Geodesic
Convergence of Fourier double series and Fourier integrals of functions on $T^2$ and $\mathbb R^2$ after rotations of coordinates
Sbornik. Mathematics, Tome 191 (2000) no. 11, pp. 1587-1606 
O. S. Dragoshanskii. Convergence of Fourier double series and Fourier integrals of functions on $T^2$ and $\mathbb R^2$ after rotations of coordinates. Sbornik. Mathematics, Tome 191 (2000) no. 11, pp. 1587-1606. http://geodesic.mathdoc.fr/item/SM_2000_191_11_a0/
@article{SM_2000_191_11_a0,
     author = {O. S. Dragoshanskii},
     title = {Convergence of {Fourier} double series and {Fourier} integrals of functions on $T^2$ and $\mathbb R^2$ after rotations of coordinates},
     journal = {Sbornik. Mathematics},
     pages = {1587--1606},
     year = {2000},
     volume = {191},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_11_a0/}
}
TY  - JOUR
AU  - O. S. Dragoshanskii
TI  - Convergence of Fourier double series and Fourier integrals of functions on $T^2$ and $\mathbb R^2$ after rotations of coordinates
JO  - Sbornik. Mathematics
PY  - 2000
SP  - 1587
EP  - 1606
VL  - 191
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/SM_2000_191_11_a0/
LA  - en
ID  - SM_2000_191_11_a0
ER  - 
%0 Journal Article
%A O. S. Dragoshanskii
%T Convergence of Fourier double series and Fourier integrals of functions on $T^2$ and $\mathbb R^2$ after rotations of coordinates
%J Sbornik. Mathematics
%D 2000
%P 1587-1606
%V 191
%N 11
%U http://geodesic.mathdoc.fr/item/SM_2000_191_11_a0/
%G en
%F SM_2000_191_11_a0

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

Let $f(\xi,\eta)$ be a function vanishing for $\xi^2+\eta^2>r^2$, where $r$ is sufficiently small, and with Fourier series (of the function considered in the square $(-\pi,\pi]^2$) or Fourier integral (of the function considered in the plane $\mathbb R^2$) convergent uniformly or almost everywhere over rectangles. It is shown that a rotation of the system of coordinates through $\pi/4$ $$ \begin{cases} \xi=(x-y)/\sqrt 2\,, \\ \eta=(y+x)/\sqrt 2 \end{cases} $$ can “damage” the convergence of the Fourier series or the Fourier integral of the resulting function.

[1] Zigmund A., Trigonometricheskie ryady, Mir, M., 1965 | MR

[2] Bari N. K., Trigonometricheskie ryady, Mir, M., 1961 | MR

[3] Saakyan A. A., “O teoreme Bora dlya kratnykh trigonometricheskikh ryadov”, Matem. zametki, 64:6 (1998), 913–924 | MR | Zbl

[4] Kakhan Zh.-P., Absolyutno skhodyaschiesya ryady Fure, Mir, M., 1970

[5] Bloshanskii I. L., “O ravnoskhodimosti razlozhenii v kratnyi trigonometricheskii ryad Fure i integral Fure”, Matem. zametki, 18:2 (1975), 153–168 | MR

[6] Luzin N. N., “Ob odnom vide skhodimosti integrala Dirikhle”, Integral i trigonometricheskii ryad, Gostekhizdat, M.–L., 1951, 320–324 | MR

[7] Bakhbukh M., Nikishin E. M., “O skhodimosti dvoinykh ryadov Fure ot nepreryvnykh funktsii”, Sib. matem. zhurn., 14:6 (1973), 1189–1199 | MR | Zbl

[8] Bloshanskii I. L., “Obobschennaya lokalizatsiya pochti vsyudu i skhodimost dvoinykh ryadov Fure”, Dokl. AN SSSR, 242:1 (1978), 11–13 | MR