Best approximation of the differentiation operator in the space L2L2 on the semiaxis

We solve the problem on the best approximation of the (first-order) differentiation operator by linear bounded operators on the class of twice differentiable functions in the space L2(0,∞)L2(0,∞). The best approximating operator is constructed. The optimal differentiation error is found for twice differentiable functions given with a known error in L2(0,∞)L2(0,∞), and the optimal method is described.