Geodesic
The construction of Dirichlet and de la Vallée–Poussin–Nikol'skiĭ kernels for j-Bessel Fourier integrals
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 76 (2015) no. 1, pp. 67-84 
L. N. Lyakhov. The construction of Dirichlet and de la Vallée–Poussin–Nikol'skiĭ kernels for j-Bessel Fourier integrals. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 76 (2015) no. 1, pp. 67-84. http://geodesic.mathdoc.fr/item/MMO_2015_76_1_a1/
@article{MMO_2015_76_1_a1,
     author = {L. N. Lyakhov},
     title = {The construction of {Dirichlet} and de la {Vall\'ee{\textendash}Poussin{\textendash}Nikol'ski\u{i}} kernels for {j-Bessel} {Fourier} integrals},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {67--84},
     year = {2015},
     volume = {76},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMO_2015_76_1_a1/}
}
TY  - JOUR
AU  - L. N. Lyakhov
TI  - The construction of Dirichlet and de la Vallée–Poussin–Nikol'skiĭ kernels for j-Bessel Fourier integrals
JO  - Trudy Moskovskogo matematičeskogo obŝestva
PY  - 2015
SP  - 67
EP  - 84
VL  - 76
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MMO_2015_76_1_a1/
LA  - ru
ID  - MMO_2015_76_1_a1
ER  - 
%0 Journal Article
%A L. N. Lyakhov
%T The construction of Dirichlet and de la Vallée–Poussin–Nikol'skiĭ kernels for j-Bessel Fourier integrals
%J Trudy Moskovskogo matematičeskogo obŝestva
%D 2015
%P 67-84
%V 76
%N 1
%U http://geodesic.mathdoc.fr/item/MMO_2015_76_1_a1/
%G ru
%F MMO_2015_76_1_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We give elementary proofs of some properties of the generalized shift generated by a spherical symmetry. We construct B-kernels for Fourier integrals with respect to Bessel j-functions (Fourier-Bessel transforms). These are designed to play the same role as Dirichlet and de la Vallée–Poussin–Nikol'skiĭ kernels in the theory of trigonometric Fourier integrals and in the theory of function approximation.

[1] Levitan B. M., “Razlozhenie po funktsiyam Besselya v ryady i integraly Fure”, UMN, 6:2 (1951), 102–143 | MR | Zbl

[2] Kipriyanov I. A., Singulyarnye ellipticheskie kraevye zadachi, Nauka, M., 1997

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

[4] Zhitomirskii Ya. I., “Zadacha Koshi dlya sistem lineinykh uravnenii v chastnykh proizvodnykh s differentsialnymi operatorami tipa Besselya”, Matem. sb., 36(78):2 (1955), 299–310 | MR | Zbl

[5] Vatson G. N., Teoriya besselevykh funktsii, v. 1, IL, M., 1949

[6] Lyakhov L. N., Sanina E. L., “Mnogochleny Shlemilkha, interpolyatsionnaya formula Rissa dlya $B$-proizvodnoi i neravenstv Bershteina dlya drobnykh $B$-proizvodnykh Veilya–Marsho”, Doklady RAN, 417:5 (2007), 592–596 | Zbl

[7] Kipriyanov I. A., Katrakhov V. V., “Ob odnom klasse odnomernykh singulyarnykh psevdodifferentsialnykh operatorov”, Matem. sb., 104:1 (1977), 49–68 | MR | Zbl

[8] Katrakhov V. I., Lyakhov L. N., “Polnoe preobrazovanie Fure–Besselya i algebra singulyarnykh psevdodifferentsialnykh operatorov”, Diff. uravneniya, 47:5 (2011), 681–685 | MR

[9] Sanina E. L., Drobnye $B$-proizvodnye Veilya $j$-besselevykh razlozhenii i neravenstvo Bernshteina dlya $B$-proizvodnykh ot chetnykh mnogochlenov Shlemilkha, Diss. ... kand. f.-m. nauk, VGU, Voronezh, 2008

[10] Kipriyanov I. A., Ivanov L. A., “Poluchenie fundamentalnykh reshenii dlya odnorodnykh uravnenii s osobennostyami po mnogim peremennym”, Tr. sem. S. L. Soboleva, 1, 1983, 55–77 | Zbl

[11] Lyakhov L. N., “O $j$-ryadakh Shlemilkha”, Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Ser. Matem. Fiz., 12(155):31 (2013), 62–73

[12] Feoktistova A. A., $B$-liuvillevskie operatsii i priblizhenie funktsii iz vesovykh klassov, Diss. ... kand. f.-m. nauk, VGU, Voronezh, 2012

[13] Tricomi F. G., Lezioni sulle equazioni a derivate parziali, ed. Gheroni T., 1954 ; Raikov V. A., Lektsii po uravneniyam v chastnykh proizvodnykh, B. M. Levitan (predislovie), IL, M., 1957 | MR

[14] Lebedev N. N., Spetsialnye funktsii i ikh prilozheniya, GIFML, M., 1953

[15] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 1, Nauka, M., 1973

[16] Lyakhov L. N., Polovinkin I. P., Shishkina E. L., “Ob odnoi zadache I. A.Kipriyanova dlya singulyarnogo ultragiperbolicheskogo uravneniya”, Diff. uravneniya, 50:4 (2014), 516–528 | DOI | Zbl

[17] Kipriyanov I. A., Klyuchantsev M. I., “O singulyarnykh integralakh, porozhdennykh operatorom obobschennogo sdviga, II”, Sib. matem. zhurn., 11:5 (1970), 1060–1083 | MR

[18] Nikolskii S. M., “Konstruktivnoe predstavlenie nulevykh klassov differentsiruemykh funktsii mnogikh peremennykh”, DAN SSSR, 170 (1966), 512–515 | MR | Zbl

[19] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 2, Nauka, M., 1974