Geodesic
Divergence almost everywhere of rectangular partial sums of multiple Fourier series of bounded functions
Matematičeskie zametki, Tome 64 (1998) no. 1, pp. 24-36 
S. Galstyan; G. A. Karagulian. Divergence almost everywhere of rectangular partial sums of multiple Fourier series of bounded functions. Matematičeskie zametki, Tome 64 (1998) no. 1, pp. 24-36. http://geodesic.mathdoc.fr/item/MZM_1998_64_1_a3/
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     author = {S. Galstyan and G. A. Karagulian},
     title = {Divergence almost everywhere of rectangular partial sums of multiple {Fourier} series of bounded functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {24--36},
     year = {1998},
     volume = {64},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_1_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we establish the following results, which are the multidimensional generalizations of well-known theorems: 1) Suppose that a function $f\in C(\mathbb T^m)$ has no intervals of constancy in $\mathbb T^m$; then there exists a homeomorphism $\varphi\colon\mathbb T^m\to\mathbb T^m$ such that the Fourier series of the superposition $F=f\circ\varphi$ is divergent with respect to rectangles almost everywhere; 2) for any integrable function $f\in L^1(\mathbb T^m)$, with $|f(\mathbf x)|\geqslant\alpha>0$, $x\in\mathbb T^m$, there exists a signum function $\varepsilon(\mathbf x)=\pm 1$, $\mathbf x\in\mathbb T^m$ such that the Fourier series of the product $f(\mathbf x)\varepsilon(\mathbf x)$ is divergent with respect to rectangles almost everywhere.

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