On the Almost Everywhere Convergence of Sequences of Multiple Rectangular Fourier Sums

In the case when a sequence of d-dimensional vectors n(k) = ( n(k)(1), n(k)(2),..., n(k)(d)) with nonnegative integer coordinates satisfies the condition n(k)(j) = alpha(j)m(k) + O(1), k is an element of N, 1 <= j <= d, where alpha(1),..., alpha(d) > 0, m(k) is an element of N, and lim(k ->infinity) m(k) = infinity, under some conditions on the function phi: [0, +infinity) -> [0, +infinity), it is proved that, if the trigonometric Fourier series of any function from phi(L)([-pi, pi) converges almost everywhere, then, for any d is an element of N and all f is an element of phi(L)(ln(+) L)(d-1)([-pi, pi)(d)), the sequence S(nk) (f, x) of the rectangular partial sums of the multiple trigonometric Fourier series of the function f, as well as the corresponding sequences of partial sums of all of its conjugate series, converges almost everywhere.