Geodesic
Approximation of functions by asymmetric two-point hermite polynomials and its optimization
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 12, pp. 1999-2014 
V. V. Shustov. Approximation of functions by asymmetric two-point hermite polynomials and its optimization. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 12, pp. 1999-2014. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_12_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A function is approximated by two-point Hermite interpolating polynomials with an asymmetric orders-of-derivatives distribution at the endpoints of the interval. The local error estimate is examined theoretically and numerically. As a result, the position of the maximum of the error estimate is shown to depend on the ratio of the numbers of conditions imposed on the function and its derivatives at the endpoints of the interval. The shape of a universal curve representing a reduced error estimate is found. Given the sum of the orders of derivatives at the endpoints of the interval, the ordersof-derivatives distribution is optimized so as to minimize the approximation error. A sufficient condition for the convergence of a sequence of general two-point Hermite polynomials to a given function is given.

[1] Hermite Sh., “Sur la formule d'interpolation de Lagrange”, J. fur die reine und angewandte Math., 84 (1878), 70–79 | DOI

[2] Berezin I. S., Zhidkov N. P., Metody vychislenii, v. 1, Fizmatlit, M., 1962 | MR

[3] Kalitkin H. H., Chislennye metody, ucheb. posobie, BKhV-Peterburg, SPb., 2011, 592 pp.

[4] Verzhbitskii V. M., Chislennye metody. (Matematicheskii analiz i obyknovennye differentsialnye uravneniya), Ucheb. posobie dlya vuzov, Izd. 2-e ispr., OOO Izdatelskii dom “ONIKS 21 vek”, M., 2005, 400 pp. | MR

[5] Samarskii A. A., Gulin A. V., Chislennye metody, Ucheb. posobie dlya vuzov, Nauka. Fizmatgiz, M., 1989, 432 pp.

[6] Shakhov Yu. N., Deza E. I., Chislennye metody, Uchebnoe posobie, Izd. 2-e, ispr. i dop., Knizhnyi dom “Librokom”, M., 2010, 248 pp.

[7] Boglaev Yu. P., Vychislitelnaya matematika i programmirovanie, Vyssh. shkola, M., 1990, 544 pp.

[8] Kakhaner D., Mouler K., Nesh S., Chislennye metody i programmnoe obespechenie, Per s angl., Mir, M., 1998, 575 pp.

[9] Zavyalov Yu. S., Leus V. A., Skorospelov V. A., Splainy v inzhenernoi geometrii, Mashinostroenie, M., 1985, 224 pp. | MR

[10] Bogachev K. U., Praktikum na EVM. Metody priblizheniya funktsii, Izd. mekhan.-matem. f-ta MGU, M., 1998, 176 pp.

[11] Spitzbart A., “A generalization of Hermite's interpolation formula”, Amer. Math. Monthly, 67 (1960), 42–46 | DOI | MR | Zbl

[12] Uteshev A. Yu., Tamasyan G. Sh., “K zadache polinominalnogo interpolirovaniya s kratnymi uzlami”, Vestn. S.-Peterburgskogo un-ta, Ser. 10, 2010, no. 3, 76–85 | MR

[13] Shustov V. V., “O priblizhenii funktsii dvukhtochechnymi interpolyatsionnymi mnogochlenami Ermita”, Zh. vychisl. matem. i matem. fiz., 2015, no. 7, 1091–1108 | DOI

[14] Bronshtein I. N., Semendyaev K. A., Spravochnik po matematike dlya inzhenerov i uchaschikhsya vtuzov, Gostekhizdat, M., 1954, 608 pp.

[15] Kudryavtsev L. D., Matematicheskii analiz, v. 1, Vyssh. shkola, M., 1970, 592 pp.