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

Abstract

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.