The best approximation to a class of functions of several variables by another class and related extremum problems

V. V. Arestov

Matematičeskie zametki, Tome 64 (1998) no. 3, pp. 323-340.

Résumé

We study the relationship between several extremum problems for unbounded linear operators of convolution type in the spaces $L_\gamma=L_\gamma(\mathbb R^m)$, $m\ge1$, $1\le\gamma\le\infty$. For the problem of calculating the modulus of continuity of the convolution operator $A$ on the function class $Q$ defined by a similar operator and for the Stechkin problem on the best approximation of the operator $A$ on the class $Q$ by bounded linear operators, we construct dual problems in dual spaces, which are the problems on, respectively, the best and the worst approximation to a class of functions by another class.

Bibliographie

[1] Stechkin S. B., “Nailuchshee priblizhenie lineinykh operatorov”, Matem. zametki, 1:2 (1967), 137–148 | MR | Zbl

[2] Arestov V. V., “O nekotorykh ekstremalnykh zadachakh dlya differentsiruemykh funktsii odnoi peremennoi”, Tr. MIAN, 138, Nauka, M., 1975, 3–28 | MR | Zbl

[3] Klots B. E., “Priblizheniya differentsiruemykh funktsii funktsiyami bolshei gladkosti”, Matem. zametki, 21:1 (1977), 21–32 | MR | Zbl

[4] Arestov V. V., Gabushin V. N., “Nailuchshee priblizhenie neogranichennykh operatorov ogranichennymi”, Izv. vuzov. Matem., 1995, no. 11, 42–68 | MR | Zbl

[5] Arestov V. V., “Priblizhenie neogranichennykh operatorov ogranichennymi i rodstvennye ekstremalnye zadachi”, UMN, 51:6 (1996), 89–124 | MR | Zbl

[6] Shilov G. E., Matematicheskii analiz. Vtoroi spetsialnyi kurs, Nauka, M., 1965

[7] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[8] Larsen R., An Introduction to the Theory of Multipliers, Springer, Berlin, 1971 | Zbl

[9] Arestov V. V., “Priblizhenie operatorov, invariantnykh otnositelno sdviga”, Tr. MIAN, 138, Nauka, M., 1975, 43–70 | MR | Zbl

[10] Arestov V. V., “Priblizhenie operatorov tipa svertki lineinymi ogranichennymi operatorami”, Tr. MIAN, 145, Nauka, M., 1980, 3–19 | MR | Zbl

[11] Arestov V. V., “Priblizhenie invariantnykh operatorov”, Matem. zametki, 34:1 (1983), 9–29 | MR

[12] Arestov V. V., “Nailuchshee priblizhenie neogranichennykh operatorov, invariantnykh otnositelno sdviga, lineinymi ogranichennymi operatorami”, Tr. MIAN, 198, Nauka, M., 1992, 3–20 | Zbl

[13] Arestov V. V., “Nailuchshee vosstanovlenie operatorov i rodstvennye zadachi”, Tr. MIAN, 189, Nauka, M., 1989, 3–20 | MR

[14] Taikov L. V., “O nailuchshem priblizhenii v srednem nekotorykh klassov analiticheskikh funktsii”, Matem. zametki, 1:2 (1967), 155–162 | MR | Zbl

[15] Subbotin Yu. N., “Nailuchshee priblizhenie klassa funktsii drugim klassom”, Matem. zametki, 2:5 (1967), 495–504 | MR | Zbl

[16] Subbotin Yu. N., Taikov L. V., “Nailuchshee priblizhenie operatora differentsirovaniya v prostranstve $L_2$”, Matem. zametki, 3:2 (1968), 157–164 | MR

[17] Arestov V. V., Gabushin V. N., “O priblizhenii klassov differentsiruemykh funktsii”, Matem. zametki, 9:2 (1971), 105–112 | MR | Zbl

[18] Subbotin Yu. N., “Svyaz splain-priblizhenii s zadachei priblizheniya klassa klassom”, Matem. zametki, 9:5 (1971), 501–510 | MR | Zbl

[19] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977

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

[21] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973

[22] Gabushin V. N., “Neravenstva dlya norm funktsii i ee proizvodnykh v metrikakh $L_p$”, Matem. zametki, 1:3 (1967), 291–298 | MR | Zbl

[23] Timoshin O. A., “O nailuchshem priblizhenii differentsialnykh operatorov s chastnymi proizvodnymi”, Matem. zametki, 46:1 (1989), 78–87 | MR | Zbl

[24] Arestov V. V., “Priblizhenie lineinykh operatorov i rodstvennye ekstremalnye zadachi”, Tr. MIAN, 138, Nauka, M., 1975, 29–42 | MR | Zbl

[25] Gabushin V. N., “Nailuchshee priblizhenie funktsionalov na nekotorykh mnozhestvakh”, Matem. zametki, 8:5 (1970), 551–562 | MR | Zbl

[26] Taikov L. V., “Neravenstva tipa Kolmogorova i nailuchshie formuly chislennogo differentsirovaniya”, Matem. zametki, 4:2 (1968), 233–238 | MR | Zbl

[27] Magaril-Ilyaev G. G., Tikhomirov V. M., “O neravenstvakh dlya proizvodnykh kolmogorovskogo tipa”, Matem. sb., 188:12 (1997), 73–106 | MR | Zbl

[28] Babenko V. F., Kofanov V. A., Pichugov S. A., “Exact inequalities of Kolmogorov type for multivariable functions and their applications”, East J. Approx., 3:2 (1997), 155–186 | MR | Zbl