Approximation of differentiation operators by bounded linear operators in lebesgue spaces on the axis and related problems in the spaces of $(p,q)$-multipliers and their predual spaces

We consider a variant $E_{n,k}(N;r,r;p,p)$ of the four-parameter Stechkin problem $E_{n,k}(N;r,s;p,q)$ on the best approximation of differentiation operators of order $k$ on the class of $n$ times differentiable functions $(0<k<n)$ in Lebesgue spaces on the real axis. We discuss the state of research in this problem and related problems in the spaces of multipliers of Lebesgue spaces and their predual spaces. We give two-sided estimates for $E_{n,k}(N;r,r;p,p)$. The paper is based on the author's talk at the S.B.Stechkin's International Workshop-Conference on Function Theory (Kyshtym, Chelyabinsk region, August 1–10, 2023).