Geodesic
On the mean convergence of multiple Fourier series and the asymptotics of the Dirichlet kernel of spherical means
Eurasian mathematical journal, Tome 9 (2018) no. 4, pp. 22-60

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Preprint no. 52 (1971), Institut Prikladnoi Matematiki Akademii Nauk USSR, Moscow (in Russian). Translated into English by E. Liflyand. See comments to this paper by E. Liflyand in this volume of the Eurasian Mathematical Journal. 
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     author = {K. I. Babenko},
     title = {On the mean convergence of multiple {Fourier} series and the asymptotics of the {Dirichlet} kernel of spherical means},
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K. I. Babenko. On the mean convergence of multiple Fourier series and the asymptotics of the Dirichlet kernel of spherical means. Eurasian mathematical journal, Tome 9 (2018) no. 4, pp. 22-60. http://geodesic.mathdoc.fr/item/EMJ_2018_9_4_a2/