Inequalities of Kolmogorov's type for derived functions in two variables and application to approximation by an ``angle''

S. B. Vakarchuk ; A. V. Shvachko

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2015), pp. 3-22.

Résumé

For functions of two variables we obtain sharp inequalities of Kolmogorov's type for partial and mixed intermediate derivatives. We also consider applications of the results to some problems of approximation of functions of two variables by angles and obtain a series of relations which are exact in definite sense.

Détail

Keywords: Hermite polynomials, Fourier–Hermite series, inequalities of Kolmogorov's type, the best approximation by angle, generalized polynomial.

Bibliographie

[1] Magaril-Ilyaev G. G., Tikhomirov V. M., Vypuklyi analiz i ego prilozheniya, Editorial URSS, M., 2000

[2] Konovalov V. N., “Tochnye neravenstva dlya norm funktsii, tretikh chastnykh, vtorykh smeshannykh i kosykh proizvodnykh”, Matem. zametki, 23:1 (1978), 67–78 | MR | Zbl

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

[4] Buslaev A. P., “O priblizhenii operatora differentsirovaniya”, Matem. zametki, 29:5 (1981), 731–742 | MR | Zbl

[5] Timoshin O. A., “Nailuchshee priblizhenie operatora differentsirovaniya vtoroi smeshannoi proizvodnoi v metrikakh $L$ i $C$ na ploskosti”, Matem. zametki, 36:3 (1984), 369–375 | MR | Zbl

[6] Timofeev V. G., “Neravenstvo tipa Landau dlya funktsii neskolkikh peremennykh”, Matem. zametki, 37:5 (1985), 676–689 | MR | Zbl

[7] Berdnikova I. V., Rafalson S. Z., “Nekotorye neravesntva mezhdu normami funktsii i ee proizvodnykh v integralnykh metrikakh”, Izv. vuzov. Matem., 1985, no. 12, 3–6 | MR | Zbl

[8] Rafalson S. Z., “O priblizhenii funktsii v srednem summami Fure–Ermita”, Izv. vuzov. Matem., 1968, no. 7, 78–84 | MR | Zbl

[9] Smirnov V. I., Kurs vysshei matematiki, V 5-ti tomakh, v. 5, Fizmatlit, M., 1959

[10] Suetin P. K., Klassicheskie ortogonalnye mnogochleny, 3-e izd., Fizmatlit, M., 2007 | MR

[11] Vakarchuk S. B., Vakarchuk M. B., “Neravenstva tipa Kolmogorova dlya analiticheskikh funktsii odnoi i dvukh kompleksnykh peremennykh i ikh primenenie k teorii approksimatsii”, Ukr. matem. zhurn., 63:12 (2011), 1579–1601 | MR

[12] Rzhavinskaya E. V., O priblizhenii algebraicheskimi mnogochlenami v metrike $L_p$ s vesom, Diss. $\dots$ kand. fiz.-matem. nauk, M., 1980

[13] Potapov M. K., “O priblizhenii ‘uglom’ ”, Proceedings of the conference on constructive theory of functions, Akadémiai Kidaó, Budapest, 1972, 371–399 | MR

[14] Tomich M., “O priblizhenii uglom funktsii s dominiruyuschim modulem gladkosti”, Publ. De L'inst. Math. (Beograd), 23:37 (1978), 193–206

[15] Haussmann W., Zeller K., “Uniqueness and non-uniqueness in bivariate $L^1$-approximation”, Approximation Theory IV, Proc. of Intern. Symp. (College Station, Tex., January 10–14, 1983), Academic Press, New York, 1983, 509–514 | MR

[16] Temlyakov V. N., “Priblizhenie funktsii s ogranichennoi smeshannoi proizvodnoi”, Tr. matem. in-ta AN SSSR, 178, 1986, 3–113 | MR | Zbl

[17] Gonska H., Jetter K., “Jackson-type theorems on approximation by trigonometric and algebraic pseudopolynomials”, J. Approxim. Theory, 48:4 (1986), 396–406 | DOI | MR | Zbl

[18] Vakarchuk S. B., “O nailuchshem priblizhenii obobschennymi polinomami v odnom prostranstve analiticheskikh funktsii dvukh kompleksnykh peremennykh”, Izv. vuzov. Matem., 1991, no. 7, 14–25 | MR | Zbl