On the approximation of the differentiation operator by linear bounded operators on the class of twice differentiable functions in the space $L_2(0,\infty)$

We give an upper bound for the error of the best approximation of the (first-order) differentiation operator by linear bounded operators on the class of twice differentiable functions in the space $L_2(0,\infty)$. This upper bound is close to a known lower bound and improves the previous upper bounds. To prove the upper estimate, we consider a specific family of operators; in this family, we choose an operator that provides the least bound for the error of the best approximation.