Widths of classes of finitely smooth functions in Sobolev spaces

S. N. Kudryavtsev

Matematičeskie zametki, Tome 77 (2005) no. 4, pp. 535-539.

Résumé

We describe the weak asymptotics of the behavior of the Kolmogorov, Gelfand, linear, Aleksandrov, and entropy widths of the unit ball of the space $W_p^lH^\omega(I^d)$ in the space $W_q^m(I^d)$.

Bibliographie

[1] Kolmogoroff A., “Über die beste Annäherung von Funktionen einer gegebehen Funktionenklasse”, Ann. Math., 37 (1936), 107–111 | DOI | MR

[2] Stechkin S. B., “O nailuchshikh priblizheniyakh zadannykh klassov funktsii lyubymi polinomami”, UMN, 9:1 (1954), 133–134 | Zbl

[3] Kolmogorov A. N., “Otsenki minimalnogo chisla elementov – setei v razlichnykh funktsionalnykh prostranstvakh i ikh primenenie k voprosu o predstavimosti funktsii neskolkikh peremennykh superpozitsiyami funktsii menshego chisla peremennykh”, UMN, 10:1 (1955), 192–193

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

[5] Babenko K. I., “Priblizhenie trigonometricheskimi polinomami v nekotorykh klassakh periodicheskikh funktsii neskolkikh peremennykh”, Dokl. AN SSSR, 132:2 (1960), 982–985 | MR | Zbl

[6] Mityagin B. S., “Priblizheniya funktsii v prostranstvakh $L^p$ i $C$ na tore”, Matem. sb., 58 (100):4 (1962), 397–414 | Zbl

[7] Birman M. Sh., Solomyak M. Z., “Kusochno-polinomialnye priblizheniya funktsii klassov $W_p^\alpha$”, Matem. sb., 73:3 (1967), 331–355 | Zbl

[8] Tikhomirov V. M., Nekotorye voprosy teorii priblizheniya, Diss. ... d. f.-m. n., MGU, M., 1969

[9] Makovoz Yu. I., “Ob odnom prieme otsenki snizu poperechnikov mnozhestv v banakhovom prostranstve”, Matem. sb., 87:1 (1972), 136–146 | Zbl

[10] Ismagilov R. S., “Poperechniki mnozhestv v lineinykh normirovannykh prostranstvakh i priblizhenie funktsii trigonometricheskimi mnogochlenami”, UMN, 29:3 (1974), 161–178 | Zbl

[11] Maiorov V. E., “Diskretizatsiya zadachi o poperechnikakh”, UMN, 30:6 (1975), 179–180 | Zbl

[12] Stesin M. I., “Aleksandrovskie poperechniki konechnomernykh mnozhestv i klassov gladkikh funktsii”, Dokl. AN SSSR, 220:6 (1975), 1278–1281 | Zbl

[13] Tikhomirov V. M., Nekotorye voprosy teorii priblizheniya, Izd-vo MGU, M., 1976

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

[15] Maiorov V. E., “O lineinykh poperechnikakh sobolevskikh klassov”, Dokl. AN SSSR, 243:5 (1978), 1127–1130 | MR | Zbl

[16] Höllig K., “Diameters of classes of smooth functions”, Quantitative Approximation, Springer, New York, 1980, 163–176 | MR

[17] 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

[18] Din Zung, Priblizhenie gladkikh funktsii mnogikh peremennykh sredstvami garmonicheskogo analiza, Diss. ... d. f.-m. n., MGU, M., 1985

[19] Galeev E. M., “Poperechniki po Kolmogorovu klassov periodicheskikh funktsii odnoi i neskolkikh peremennykh”, Izv. AN SSSR. Ser. matem., 54:2 (1990), 418–429

[20] Kudryavtsev S. N., “Poperechniki klassov gladkikh funktsii”, Izv. RAN. Ser. matem., 59:4 (1995), 81–104 | Zbl