Geodesic
Optimal recovery of values of functions and their derivatives from inaccurate data on the Fourier transform
Sbornik. Mathematics, Tome 195 (2004) no. 10, pp. 1461-1476 
G. G. Magaril-Il'yaev; K. Yu. Osipenko. Optimal recovery of values of functions and their derivatives from inaccurate data on the Fourier transform. Sbornik. Mathematics, Tome 195 (2004) no. 10, pp. 1461-1476. http://geodesic.mathdoc.fr/item/SM_2004_195_10_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The problems of the optimal recovery of the derivatives of functions from inaccurate information about the Fourier transforms of these functions on a finite interval or the entire number line are considered. The Stechkin problem of the approximation of derivatives by bounded linear functionals, which is closely connected to this range of problems, is also studied. Precise Kolmogorov-type inequalities for derivatives corresponding to these problems are obtained.

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

[2] Micchelli C. A., Rivlin T. J., “A survey of optimal recovery”, Optimal estimation in approximation theory, eds. C. A. Micchelli, 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”, Numerical analysis, Proc. SERC Summer Sch. (Lancaster/Engl., 1984), Lecture Notes in Math., 1129, Springer-Verlag, Berlin, 1984, 21–93 | MR

[5] Arestov V. V., “Nailuchshee vosstanovlenie operatorov i rodstvennye zadachi”, Trudy MIAN, 189, 1989, 3–20 | MR

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

[7] Osipenko K. Yu., Optimal recovery of analytic functions, Nova Science Publ., Inc., Huntington, New York, 2000

[8] Magaril-Ilyaev G. G., Osipenko K. Yu., “Optimalnoe vosstanovlenie funktsii i ikh proizvodnykh po koeffitsientam Fure, zadannym s pogreshnostyu”, Matem. sb., 193:3 (2002), 79–100 | MR | Zbl

[9] 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 prilozh., 37:3 (2003), 51–64 | MR | Zbl

[10] Arestov V. V., “Priblizhenie neogranichennykh operatorov ogranichennymi i rodstvennye ekstremalnye zadachi”, UMN, 51:6 (1996), 89–124 | MR | Zbl

[11] Osipenko K. Yu., “Optimal recovery of periodic functions from Fourier coefficients given with an error”, J. Complexity, 12 (1996), 35–46 | DOI | MR | Zbl

[12] Taikov L. V., “Neravenstva tipa Kolmogorova i nailuchshie formuly chislennogo differentsirovaniya”, Matem. zametki, 4:2 (1968), 233–238 | MR | Zbl

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

[14] Magaril-Ilyaev G. G., Tikhomirov V. M., Vypuklyi analiz i ego prilozheniya, Editorial URSS, M., 2003

[15] Magaril-Il'yaev G. G., Osipenko K. Yu., Tikhomirov V. M., “Optimal recovery and extremum theory”, Comput. Methods Funct. Theory, 2 (2002), 87–112 | MR | Zbl