Best Approximation and Symmetric Decreasing Rearrangements of Functions

N. P. Korneichuk

Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 179-193.

Résumé

The problem of estimating the best approximation by a subspace of classes of functions of $n$ variables defined by restrictions imposed on the modulus of continuity is considered on the basis of the duality principle. An approach is analyzed that is connected with the representation of a function of $n$ variables as a countable sum of simple functions and the subsequent transition to spatial symmetric decreasing rearrangements.

Bibliographie

[1] Favard J., “Application de la formule sommatoire d'Euler a la demonstration de quelques proprietes extremales des integrales des fonctions periodiques”, Mat. Tidsskrift, 1936, no. 4, 81–84

[2] Favard J., “Sur l'approximation des fonctions periodiques par des polynomes trigonometriques”, C. R. Acad. Sci. Paris, 203 (1936), 1122–1124 | Zbl

[3] Favard J., “Sur les meilleurs procedes d'approximation de certaines classes de fonctions par des polynomes trigonometriques”, Bull. Sci. Math. Sér. 2, 60 (1937), 209–224; 243–256

[4] Nikolskii S. M., “Priblizhenie funktsii trigonometricheskimi polinomami v srednem”, Izv. AN SSSR. Ser. mat., 10:3 (1946), 207–256

[5] Korneichuk N. P., “O suschestvovanii lineinogo polinomialnogo operatora, dayuschego na klasse funktsii nailuchshee priblizhenie”, DAN SSSR, 143:1 (1962), 25–27

[6] Korneichuk N. P., Ekstremalnye zadachi teorii priblizheniya, Nauka, M., 1976, 320 pp. | MR

[7] Korneichuk N. P., “K zadache o nailuchshem lineinom metode priblizheniya dlya klassa $H^\alpha$”, Sovremennye problemy deistvitelnogo i kompleksnogo analiza, Izd. In-ta matematiki AN USSR, Kiev, 1984, 87–90 | MR

[8] Korneichuk N. P., “Verkhnie grani nailuchshikh priblizhenii na klassakh differentsiruemykh periodicheskikh funktsii v metrikakh $C$ i $L$”, DAN SSSR, 190:2 (1970), 269–271

[9] Korneichuk N. P., “Ekstremalnye znacheniya funktsionalov i nailuchshee priblizhenie na klassakh periodicheskikh funktsii”, Izv. AN SSSR. Ser. mat., 35:1 (1971), 93–124 | MR

[10] Khardi G. G., Littlvud Dzh. E., Polia G., Neravenstva, Izd-vo inostr. lit., M., 1948, 456 pp.

[11] Korneichuk N. P., “O ravnomernom priblizhenii periodicheskikh funktsii podprostranstvami konechnoi razmernosti”, DAN SSSR, 213:3 (1973), 525–528

[12] Korneichuk N. P., “Nailuchshee priblizhenie splainami na klassakh periodicheskikh funktsii v metrike $L$”, Mat. zametki, 20:5 (1976), 655–664 | MR

[13] Korneichuk N. P., “Neravenstva dlya differentsiruemykh periodicheskikh funktsii i nailuchshee priblizhenie odnogo klassa funktsii drugim”, Izv. AN SSSR. Ser. mat., 36:2 (1972), 423–434

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

[15] Korneichuk N. P., “O nailuchshem priblizhenii funktsii $n$ peremennykh”, Ukr. mat. zhurn., 51:10 (1999), 1352–1359 | MR

[16] Ivanov L. D., Variatsii mnozhestv i funktsii, Nauka, M., 1975, 352 pp. | MR

[17] Korneichuk N. P., “Informativnost funktsionalov”, Ukr. mat. zhurn., 46:9 (1994), 1156–1163 | MR | Zbl

[18] Babenko K. I., “O priblizhenii periodicheskikh funktsii mnogikh peremennykh trigonometricheskimi mnogochlenami”, DAN SSSR, 132:2 (1960), 247–250 | MR | Zbl

[19] Potapov M. K., “Issledovanie nekotorykh klassov funktsii pri pomoschi priblizheniya “uglom””, Tr. MIAN, 117, 1972, 256–291 | MR | Zbl

[20] Korneichuk N. P., “Nailuchshee priblizhenie polinomialnymi splainami periodicheskikh funktsii dvukh peremennykh”, Ukr. mat. zhurn., 52:1 (2000), 52–57 | MR