Geodesic
Optimal Recovery of Functions and Their Derivatives from Inaccurate Information about the Spectrum and Inequalities for Derivatives
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 3, pp. 51-64

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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.
Keywords: optimal recovery, inequality for derivatives.
Mots-clés : Fourier transform 
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     author = {G. G. Magaril-Il'yaev and K. Yu. Osipenko},
     title = {Optimal {Recovery} of {Functions} and {Their} {Derivatives} from {Inaccurate} {Information} about the {Spectrum} and {Inequalities} for {Derivatives}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {51--64},
     publisher = {mathdoc},
     volume = {37},
     number = {3},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2003_37_3_a3/}
}
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G. G. Magaril-Il'yaev; K. Yu. Osipenko. Optimal Recovery of Functions and Their Derivatives from Inaccurate Information about the Spectrum and Inequalities for Derivatives. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 3, pp. 51-64. http://geodesic.mathdoc.fr/item/FAA_2003_37_3_a3/