Geodesic
Optimal recovery of the derivative of the function from its inaccurately given other orders of derivatives and the function itself
Vladikavkazskij matematičeskij žurnal, Tome 18 (2016) no. 3, pp. 60-71 
S. A. Unuchek. Optimal recovery of the derivative of the function from its inaccurately given other orders of derivatives and the function itself. Vladikavkazskij matematičeskij žurnal, Tome 18 (2016) no. 3, pp. 60-71. http://geodesic.mathdoc.fr/item/VMJ_2016_18_3_a6/
@article{VMJ_2016_18_3_a6,
     author = {S. A. Unuchek},
     title = {Optimal recovery of the derivative of the function from its inaccurately given other orders of derivatives and the function itself},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {60--71},
     year = {2016},
     volume = {18},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2016_18_3_a6/}
}
TY  - JOUR
AU  - S. A. Unuchek
TI  - Optimal recovery of the derivative of the function from its inaccurately given other orders of derivatives and the function itself
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2016
SP  - 60
EP  - 71
VL  - 18
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VMJ_2016_18_3_a6/
LA  - ru
ID  - VMJ_2016_18_3_a6
ER  - 
%0 Journal Article
%A S. A. Unuchek
%T Optimal recovery of the derivative of the function from its inaccurately given other orders of derivatives and the function itself
%J Vladikavkazskij matematičeskij žurnal
%D 2016
%P 60-71
%V 18
%N 3
%U http://geodesic.mathdoc.fr/item/VMJ_2016_18_3_a6/
%G ru
%F VMJ_2016_18_3_a6

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

The paper deals with the problem of simultaneous recovery of the $k_1$-th and $k_2$-th order derivatives of a function in the mean square norm from inaccurately given derivatives of $n_1$-th and $n_2$-th order and the function itself. The solution is given under some conditions on the errors of given derivatives and the function itself. The problem is solved completely for the case $k_1=k$, $n_1=2k$, $k_2=3k$, $n_2=4k$, $k\in\mathbb N$. It turns out that in contrast to previously encountered situations in the general case, the error of recovery depends on errors of all three errors of input data.

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

[2] Nikolskii S. M., Kvadraturnye formuly, Nauka, M., 1988, 254 pp. | MR

[3] Marchuk A. G., Osipenko K. Yu., “Nailuchshee priblizhenie funktsii, zadannykh s pogreshnostyu v konechnom chisle tochek”, Mat. zametki, 17:3 (1975), 359–368 | MR | Zbl

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

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

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