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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 
@article{FAA_2003_37_3_a3,
     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/}
}
TY  - JOUR
AU  - G. G. Magaril-Il'yaev
AU  - K. Yu. Osipenko
TI  - Optimal Recovery of Functions and Their Derivatives from Inaccurate Information about the Spectrum and Inequalities for Derivatives
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2003
SP  - 51
EP  - 64
VL  - 37
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2003_37_3_a3/
LA  - ru
ID  - FAA_2003_37_3_a3
ER  - 
%0 Journal Article
%A G. G. Magaril-Il'yaev
%A K. Yu. Osipenko
%T Optimal Recovery of Functions and Their Derivatives from Inaccurate Information about the Spectrum and Inequalities for Derivatives
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2003
%P 51-64
%V 37
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2003_37_3_a3/
%G ru
%F FAA_2003_37_3_a3
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/

[1] Smolyak S. A., Ob optimalnom vosstanovlenii funktsii i funktsionalov ot nikh, Kand. diss., MGU, M., 1965

[2] Micchelli C. A., Rivlin T. J., “A survey of optimal recovery”, Optimal Estimation in Approximation Theory, eds. C. A. Micchelli and T. J. Rivlin, Plenum Press, New York, 1977, 1–54 | MR

[3] Traub J. F., Woźniakowski H., A General Theory of Optimal Algorithms, Academic Press, New York, 1980 | MR

[4] Micchelli C. A., Rivlin T. J., “Lectures on Optimal Recovery”, Lect. Notes in Math., 1129, Springer-Verlag, Berlin, 1985, 21–93 | DOI | MR

[5] Osipenko K. Yu., Optimal Recovery of Analytic Functions, Nova Science Publ., Huntington, New York, 2000

[6] Magaril-Ilyaev G. G., Tikhomirov V. M., “O neravenstvakh dlya proizvodnykh kolmogorovskogo tipa”, Matem. sb., 188:12 (1997), 73–106 | DOI | MR

[7] Magaril-Ilyaev G. G., Tikhomirov V. M., Vypuklyi analiz i ego prilozheniya, Editorial URSS, M., 2000

[8] Magaril-Ilyaev G. G., Osipenko K. Yu., Tikhomirov V. M., “Optimalnoe vosstanovlenie i teoriya ekstremuma”, Dokl. RAN, 379:2 (2001), 161–164 | MR

[9] Melkman A. A., Micchelli C. A., “Optimal estimation of linear operators in Hilbert spaces from inaccurate data”, SIAM J. Numer. Anal., 16 (1979), 87–105 | DOI | MR | Zbl

[10] Magaril-Ilyaev G. G., Osipenko K. Yu., “Optimalnoe vosstanovlenie funktsii i ikh proizvodnykh po koeffitsientam Fure, zadannym s pogreshnostyu”, Matem. sb., 193 (2002), 79–100 | DOI | MR

[11] “Tikhomirov V. M., Magaril-Ilyaev G. G.”, Izbrannye trudy A. N. Kolmogorova. Matematika i mekhanika, Nauka, M., 1985, 386–390 | MR

[12] Arestov V. V., “Priblizhenie neogranichennykh operatorov ogranichennymi i rodstvennye ekstremalnye zadachi”, UMN, 51 (1996), 89–124 | DOI | MR | Zbl