Geodesic
Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series
Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 723-731

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It follows from results of A. Yudin, V. Yudin, E. Belinskii, and I. Liflyand that if $m\ge2$ and a $2\pi$-periodic (in each variable) function $f(\mathbf x)\in C(T^m)$ belongs to the Nikol'skii class $h_\infty^{(m-1)/2}(T^m)$, then its multiple Fourier series is uniformly convergent over hyperbolic crosses. In this paper, we establish the finality of this result. More precisely, there exists a function in the class $h_\infty^{(m-1)/2}(T^m)$ whose Fourier series is divergent over hyperbolic crosses at some point. 
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     author = {M. I. Dyachenko},
     title = {Uniform {Convergence} of {Hyperbolic} {Partial} {Sums} of {Multiple} {Fourier} {Series}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {723--731},
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     number = {5},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a7/}
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M. I. Dyachenko. Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series. Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 723-731. http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a7/