Approximation of the differentiation operator by linear bounded operators on the class of twice differentiable functions in the space L2(0,∞)

We give an upper estimate for the value of the best approximation of the (firstorder) differentiation operator by linear bounded operators on the class of twice differentiable functions in the space L2(0,∞). This upper estimate is close to a known lower estimate and improves previously known upper estimates. To prove the upper estimate, we consider a specific family of operators; in this family, we choose an operator that provides the least estimate for the value of the best approximation.