Approximation by Sums of the Form $\sum_k\lambda_kh(\lambda_kz)$ in the Disk

Given a function $h$ analytic in the unit disk $D$, we study the density in the space $A(D)$ of functions analytic inside $D$ of the set $S(h,E)$ of sums of the form $\sum_k\lambda_kh(\lambda_kz)$ with parameters $\lambda_k\in E$, where $E$ is a compact subset of $\overline D$. It is proved, in particular, that if the compact set $E$ “surrounds” the point $0$ and all Taylor coefficients of the function $h$ are nonzero, then $S(h,E)$ is dense in $A(D)$.