Geodesic
On the Norms of the Integral Means of Spherical Fourier Sums
Matematičeskie zametki, Tome 96 (2014) no. 5, pp. 701-708.

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

The paper deals with the spherical Fourier sums $S_r(f,x)=\sum_{\|k\|\le r}\widehat f(k)e^{ik\cdot x}$ of a periodic function $f$ in $m$ variables and the strong integral means of these sums $((\int_0^R |S_r(f,x)|^p \,dr)/R)^{1/p}$ for $p\ge1$. We establish the exact growth order as $R\to+\infty$ of the corresponding operators, i.e., the growth order of the quantities $\sup_{|f|\le 1}((\int_0^R |S_r(f,0)|^p\, dr)/R)^{1/p}$. The upper and lower bounds differ by their coefficients, which depend only on the dimension $m$. A sufficient condition on the function ensuring the uniform strong $p$-summability of its Fourier series is given.
Keywords: periodic function of several variables, spherical Fourier sums, exact growth order of operators, $p$-summability of Fourier series. 
@article{MZM_2014_96_5_a6,
     author = {O. I. Kuznetsova and A. N. Podkorutov},
     title = {On the {Norms} of the {Integral} {Means} of {Spherical} {Fourier} {Sums}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {701--708},
     publisher = {mathdoc},
     volume = {96},
     number = {5},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_96_5_a6/}
}
TY  - JOUR
AU  - O. I. Kuznetsova
AU  - A. N. Podkorutov
TI  - On the Norms of the Integral Means of Spherical Fourier Sums
JO  - Matematičeskie zametki
PY  - 2014
SP  - 701
EP  - 708
VL  - 96
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2014_96_5_a6/
LA  - ru
ID  - MZM_2014_96_5_a6
ER  - 
%0 Journal Article
%A O. I. Kuznetsova
%A A. N. Podkorutov
%T On the Norms of the Integral Means of Spherical Fourier Sums
%J Matematičeskie zametki
%D 2014
%P 701-708
%V 96
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2014_96_5_a6/
%G ru
%F MZM_2014_96_5_a6
O. I. Kuznetsova; A. N. Podkorutov. On the Norms of the Integral Means of Spherical Fourier Sums. Matematičeskie zametki, Tome 96 (2014) no. 5, pp. 701-708. http://geodesic.mathdoc.fr/item/MZM_2014_96_5_a6/

[1] G. H. Hardy, J. E. Littlewood, “Sur la série de Fourier d'une function á carré sommable”, C. R. Acad. Sci. Paris, 156 (1913), 1307–1309 | Zbl

[2] O. I. Kuznetsova, A. N. Podkorytov, “O silnykh srednikh sfericheskikh summ Fure”, Algebra i analiz, 25:3 (2013), 121–130 | MR

[3] O. I. Kuznetsova, “Silnye sfericheskie srednie kratnykh ryadov Fure”, Izv. NAN Armenii. Matem., 44:4 (2009), 27–40 | MR | Zbl

[4] K. I. Babenko, “O skhodimosti v srednem kratnykh ryadov Fure i asimptotike yadra Dirikhle sfericheskikh srednikh”, Preprinty IPM, 1971, 052

[5] Sh. A. Alimov, V. A. Ilin, E. M. Nikishin, “Voprosy skhodimosti kratnykh trigonometricheskikh ryadov i spektralnykh razlozhenii. I”, UMN, 31:6 (1976), 28–83 | MR | Zbl

[6] I. Stein, G. Veis, Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl