Sufficient conditions for the convergence (almost everywhere) of multiple trigonometric Fourier series of functions f in the classes Lp, p > 1, are obtained in the case where rectangular partial sums Sn(x; f) of this series have numbers n = (n1, …, nN) ∊ ℤN, N ≥ 3, such that of N components only k (1 ≤ k ≤ N – 2) are elements of some lacunary sequences. Earlier, in the case where N – 1 components of the number n are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained for functions in the classes Lp, p > 1, by M. Kojima (1977), and for functions in Orlizc classes by D. K. Sanadze, Sh. V. Kheladze (1977) and N. Yu. Antonov (2014). Note that presence of two or more “free” components in the number n, as follows from the results by C. Fefferman (1971) and M. Kojima (1977), does not guarantee the convergence almost everywhere of Sn(x; f) for N ≥ 3 even in the class of continuous functions. Mathematical Reviews subject classification: Primary: 42B05 Key words: multiple trigonometric Fourier series, convergence almost everywhere, lacunary sequence
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