Geodesic
Convergence of multiple Fourier series for functions of bounded variation
Matematičeskie zametki, Tome 61 (1997) no. 4, pp. 583-595

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For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy. 
@article{MZM_1997_61_4_a9,
     author = {S. A. Telyakovskii and V. N. Temlyakov},
     title = {Convergence of multiple {Fourier} series for functions of bounded variation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {583--595},
     publisher = {mathdoc},
     volume = {61},
     number = {4},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_61_4_a9/}
}
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S. A. Telyakovskii; V. N. Temlyakov. Convergence of multiple Fourier series for functions of bounded variation. Matematičeskie zametki, Tome 61 (1997) no. 4, pp. 583-595. http://geodesic.mathdoc.fr/item/MZM_1997_61_4_a9/