Optimal recovery of functions and their derivatives from inaccurate information about the spectrum and inequalities for derivatives. (English. Russian original) Zbl 1048.41007

Magaril-Il’yaev, G. G.; Osipenko, K. Yu.

Funct. Anal. Appl. 37, No. 3, 203-214 (2003); translation from Funkts. Anal. Prilozh. 37, No. 3, 51-64 (2003).

The paper is devoted to 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 whole line. Exact values of optimal recovery errors and closed-form expessions for optimal recovery methods are obtained. The authors also prove a sharp inequality for derivatives 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.

Reviewer: Boris Rubin (Jerusalem)

MSC:

41A30	Approximation by other special function classes
41A17	Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)

Keywords:

optimal recivery; Fourier transform; inequality for derivatives