Kolmogorov widths of classes of periodic functions of one and several variables

È. M. Galeev

Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 435-448.

Résumé

The order of Kolmogorov widths $d_N(\widetilde W_{\bar p}^{\bar\alpha},\widetilde L_q)$ are determined for the class $\widetilde W_{\bar p}^{\bar\alpha}=\bigcap\limits_{i=1}^m\widetilde W_{p^i}^{\alpha^i}$ that is the intersection of classes of periodic functions of one variable of “higher” smoothness, in the space $\widetilde L_q$ for $1$, and estimates from above for “low” smoothness, and also the order of Kolmogorov widths $d_N(\widetilde H_p^r,\widetilde L_q)$ is calculated for periodic functions of several variables in the space $\widetilde L_q$ for $1$. The estimate from below for $d_N(\widetilde H_p^r,\widetilde L_q)$ reduces to the estimate from below of the width of a finite-dimensional set whose width is determined.

Bibliographie

[1] Nikolskaya N. S., “Priblizhenie periodicheskikh funktsii klassa $SH_p^r$ summami Fure”, Sib. matem. zhurn., 15:4 (1975), 761–780

[2] Analiz – 2, Itogi nauki i tekhniki: Sovremennye problemy matematiki. Fundamentalnye napravleniya, 14, VINITI, M., 1987

[3] Kulanin E. D., “Otsenki poperechnikov klassov Soboleva maloi gladkosti”, Vest. MGU. Ser. matem., 1983, no. 2, 24–30 | MR | Zbl

[4] Kashin B. S., “O poperechnikakh klassov Soboleva maloi gladkosti”, Vest. MGU. Ser. matem., 1981, no. 5, 50–54 | MR | Zbl

[5] Galeev E. M., “Priblizhenie summami Fure klassov funktsii s neskolkimi ogranichennymi proizvodnymi”, Matem. zametki, 23:2 (1978), 197–212 | MR | Zbl

[6] Galeev E. M., “Poryadki ortoproektsionnykh poperechnikov klassov periodicheskikh funktsii odnoi i neskolkikh peremennykh”, Matem. zametki, 43:2 (1988), 197–211 | MR | Zbl

[7] Galeev E. M., “Nekotorye otsenki poperechnikov peresecheniya klassov funktsii”, UMN, 37:6 (1982), 153–154 | MR | Zbl

[8] Galeev E. M., “Poperechniki po Kolmogorovu peresecheniya klassov periodicheskikh funktsii i konechnomernykh mnozhestv”, Matem. zametki, 29:5 (1981), 749–760 | MR | Zbl

[9] Temlyakov V. N., “Priblizhenie periodicheskikh funktsii neskolkikh peremennykh trigonometricheskimi polinomami i poperechniki nekotorykh klassov funktsii”, Izv. AN SSSR. Ser. matem., 49:5 (1985), 986–1030 | MR

[10] Galeev E. M., “Poperechniki po Kolmogorovu klassov periodicheskikh funktsii mnogikh peremennykh $\widetilde{W}{}^{\bar\alpha}_p$ i $H^{\bar\alpha}_p$ v prostranstve $\widetilde{L}_q$”, Izv. AN SSSR. Ser. matem., 49:5 (1985), 916–934 | MR | Zbl

[11] Temlyakov V. N., “Priblizhenie funktsii s ogranichennoi smeshannoi proizvodnoi”, Tr. MIAN SSSR, 178, 1986, 3–113 | MR | Zbl

[12] Temlyakov V. N., “Poperechniki nekotorykh klassov funktsii neskolkikh peremennykh”, Dokl. AN SSSR, 267:3 (1982), 314–317 | MR

[13] Gluskin E. D., “Peresecheniya kuba s oktaedrom plokho approksimiruyutsya podprostranstvami maloi razmernosti”, Priblizhenie funktsii spetsialnymi klassami operatorov, Arkhangelsk, 1987, 35–41 | Zbl

[14] Gluskin E. D., “O nekotorykh konechnomernykh zadachakh teorii poperechnikov”, Vest. LGU, 1981, no. 13, 5–10 | MR | Zbl

[15] Gluskin E. D., Geometricheskie kharakteristiki operatorov v simmetrichnykh prostranstvakh, Dis. ... kand. fiz.-matem. nauk, L., 1981

[16] Zigmund A., Trigonometricheskie ryady, t. I, II, Mir, M., 1965 | MR

[17] Galeev E. M., “O lineinykh poperechnikakh klassov periodicheskikh funktsii mnogikh peremennykh”, Vest. MGU. Ser. matem., 1987, no. 4, 13–16 | MR | Zbl

[18] Kashin B. S., “Poperechniki nekotorykh konechnomernykh mnozhestv i klassov gladkikh funktsii”, Izv. AN SSSR. Ser. matem., 41:2 (1977), 334–351 | MR | Zbl

[19] Gluskin E. D., “Normy sluchainykh matrits i poperechniki konechnomernykh mnozhestv”, Matem. sb., 120:2 (1983), 180–189 | MR | Zbl

[20] Tikhomirov V. M., Nekotorye voprosy teorii priblizhenii, MGU, M., 1976 | MR

[21] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[22] Khardi G., Littlvud D., Polia G., Neravenstva, IL, M., 1948

[23] Milman V. D., Schectman G., Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes Math., 1200, 1986 | MR | Zbl

[24] Gluskin E. D., “Ob otsenkakh norm nekotorykh $p$-absolyutno summiruyuschikh operatorov”, Funkts. analiz, 12:2 (1978), 23–31 | MR

[25] Korneichuk N. P., Ekstremalnye zadachi teorii priblizhenii, Nauka, M., 1976

[26] Galeev E. M., “Priblizhenie periodicheskikh funktsii odnoi i mnogikh peremennykh”, Konstruktivnaya teoriya funktsii, Tezisy Mezhdunarodnoi konferentsii po konstruktivnoi teorii funktsii, Sofiya, 1987, 102 | MR

[27] Din Zung, “O priblizhenii periodicheskikh funktsii mnogikh peremennykh”, UMN, 38:6 (1983), 111–112 | MR | Zbl

[28] Din Zung, “Priblizhenie klassov gladkikh funktsii mnogikh peremennykh”, Tr. seminara im. I. G. Petrovskogo, no. 10, 1984, 207–226 | Zbl