Geodesic
On the Uniqueness and Integrability of Multiple Trigonometric Series
Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 761-775

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Under minimal constraints on the coefficients, we prove uniqueness theorems for multiple trigonometric series in which, instead of pointwise convergence, we consider the convergence of integral means of spherical, cubic, and other partial sums. We also obtain sufficient conditions for the integrability of multiple trigonometric series, i.e., conditions under which these series are Fourier series.
Keywords: multiple trigonometric series, Fourier series, integral means, spherical and cubic partial sums, periodic function, binary net of cubes, set of the second category. 
@article{MZM_2009_86_5_a12,
     author = {A. A. Talalyan},
     title = {On the {Uniqueness} and {Integrability} of {Multiple} {Trigonometric} {Series}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {761--775},
     publisher = {mathdoc},
     volume = {86},
     number = {5},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a12/}
}
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A. A. Talalyan. On the Uniqueness and Integrability of Multiple Trigonometric Series. Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 761-775. http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a12/