Geodesic
Exactness and optimality of methods for recovering functions from their spectrum
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 201-216.

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

Optimal methods are constructed for recovering functions and their derivatives in a Sobolev class of functions on the line from exactly or approximately defined Fourier transforms of these functions on an arbitrary measurable set. The methods are exact on certain subspaces of entire functions. Optimal recovery methods are also constructed for wider function classes obtained as the sum of the original Sobolev class and a subspace of entire functions. 
@article{TM_2016_293_a13,
     author = {G. G. Magaril-Il'yaev and K. Yu. Osipenko},
     title = {Exactness and optimality of methods for recovering functions from their spectrum},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {201--216},
     publisher = {mathdoc},
     volume = {293},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_293_a13/}
}
TY  - JOUR
AU  - G. G. Magaril-Il'yaev
AU  - K. Yu. Osipenko
TI  - Exactness and optimality of methods for recovering functions from their spectrum
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2016
SP  - 201
EP  - 216
VL  - 293
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2016_293_a13/
LA  - ru
ID  - TM_2016_293_a13
ER  - 
%0 Journal Article
%A G. G. Magaril-Il'yaev
%A K. Yu. Osipenko
%T Exactness and optimality of methods for recovering functions from their spectrum
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2016
%P 201-216
%V 293
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2016_293_a13/
%G ru
%F TM_2016_293_a13
G. G. Magaril-Il'yaev; K. Yu. Osipenko. Exactness and optimality of methods for recovering functions from their spectrum. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 201-216. http://geodesic.mathdoc.fr/item/TM_2016_293_a13/

[1] Nikolskii S.M., Kvadraturnye formuly, Nauka, M., 1988 | MR

[2] Nikolskii S.M., “K voprosu ob otsenkakh priblizhenii kvadraturnymi formulami”, UMN., 5:2 (1950), 165–177 | MR

[3] Kolmogorov A.N., “Über die beste Annäherung von Funktionen einer gegebenen Functionenklasse”, Ann. Math. Ser. 2., 37 (1936), 107–110 ; Kolmogorov A.N., “O nailuchshem priblizhenii funktsii zadannogo funktsionalnogo klassa”, Izbrannye trudy. Matematika i mekhanika, Nauka, M., 1985, 186–189 | DOI | MR | Zbl | MR

[4] Smolyak S.A., Ob optimalnom vosstanovlenii funktsii i funktsionalov ot nikh, Dis. ... kand. fiz.-mat. nauk, MGU, M., 1965

[5] Osipenko K.Yu., “Optimalnaya interpolyatsiya analiticheskikh funktsii”, Mat. zametki., 12:4 (1972), 465–476 | MR | Zbl

[6] Micchelli C.A., Rivlin T.J., “A survey of optimal recovery”, Optimal estimation in approximation theory, Proc. Int. Symp., Freudenstadt, 1976, eds. C.A. Micchelli, T.J. Rivlin, Plenum Press, New York, 1977, 1–54 | MR

[7] 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

[8] Traub J.F., Woźniakowski H., A general theory of optimal algorithms, Acad. Press, New York, 1980 | MR | Zbl

[9] Micchelli C.A., Rivlin T.J., “Lectures on optimal recovery”, Numerical analysis, Proc. SERC Summer School, Lancaster, 1984, Lect. Notes Math., 1129, Springer, Berlin, 1985, 21–93 | DOI | MR

[10] Arestov V.V., “Nailuchshee vosstanovlenie operatorov i rodstvennye zadachi”, Tr. MIAN., 189 (1989), 3–20 | MR

[11] Magaril-Ilyaev G.G., Osipenko K.Yu., “Ob optimalnom vosstanovlenii funktsionalov po netochnym dannym”, Mat. zametki., 50:6 (1991), 85–93 | MR

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

[13] Magaril-Ilyaev G.G., Osipenko K.Yu., “Optimalnoe vosstanovlenie funktsii i ikh proizvodnykh po priblizhennoi informatsii o spektre i neravenstva dlya proizvodnykh”, Funkts. analiz i ego pril., 37:3 (2003), 51–64 | DOI | MR | Zbl

[14] Magaril-Ilyaev G.G., Osipenko K.Yu., “Ob optimalnom garmonicheskom sinteze po netochno zadannomu spektru”, Funkts. analiz i ego pril., 44:3 (2010), 76–79 | DOI | MR | Zbl

[15] Magaril-Ilyaev G.G., Osipenko K.Yu., “Neravenstvo Khardi–Littlvuda–Polia i vosstanovlenie proizvodnykh po netochnoi informatsii”, DAN, 438:3 (2011), 300–302 | MR | Zbl