SUFFICIENT CONDITIONS FOR CONVERGENCE ALMOST EVERYWHERE OF MULTIPLE TRIGONOMETRIC FOURIER SERIES WITH LACUNARY SEQUENCE OF PARTIAL SUMS

摘要

Sufficient conditions for the convergence (almost everywhere) of multiple trigonometric Fourier series of functions f in the classes L_, p ＞ 1, are obtained in the case where rectangular partial sums S_n(x; f) of this series have numbers n = (n_1,... ,n_N) ∈ Z~N, 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 L_p, 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 S_n(x; f) for N ≥ 3 even in the class of continuous functions.