The Carathéodory–Fejér problem and optimal recovery of derivatives in Hardy spaces

The Carathéodory–Fejér problem in the Hardy spaces $ H_p$ is reduced to solving systems of a certain form. The optimal method of recovery of the derivative of any order of a function in $H_p$ from its values on a collection of points is expressed in terms of the solution of a system of the same type. The analogous problem of recovery is considered in the space $h_\infty$ of bounded harmonic functions.