Geodesic
On the problem of optimal interpolation of functions
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2023), pp. 59-70.

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In this work, the problem of constructing optimal interpolation formulas is discussed. Here, first, an exact upper bound for the error of the interpolation formula in the Sobolev space is calculated. The existence and uniqueness of the optimal interpolation formula, which gives the smallest error, are proved. An algorithm for finding the coefficients of the optimal interpolation formula is given. By implementing this algorithm, the optimal coefficients are found.
Keywords: Sobolev space, extremal function, composite lattice optimal cubature formulas, error functional. 
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Kh. M. Shadimetov; N. H. Mamatova. On the problem of optimal interpolation of functions. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2023), pp. 59-70. http://geodesic.mathdoc.fr/item/IVM_2023_12_a4/

[1] Sobolev S.L., “O zadache interpolirovaniya funktsii $n$ peremennykh”, DAN SSSR, 137:4 (1961), 778–781 | Zbl

[2] Holladay J.C., “Smoothest curve approximation”, Math. Tables Aids Comput., 11 (1957), 223–243 | DOI | MR

[3] Alberg Dzh., Nilson E., Uolsh Dzh., Teoriya splainov i ee prilozheniya, Mir, M., 1972, 318 pp.

[4] Stechkin S.B., Subbotin Yu.N., Splainy v vychislitelnoi matematike, Nauka, M., 1976, 248 pp. | MR

[5] Loran P.Zh., Approksimatsiya i optimizatsiya, Mir, M., 1975, 496 pp.

[6] Vasilenko V.A., Splain-funktsii: teoriya, algoritmy, programmy, Nauka, Novosibirsk, 1983, 215 pp. | MR

[7] Ignatov M.I., Pevnyi A.B., Naturalnye splainy mnogikh peremennykh, Nauka, L., 1991, 127 pp.

[8] Korneichuk N.P., Tochnye konstanty v teorii priblizheniya, Nauka, M., 1987, 424 pp. | MR

[9] Mamatova N.Kh., Khaëtov A.R., Shadimetov Kh.M., “Postroenie reshetchatykh optimalnykh interpolyatsionnykh formul v prostranstve Soboleva $\widetilde{L}_2^{(m)}(H)$ periodicheskikh funktsii $n$ peremennykh metodom Soboleva”, Ufimsk. matem. zhurn., 5:1 (2013), 90–101 | MR

[10] Cabada A., Hayotov A.R., Shadimetov Kh.M., “Construction of $D^m$-splines in $L_2^{(m)}(0,1)$ space by Sobolev method”, Appl. Math. and Comput., 244:2 (2014), 542–551 | MR | Zbl

[11] Hayotov A.R., Milovanović G.V., Shadimetov Kh.M., “Interpolation splines minimizing a semi-norm”, Calcolo, 51:2 (2014), 245–260 | DOI | MR | Zbl

[12] Hayotov A.R., “Construction of Interpolation Splines Minimizing the Semi-norm in the Space $K_2(P_m)$”, J. Siberian Federal Univ. Math. and Phys., 11:3 (2018), 383–396 | DOI | MR

[13] Babaev S.S., Hayotov A.R., “Optimal interpolation formulas in the space $W_2^{(m,m-1)}$”, Calcolo, 56 (2019), 23 | DOI | MR | Zbl

[14] Shadimetov Kh.M., Hayotov A.R., Nuraliev F.A., “Construction of optimal interpolation formulas in the Sobolev space”, J. Math. Sci., 264 (2022), 782–793 | DOI | MR | Zbl

[15] Dautov R.Z., “Tochnaya otsenka pogreshnosti nailuchshego priblizheniya algebraicheskimi polinomami v vesovom $L_2(-1,1)$”, Izv. vuzov. Matem., 2013, no. 5, 61–63 | MR | Zbl

[16] Dautov R.Z., “Pryamye i obratnye teoremy approksimatsii funktsii algebraicheskimi polinomami i splainami v normakh prostranstva Soboleva”, Izv. vuzov. Matem., 2022, no. 6, 79–86

[17] Shadimetov Kh.M., “Diskretnyi analog operatora $d^{2m}/dx^{2m}$ i ego postroenie”, Vopr. vychisl. i prikl. matem., 79 (1985), 22–35 | Zbl