Optimal recovery of values of functions and their derivatives from inaccurate data on the Fourier transform

The problems of the optimal recovery of the derivatives of functions from inaccurate information about the Fourier transforms of these functions on a finite interval or the entire number line are considered. The Stechkin problem of the approximation of derivatives by bounded linear functionals, which is closely connected to this range of problems, is also studied. Precise Kolmogorov-type inequalities for derivatives corresponding to these problems are obtained.