Geodesic
Approximation in $L_2$ by partial integrals of the multidimensional Fourier transform in the eigenfunctions of the Sturm–Liouville operator
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 4, pp. 136-152 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

For approximations in the space $L^2(\mathbb{R}^d_+)$ by partial integrals of the multidimensional Fourier transform in the eigenfunctions of the Sturm–Liouville operator, we prove the Jackson inequality with exact constant and optimal argument in the modulus of continuity. The multidimensional weight that defines the Sturm–Liouville operator is the product of one-dimensional weights. The one-dimensional weights can be, in particular, power and hyperbolic weights with various parameters. The optimality of the argument in the modulus of continuity is established by means of the multidimensional Gauss quadrature formula over zeros of an eigenfunction of the Sturm–Liouville operator. The obtained results are complete; they generalize a number of known results.
Keywords: Sturm–Liouville operator, $L^2$-space, Jackson inequality
Mots-clés : Fourier transform, Gauss quadrature formula. 
@article{TIMM_2016_22_4_a13,
     author = {D. V. Gorbachev and V. I. Ivanov and R. A. Veprintsev},
     title = {Approximation in $L_2$ by partial integrals of the multidimensional {Fourier} transform in the eigenfunctions of the {Sturm{\textendash}Liouville} operator},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {136--152},
     year = {2016},
     volume = {22},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a13/}
}
TY  - JOUR
AU  - D. V. Gorbachev
AU  - V. I. Ivanov
AU  - R. A. Veprintsev
TI  - Approximation in $L_2$ by partial integrals of the multidimensional Fourier transform in the eigenfunctions of the Sturm–Liouville operator
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2016
SP  - 136
EP  - 152
VL  - 22
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a13/
LA  - ru
ID  - TIMM_2016_22_4_a13
ER  - 
%0 Journal Article
%A D. V. Gorbachev
%A V. I. Ivanov
%A R. A. Veprintsev
%T Approximation in $L_2$ by partial integrals of the multidimensional Fourier transform in the eigenfunctions of the Sturm–Liouville operator
%J Trudy Instituta matematiki i mehaniki
%D 2016
%P 136-152
%V 22
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a13/
%G ru
%F TIMM_2016_22_4_a13
D. V. Gorbachev; V. I. Ivanov; R. A. Veprintsev. Approximation in $L_2$ by partial integrals of the multidimensional Fourier transform in the eigenfunctions of the Sturm–Liouville operator. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 4, pp. 136-152. http://geodesic.mathdoc.fr/item/TIMM_2016_22_4_a13/

[1] Chernykh N.I., “O neravenstve Dzheksona v $L_2$”, Tr. MIAN, 88 (1967), 71–74

[2] Arestov V.V., Chernykh N.I., “On the $L_2$-approximation of periodic functions by trigonometric polynomials”, Approximation and functions spaces, Proc. Intern. Conf. (Gdansk, 1979), North-Holland, Amsterdam, 1981, 25–43 | MR

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

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

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

[6] Gorbachev D.V., Ivanov V.I., Veprintsev R.A., “Optimal argument in sharp Jackson's inequality in the space $L_2$ with the hyperbolic weight”, Math. Notes, 96:5 (2014), 904–913 | DOI | MR | Zbl

[7] Veprintsev R.A., “Priblizhenie v $L_2$ chastichnymi integralami mnogomernogo preobrazovaniya Yakobi”, Mat. zametki, 97:6 (2015), 815–831 | DOI | MR | Zbl

[8] Gorbachev D.V., Ivanov V.I., “Priblizhenie v $L_2$ chastichnymi integralami preobrazovaniya Fure po sobstvennym funktsiyam operatora Shturma–Liuvillya”, Mat. zametki, 100:4 (2016), 519–530 | DOI

[9] Gorbachev D.V., Ivanov V.I., “Kvadraturnye formuly Gaussa i Markova po nulyam sobstvennykh funktsii zadachi Shturma–Liuvillya, tochnye dlya tselykh funktsii eksponentsialnogo tipa”, Mat. sb., 206:8 (2015), 63–98 | DOI | MR | Zbl

[10] Arestov V.V., Popov V.Yu., “Neravenstvo Dzheksona na sfere v $L_2$”, Izv. vuzov. Matematika, 1995, no. 8, 13–20 | MR | Zbl

[11] Ivanov V.I., Yunpin Lyu, Smirnov O.I., “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