Convergence of Fourier double series and Fourier integrals of functions on $T^2$ and $\mathbb R^2$ after rotations of coordinates

O. S. Dragoshanskii

Sbornik. Mathematics, Tome 191 (2000) no. 11, pp. 1587-1606

Résumé

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.