Geodesic
Some new estimates of the Fourier transform in $\mathbb{L}_2(\mathbb{R})$
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 9, pp. 1419-1426 
V. A. Abilov; F. V. Abilova; M. K. Kerimov. Some new estimates of the Fourier transform in $\mathbb{L}_2(\mathbb{R})$. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 9, pp. 1419-1426. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_9_a0/
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     title = {Some new estimates of the {Fourier} transform in $\mathbb{L}_2(\mathbb{R})$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Given a function $\mathbb{L}_2(\mathbb{R})$, its Fourier transform $$ g(x)=\hat{f}(x)=F[f](x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}f(x)e^{-ixt}dt,\quad f(t)=F^{-1}[g](t)=\frac1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}g(x)e^{ixt}dx $$ and the inverse Fourier transform are considered in the space $f\in\mathbb{L}_2(\mathbb{R})$. New estimates are presented for the integral $ \int_{|t|\geqslant N}|g(t)|^2dt=\int_{|t|\geqslant N}|\hat{f}(t)|^2dt, \quad N\geqslant1 $, in the vase of $f\in\mathbb{L}_2(\mathbb{R})$ characterized by the generalized modulus of continuity of the $k$th order constructed with the help of the Steklov function. Some other estimates associated with this integral are proved.

[1] Titchmarsh E., Vvedenie v teoriyu integralov Fure, Kom Kniga, M., 2005

[2] Bokhner S., Lektsii ob integralakh Fure, Fizmatlit, M., 1962

[3] Dzhrbashyan M. M., Integralnye preobrazovaniya i predstavlenie funktsii v kompleksnoi oblasti, Nauka, M., 1966

[4] Sveshnikov A. G., Bogolyubov A. N., Kravtsov V. V., Lektsii po matematicheskoi fizike, Nauka, M., 2004

[5] Zhukov A. I., Metod Fure v vychislitelnoi matematike, Fizmatlit, M., 1992 | MR | Zbl

[6] Abilov V. A., Abilova F. V., Kerimov M. K., “Neskolko zamechanii o preobrazovanii Fure v prostranstve $\mathbb{L}_2(\mathbb{R})$”, Zh. vychisl. matem. i matem. fiz., 48:6 (2008), 939–945 | MR | Zbl

[7] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR

[8] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960

[9] Abilov V. A., Abilova F. I., “Nekotorye voprosy priblizheniya $2\pi$-periodicheskikh funktsii v prostranstve $\mathbb{L}_2(2\pi)$”, Matem. zametki, 76:6 (2004), 803–811 | DOI | MR | Zbl

[10] Kharshiladze F. I., “Priblizhenie funktsii srednimi V. A. Steklova”, Tr. Tbilisskogo matem. in-ta AN Gruzii, 31, 1966, 55–70 | Zbl

[11] Kharshiladze F. I., “O funktsiyakh Steklova”, Soobsch. AN Gruz. SSR, 14:3 (1953), 139–144

[12] Lanina E. G., “Nailuchshie priblizheniya funktsii i priblizheniya funktsiyami Steklova”, Vestn. MGU. Ser. 1: Matem. i mekhan., 2000, no. 2, 49–52 | MR | Zbl

[13] Abilov V. A., “On the convergence of multiple Fourier series and quadratur formulae”, Math. Balcanica. New Series, 16:1–4 (2002), 73–94 | MR | Zbl