Geodesic
$u$-convergence of multiple Fourier series
Izvestiya. Mathematics , Tome 59 (1995) no. 2, pp. 353-366

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The $u$-convergence of multiple Fourier series is studied, generalizing convergence in the Pringsheim sense, with respect to spheres. A definitive condition in terms of moduli of smoothness is found on a functional class that implies the $u$-convergence of Fourier series in the metrics $L_p(T^m)$, where $1\leqslant p\leqslant\infty$, $p\ne 2$ and $m\geqslant 2$. 
@article{IM2_1995_59_2_a5,
     author = {M. I. Dyachenko},
     title = {$u$-convergence of multiple {Fourier} series},
     journal = {Izvestiya. Mathematics },
     pages = {353--366},
     publisher = {mathdoc},
     volume = {59},
     number = {2},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_59_2_a5/}
}
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M. I. Dyachenko. $u$-convergence of multiple Fourier series. Izvestiya. Mathematics , Tome 59 (1995) no. 2, pp. 353-366. http://geodesic.mathdoc.fr/item/IM2_1995_59_2_a5/