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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@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/}
}
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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/

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