Lower estimates of the widths of the classes of functions defined by a modulus of continuity

V. T. Shevaldin

Izvestiya. Mathematics , Tome 45 (1995) no. 2, pp. 399-415.

Résumé

For kernels $K$ satisfying the condition $B_{2m}$ introduced by the author, lower bounds are found for the Kolmogorov widths of classes of convolutions $K\ast H^\omega$, $\omega$ convex, in the uniform metric. In a number of cases these bounds are sharp.

Bibliographie

[1] V. T. Shevaldin, “Otsenki snizu poperechnikov klassov istokoobrazno predstavimykh funktsii”, Tr. MIAN, 189, 1989, 185–200 | MR | Zbl

[2] N. P. Korneichuk, Ekstremalnye zadachi teorii priblizheniya, Nauka, M, 1976 | MR

[3] V. M. Tikhomirov, Nekotorye voprosy teorii priblizhenii, Izd-vo MGU, M., 1976 | MR

[4] N. P. Korneichuk, Splainy v teorii priblizheniya, Nauka, M., 1984 | MR

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

[6] I. C. Mairhuber, I. J. Schoenberg, R. E. Williamson, “On variation diminishing transformations on the Circle”, Rend. Circ. Mat. Palermo, 1959, no. 8, 241–270 | DOI | MR | Zbl

[7] V. F. Babenko, “Priblizhenie klassov funktsii, zadavaemykh s pomoschyu modulya nepreryvnosti”, DAN SSSR, 298:6 (1988), 1296–1299 | MR

[8] V. T. Shevaldin, Otsenki snizu poperechnikov nekotorykh klassov periodicheskikh funktsii, Preprint, UrO AN SSSR, Sverdlovsk, 1989 | MR | Zbl

[9] V. T. Shevaldin, “Otsenki snizu poperechnikov nekotorykh klassov periodicheskikh funktsii”, Tr. MIAN, 198, 1992, 242–267 | MR | Zbl

[10] V. M. Tikhomirov, “Poperechniki mnozhestv v funktsionalnykh prostranstvakh i teoriya nailuchshikh priblizhenii”, UMN, 15:3(93) (1960), 81–120 | MR | Zbl

[11] Yu. N. Subbotin, “Priblizhenie “splain”-funktsiyami i otsenki poperechnikov”, Tr. MIAN, 109, 1971, 35–60 | MR | Zbl

[12] V. T. Shevaldin, “Nekotorye zadachi ekstremalnoi interpolyatsii v srednem dlya lineinykh differentsialnykh operatorov”, Tr. MIAN, 164, 1983, 203–240 | MR | Zbl

[13] S. I. Novikov, V. T. Shevaldin, “Otsenka snizu dlya chetnogo poperechnika klassa periodicheskikh funktsii, opredelyaemogo differentsialnym operatorom i modulem nepreryvnosti”, Approksimatsiya v konkretnykh i abstraktnykh banakhovykh prostranstvakh, UNTs AN SSSR, Sverdlovsk, 1987, 107–112 | MR

[14] V. M. Tikhomirov, “Nailuchshie metody priblizheniya i interpolirovaniya differentsiruemykh funktsii v prostranstve $C[-1;1]$”, Matem. sb., 80:2 (1969), 290–304 | Zbl

[15] V. F. Babenko, O. V. Polyakov, “Tochnye otsenki nekotorykh kharakteristik klassov periodicheskikh funktsii”, Teoriya priblizhenii i smezhnye voprosy analiza i topologii, In-t matematiki AN USSR, Kiev, 1987, 9–14 | MR

[16] A. K. Kushpel, “Tochnye otsenki poperechnikov klassov svertok”, Izv. AN SSSR. Ser. matem., 52:6 (1988), 1305–1322

[17] V. K. Dzyadyk, “O nailuchshem priblizhenii na klassakh periodicheskikh funktsii, opredelyaemykh integralami ot lineinoi kombinatsii absolyutno monotonnykh yader”, Matem. zametki, 16:5 (1974), 691–701 | Zbl

[18] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integraly i ryady, Nauka, M., 1981 | MR | Zbl

[19] G. G. Lorentz, Approximation of functions, Holt, Rinehalt and Winston, N.Y., 1966 | MR | Zbl

[20] V. I. Ruban, “Chetnye poperechniki klassov $W^rH^\omega$ v prostranstve $C_{2\pi}$”, Matem. zametki, 15:3 (1974), 387–392 | MR | Zbl

[21] N. P. Korneichuk, “Neravenstva dlya differentsiruemykh funktsii i nailuchshee priblizhenie odnogo klassa funktsii drugim”, Izv. AN SSSR. Ser. matem., 36 (1972), 423–434

[22] I. Feschiev, M. Khemeamin, “Verkhnie grani nailuchshikh priblizhenii trigonometricheskimi polinomami na klassakh $W^rH^\omega$ v prostranstve $C$”, Nauchn. tr. Plovdiv. un-ta. Mat., 22, no. 1, 1984, 45–67 | MR | Zbl

[23] I. Feschiev, M. Khemeamin, “Verkhnie grani nailuchshikh priblizhenii trigonometricheskimi polinomami na klassakh $W^rH^\omega$ prostranstve $L$”, Nauchn. tr. Plovdiv. un-ta. Mat., 22, no. 2, 1984, 105–123 | MR | Zbl