Approximation of the Multidimensional Jacobi Transform in~$L_2$ by Partial Integrals

R. A. Veprintsev

Matematičeskie zametki, Tome 97 (2015) no. 6, pp. 815-831.

Résumé

In the space $L_2(\mathbb{R}^d_+)$ with hyperbolic weight, the exact Jackson inequality with the optimal argument in the modulus of continuity is proved. The optimal argument is the least value of the argument in the modulus of continuity for which the exact constant in the Jackson inequality takes the minimum value. The approximation is carried out by partial integrals of the multidimensional Jacobi transform. In the study of the optimal argument, the geometry of the domain of the partial integral and the geometry of the neighborhood of zero in the definition of the modulus of continuity are taken into account. The optimal argument is obtained for the case in which the first skew field is an $l_p$-ball for $1\leq p\leq 2$ and the second, a parallelepiped.

Détail

Mots-clés : Jacobi transform
Keywords: Jackson inequality, modulus of continuity, $l_p$-ball, parallelepiped, Jacobi function, Lebesgue-measurable function, Bernstein's inequality.

Bibliographie

[1] V. V. Arestov, N. I. Chernykh, “On the $L_2$-approximation of periodic functions by trigonometric polynomials”, Approximation and Function Spaces, North-Holland, Amsterdam, 1981, 25–43 | MR | Zbl

[2] N. I. Chernykh, “O neravenstve Dzheksona v $L_2$”, Priblizhenie funktsii v srednem, Sbornik rabot, Tr. MIAN SSSR, 88, 1967, 71–74 | MR | Zbl

[3] E. E. Berdysheva, “Dve vzaimosvyazannye ekstremalnye zadachi dlya tselykh funktsii mnogikh peremennykh”, Matem. zametki, 66:3 (1999), 336–350 | DOI | MR | Zbl

[4] D. V. Gorbachev, “Ekstremalnye zadachi dlya tselykh funktsii eksponentsialnogo sfericheskogo tipa”, Matem. zametki, 68:2 (2000), 179–187 | DOI | MR | Zbl

[5] A. V. Ivanov, “Nekotorye ekstremalnye zadachi dlya tselykh funktsii v vesovykh prostranstvakh”, Izv. TulGU. Estestvennye nauki, 2010, no. 1, 26–44

[6] A. V. Ivanov, V. I. Ivanov, “Teorema Dzheksona v prostranstve $L_2(\mathbb{R}^d)$ so stepennym vesom”, Matem. zametki, 88:1 (2010), 148–151 | DOI | MR | Zbl

[7] A. V. Ivanov, V. I. Ivanov, “Teoriya Danklya i teorema Dzheksona v prostranstve $L_2(\mathbb R^d)$ so stepennym vesom”, Tr. IMM UrO RAN, 16, no. 4, 2010, 180–192

[8] A. V. Ivanov, “Zadacha Logana dlya tselykh funktsii mnogikh peremennykh i konstanty Dzheksona v vesovykh prostranstvakh”, Izv. TulGU. Estestvennye nauki, 2011, no. 2, 29–58

[9] A. V. Ivanov, V. I. Ivanov, “Optimalnye argumenty v neravenstve Dzheksona v prostranstve $L_2(\mathbb{R}^d)$ so stepennym vesom”, Matem. zametki, 94:3 (2013), 338–348 | DOI | MR | Zbl

[10] V. I. Ivanov, A. V. Ivanov, “Optimalnye argumenty v neravenstve Dzheksona–Stechkina v $L_2(\mathbb{R}^d)$ s vesom Danklya”, Matem. zametki, 96:5 (2014), 674–686 | DOI

[11] D. V. Gorbachev, V. I. Ivanov, R. A. Veprintsev, “Optimal Argument in the Sharp Jackson Inequality in the Space $L_2$ with Hyperbolic Weight”, Math. Notes, 96:6 (2014), 904–913 | DOI

[12] M. Flensted-Jensen, T. H. Koornwinder, “The convolution structure for Jacobi function expansions”, Ark. Mat., 11 (1973), 245–262 | DOI | MR | Zbl

[13] M. Flensted-Jensen, T. H. Koornwinder, “Jacobi functions: The addition formula and the positivity of dual convolution structure”, Ark. Mat., 17 (1979), 139–151 | DOI | MR | Zbl

[14] T. Koornwinder, “A new proof of a Paley–Wiener type theorem for the Jacobi transform”, Ark. Mat., 13 (1975), 145–159 | DOI | MR | Zbl

[15] B. M. Levitan, I. S. Sargsyan, Operatory Shturma–Liuvillya i Diraka, Nauka, M., 1988 | MR | Zbl

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

[17] D. V. Gorbachev, Izbrannye zadachi teorii funktsii i teorii priblizhenii, Grif i K, Tula, 2005

[18] D. V. Gorbachev, V. I. Ivanov, “Kvadraturnye formuly Gaussa i Markova po nulyam funktsii Yakobi”, Sovremennye problemy matematiki, mekhaniki, informatiki, TulGU, Tula, 2014, 31–37

[19] V. I. Ivanov, Yunpin Lyu, O. I. Smirnov, “O nekotorykh klassakh tselykh funktsii eksponentsialnogo tipa v prostranstvakh $L_p(\mathbb{R}^d)$ so stepennym vesom”, Izv. TulGU. Estestvennye nauki, 2011, no. 2, 70–80

[20] N. I. Akhiezer, Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR | Zbl

[21] V. A. Yudin, “Mnogomernaya teorema Dzheksona v $L_2$”, Matem. zametki, 29:2 (1981), 309–315 | MR | Zbl