Optimal Recovery of Functions and Their Derivatives from Inaccurate Information about the Spectrum and Inequalities for Derivatives

We study problems of optimal recovery of functions and their derivatives in the $L_2$ metric on the line from information about the Fourier transform of the function in question known approximately on a finite interval or on the entire line. Exact values of optimal recovery errors and closed-form expressions for optimal recovery methods are obtained. We also prove a sharp inequality for derivatives (closely related to these recovery problems), which estimates the $k$th derivative of a function in the $L_2$-norm on the line via the $L_2$-norm of the $n$th derivative and the $L_p$-norm of the Fourier transform of the function.