Approximation of the Hardy–Sobolev class of functions analytic in a half-plane by entire functions of exponential type

We study the value $\mathcal E_\sigma(H^p_n)_{H^p}$ of the best approximation in the norm of the Hardy space $H^p$ for $1\le p\le\infty$ of the Hardy–Sobolev class $H_n^p$ of functions analytic in a half-plane with bounded $H^p$-norm of the $n$th-order derivative by entire functions of exponential type not exceeding $\sigma$. The equality $\mathcal E_\sigma(H^p_n)_{H^p}=\sigma^{-n}$ is proved. A linear method providing the best approximation of the class is constructed.