Geodesic
Optimal Recovery of Linear Functionals on Sets of Finite Dimension
Matematičeskie zametki, Tome 84 (2008) no. 4, pp. 602-608.

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

Suppose that $X$ is a linear space and $L_1,\dots,L_n$ is a system of linearly independent functionals on $P$, where $P\subset X$ is a bounded set of dimension $n+1$. Suppose that the linear functional $L_0$ is defined in $X$. In this paper, we find an algorithm that recovers the functional $L_0$ on the set $P$ with the least error among all linear algorithms using the information $L_1f,\dots,L_nf$, $f\in P$.
Keywords: optimal recovery of a linear functional, optimal complexity, information operator, information radius, problem complexity, Chebyshev polynomial.
Mots-clés : optimal interpolation 
@article{MZM_2008_84_4_a11,
     author = {S. P. Sidorov},
     title = {Optimal {Recovery} of {Linear} {Functionals} on {Sets} of {Finite} {Dimension}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {602--608},
     publisher = {mathdoc},
     volume = {84},
     number = {4},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_84_4_a11/}
}
TY  - JOUR
AU  - S. P. Sidorov
TI  - Optimal Recovery of Linear Functionals on Sets of Finite Dimension
JO  - Matematičeskie zametki
PY  - 2008
SP  - 602
EP  - 608
VL  - 84
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2008_84_4_a11/
LA  - ru
ID  - MZM_2008_84_4_a11
ER  - 
%0 Journal Article
%A S. P. Sidorov
%T Optimal Recovery of Linear Functionals on Sets of Finite Dimension
%J Matematičeskie zametki
%D 2008
%P 602-608
%V 84
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2008_84_4_a11/
%G ru
%F MZM_2008_84_4_a11
S. P. Sidorov. Optimal Recovery of Linear Functionals on Sets of Finite Dimension. Matematičeskie zametki, Tome 84 (2008) no. 4, pp. 602-608. http://geodesic.mathdoc.fr/item/MZM_2008_84_4_a11/

[1] P. W. Gaffney, M. J. D. Powell, “Optimal interpolation”, Numerical Analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975), Lecture Notes in Math., 506, Springer-Verlag, Berlin, 1976, 90–99 | DOI | MR | Zbl

[2] P. W. Gaffney, Optimal Interpolation, Thesis, Oxford Univ. Press, Oxford, 1976

[3] P. W. Gaffney, “To compute the optimal interpolation formula”, Math. Comp., 32:143 (1978), 763–777 | DOI | MR | Zbl

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

[5] S. A. Smolyak, “Interpolyatsionnye i kvadraturnye formuly na klassakh $W_s^\alpha$ i $E_s^\alpha$”, Dokl. AN SSSR, 131:5 (1960), 1028–1031 | MR | Zbl

[6] R. E. Barnhill, J. A. Wixom, “An error analysis for interpolation on analytic functions”, SIAM J. Numer. Anal., 5:3 (1968), 522–529 | DOI | MR | Zbl

[7] N. S. Bakhvalov, “Otsenki snizu asimptoticheskikh kharakteristik klassov funktsii s dominiruyuschei smeshannoi proizvodnoi”, Matem. zametki, 12:6 (1972), 655–664 | MR | Zbl

[8] B. D. Boyanov, “Nailuchshie metody interpolirovaniya dlya nekotorykh klassov differentsiruemykh funktsii”, Matem. zametki, 17:4 (1975), 511–524 | MR | Zbl

[9] B. D. Bojanov, V. G. Chernogorov, “An optimal interpolation formula”, J. Approx. Theory, 20:3 (1977), 264–274 | DOI | MR | Zbl

[10] V. L. Velikin, “Optimalnaya interpolyatsiya periodicheskikh differentsiruemykh funktsii s ogranichennoi starshei proizvodnoi”, Matem. zametki, 22:5 (1977), 663–670 | MR | Zbl

[11] M. Golomb, “Interpolation operators as optimal recovery schemes for classes of analytic functions”, Optimal Estimation in Approximation Theory (Proc. Internat. Sympos., Freudenstadt, 1976), Plenum Press, New York, 1977, 93–138 | MR | Zbl

[12] N. M. Korobov, Teoretiko-chislovye metody v priblizhennom analize, Fizmatlit, M., 1963 | MR | Zbl

[13] L. F. Meyers, A. Sard, “Best interpolation formulas”, J. Math. Phys., 29 (1950), 198–206 | MR | Zbl

[14] A. A. Melkman, “$n$-width and optimal interpolation of time- and band-limited functions”, Optimal Estimation in Approximation Theory (Proc. Internat. Sympos., Freudenstadt, 1976), Plenum Press, New York, 1977, 55–68 | MR | Zbl

[15] K. Yu. Osipenko, “Nailuchshee priblizhenie analiticheskikh funktsii po informatsii ob ikh znacheniyakh v konechnom chisle tochek”, Matem. zametki, 19:1 (1976), 29–40 | MR | Zbl

[16] K. Yu. Osipenko, “Ob optimalnykh metodakh vosstanovleniya v prostranstvakh Khardi–Soboleva”, Matem. sb., 192:2 (2001), 67–86 | MR | Zbl

[17] W. Forst, “Optimale Hermite-interpolation differenzierbaren periodischer funktionen”, J. Approx. Theory, 20:4 (1977), 333–347 | DOI | MR | Zbl

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

[19] A. Sard, “Best approximate integration formulas; best approximation formulas”, Amer. J. Math., 71:1 (1949), 80–91 | DOI | MR | Zbl

[20] Dzh. Traub, Kh. Vozhnyakovskii, Obschaya teoriya optimalnykh algoritmov, Mir, M., 1983 | MR | Zbl