On the rate of approximation in the unit disc of $H^1$-functions by logarithmic derivatives of polynomials with zeros on the boundary

We study uniform approximation in the open unit disc $D=\{z\colon |z|<1\}$ by logarithmic derivatives of $C$-polynomials, that is, polynomials whose zeros lie on the unit circle $C=\{z\colon |z|\,{=}\,1\}$. We find bounds for the rate of approximation for functions in Hardy class $H^1(D)$ and certain subclasses. We prove bounds for the rate of uniform approximation (either in $D$ or its closure) by $h$-sums $\sum_k \lambda_k h(\lambda_k z)$ with parameters $\lambda_k\in C$.