On optimal recovery and information complexity in numerical differentiation and summation

In this paper, we study the problems of numerical differentiation and summation of univariate functions from the weighted Wiener classes. To solve these problems, we propose an approach based on the truncation method. The essence of this method is to replace the infinite Fourier series with a finite sum. It is only necessary to properly select the order of this sum, which plays the role of a regularization parameter here. The results show that the proposed approach not only ensures a stability of approximations and does not require cumbersome computational procedures, but also constructs algorithms that achieve the optimal order of accuracy using the minimal amount of perturbed values of Fourier-Chebyshev coefficients. Moreover, we establish under what conditions the summation problem is well-posed on the considered function classes.