Geodesic
On Multipliers of Fourier Series in the Haar System
Matematičeskie zametki, Tome 109 (2021) no. 6, pp. 912-920.

Voir la notice de l'article provenant de la source Math-Net.Ru

A complete description of multipliers of Fourier–Haar series in Lorentz spaces is given. Necessary and sufficient conditions for sequences of complex numbers to belong to the class $m(L_{p,r}\to L_{q,s})$ of multipliers of Fourier–Haar series are obtained.
Keywords: Fourier series, Haar system, Lorentz space.
Mots-clés : multipliers of Fourier series, net spaces 
@article{MZM_2021_109_6_a9,
     author = {N. T. Tleukhanova and A. N. Bashirova},
     title = {On {Multipliers} of {Fourier} {Series} in the {Haar} {System}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {912--920},
     publisher = {mathdoc},
     volume = {109},
     number = {6},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_109_6_a9/}
}
TY  - JOUR
AU  - N. T. Tleukhanova
AU  - A. N. Bashirova
TI  - On Multipliers of Fourier Series in the Haar System
JO  - Matematičeskie zametki
PY  - 2021
SP  - 912
EP  - 920
VL  - 109
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2021_109_6_a9/
LA  - ru
ID  - MZM_2021_109_6_a9
ER  - 
%0 Journal Article
%A N. T. Tleukhanova
%A A. N. Bashirova
%T On Multipliers of Fourier Series in the Haar System
%J Matematičeskie zametki
%D 2021
%P 912-920
%V 109
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2021_109_6_a9/
%G ru
%F MZM_2021_109_6_a9
N. T. Tleukhanova; A. N. Bashirova. On Multipliers of Fourier Series in the Haar System. Matematičeskie zametki, Tome 109 (2021) no. 6, pp. 912-920. http://geodesic.mathdoc.fr/item/MZM_2021_109_6_a9/

[1] A. Zygmund, Trigonometric Series, Vol. 2, Cambridge Univ. Press, Cambridge, 1959 | MR

[2] J. Marcinkiewicz, “Sur les multiplicateurs des séries de Fourier”, Studia Math., 8 (1939), 78–91 | DOI

[3] P. I. Lizorkin, “O multiplikatorakh integralov Fure v prostranstvakh $L_{p,\theta}$”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam. Chast 2, Tr. MIAN SSSR, 89, 1967, 231–248 | MR | Zbl

[4] P. I. Lizorkin, “K teorii multiplikatorov Fure”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam. Chast 11, Tr. MIAN SSSR, 173, 1986, 149–163 | MR | Zbl

[5] V. A. Yudin, “Sfericheskie summy ryadov Fure v $L_p$”, Matem. zametki, 46:2 (1989), 145–152 | MR | Zbl

[6] E. D. Nursultanov, “O multiplikatorakh ryadov Fure po trigonometricheskoi sisteme”, Matem. zametki, 63:2 (1998), 235–247 | DOI | MR | Zbl

[7] E. D. Nursultanov, N. T. Tleukhanova, “O multiplikatorakh kratnykh ryadov Fure”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam. Chast 18, Trudy MIAN, 227, Nauka, MAIK «Nauka/Interperiodika», M., 1999, 237–242 | MR | Zbl

[8] E. D. Nursultanov, N. T. Tleukhanova, “Nizhnie i verkhnie otsenki normy multiplikatorov kratnykh trigonometricheskikh ryadov Fure v prostranstvakh Lebega”, Funkts. analiz i ego pril., 34:2 (2000), 86–88 | DOI | MR | Zbl

[9] B. S. Kashin, A. A. Saakyan, Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[10] D. L. Burkholder, “A nonlinear partial differential equation and unconditional constant of the Haar system in $L_p$”, Bull. Amer. Math. Soc. (N.S.), 7:3 (1982), 591–595 | DOI | MR

[11] I. Novikov, E. Semenov, Haar Series and Linear Operators, Math. Appl., 367, Kluwer Acad. Publ., Dordrecht, 1997 | MR

[12] S. Yano, “On a lemma of Marcinkiewicz and its applications to Fourier series”, Tohoku Math. J. (2), 11 (1959), 195–215 | MR

[13] I. B. Bryskin, O. V. Lelond, E. M. Semenov, “Multiplikatory ryadov Fure–Khaara”, Sib. matem. zhurn., 41:4 (2000), 758–766 | MR | Zbl

[14] M. Girardi, “Operator-valued Fourier Haar multipliers”, J. Math. Anal. Appl., 325:2 (2007), 1314–1326 | DOI | MR

[15] O. V. Lelond, E. M. Semenov, S. N. Uksusov, “Prostranstvo multiplikatorov Fure–Khaara”, Sib. matem. zhurn., 46:1 (2005), 130–138 | MR | Zbl

[16] E. M. Semenov, S. N. Uksusov, “Multiplikatory ryadov po sisteme Khaara”, Sib. matem. zhurn., 53:2 (2012), 388–395 | MR

[17] H. M. Wark, “Operator-valued Fourier Haar multipliers on vector-valued $L_1$ spaces”, J. Math. Anal. Appl., 450 (2017), 1148–1156 | DOI | MR

[18] E. D. Nursultanov, “Setevye prostranstva i neravenstva tipa Khardi–Litlvuda”, Matem. sb., 189:3 (1998), 83–102 | DOI | MR | Zbl

[19] E. D. Nursultanov, T. U. Aubakirov, “Teorema Khardi–Littlvuda dlya ryadov Fure–Khaara”, Matem. zametki, 73:3 (2003), 340–347 | DOI | MR | Zbl

[20] B. I. Golubov, “Nailuchshie priblizheniya funktsii v metrike $L_p$ polinomami Khaara i Uolsha”, Matem. sb., 87 (129):2 (1972), 254–274 | MR | Zbl

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

[22] J. Schauder, “Eine Eigenschaft des Haarschen Orthogonal Systems”, Math. Z., 28 (1928), 317–320 | DOI | MR

[23] E. D. Nursultanov, “Neravenstvo raznykh metrik S. M. Nikolskogo i svoistva posledovatelnosti norm summ Fure funktsii iz prostranstva Lorentsa”, Funktsionalnye prostranstva, teoriya priblizhenii, nelineinyi analiz, Trudy MIAN, 255, Nauka, MAIK «Nauka/Interperiodika», M., 2006, 197–215 | MR

[24] L-E. Persson, L. Sarybekova, N. Tleukhanova, “A Lizorkin theorem on Fourier series multipliers for strong regular systems”, Analysis for Science, Engineering and Beyond, Springer Proc. Math., 6, Springer, Heidelberg, 2012, 305–317 | MR

[25] L. O. Sarybekova, T. V. Tararykova, N. T. Tleukhanova, “On a generalization of the Lizorkin theorem on Fourier multipliers”, Math. Inequal. Appl., 13:3 (2010), 613–624 | MR

[26] I. Berg, I. Lefstrem, Interpolyatsionnye prostranstva, Vvedenie. per. s angl., Mir, M., 1980