On a nontraditional method of approximation

We study the approximation of functions $f(z)$ that are analytic in a neighborhood of zero by finite sums of the form $H_n(z)=H_n(h,f,\{\lambda_k\};z)=\sum_{k=1}^n\lambda_kh(\lambda_kz)$, where $h$ is a fixed function that is analytic in the unit disk $|z|<1$ and the numbers $\lambda_k$ (which depend on $h,f$, and $n$) are calculated by a certain algorithm. An exact value of the radius of the convergence $H_n(z)\to f(z)$, $n\to\infty$, and an order-sharp estimate for the rate of this convergence are obtained; an application to numerical analysis is given.