Geodesic
Convergence problems of multiple trigonometric series and spectral decompositions.~I
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 31 (1976) no. 6, pp. 29-86

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This article is a survey of the present state of convergence problems of multiple trigonometric series and spectral decompositions corresponding to self-adjoint elliptic differential operators. Particular attention is paid to questions of localization, uniform convergence, convergence in mean, convergence almost everywhere, and also of absolute convergence. In some cases short sketches of proofs of important propositions are given. This applies mostly to Chapter III, which contains a large number of new proofs of various propositions connected with convergence problems of spectral decompositions. In addition, the article mentions unsolved questions and formulates many new problems 
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     author = {Sh. A. Alimov and V. A. Il'in and E. M. Nikishin},
     title = {Convergence problems of multiple trigonometric series and spectral {decompositions.~I}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     url = {http://geodesic.mathdoc.fr/item/RM_1976_31_6_a3/}
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Sh. A. Alimov; V. A. Il'in; E. M. Nikishin. Convergence problems of multiple trigonometric series and spectral decompositions.~I. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 31 (1976) no. 6, pp. 29-86. http://geodesic.mathdoc.fr/item/RM_1976_31_6_a3/