n(z) - f(z) | \to 0$.
Is there a chance that the decay rates of the approximating functions $f_n$ are uniform, i.e. $\exists \eps, M>0 \forall n:\; |f_n(z)| \le M (1+|z|)^{-\eps} $$\exists \epsilon, M>0 \forall n:\; |f_n(z)| \le M (1+|z|)^{-\epsilon} $?
