On almost everywhere convergence of lacunary sequences of multiple rectangular Fourier sums

Abstract

Let a sequence of d-dimensional vectors n _k = (n ¹ _k , n ² _k ,..., n ^d _k ) with positive integer coordinates satisfy the condition n ^j _k = α _j m _k +O(1), k ∈ ℕ, 1 ≤ j ≤ d, where α ₁ > 0,..., α _d > 0 and {m _k } ^∞ _k=1 is an increasing sequence of positive integers. Under some conditions on a function φ: [0,+∞) → [0,+∞), it is proved that, if the sequence of Fourier sums ${S_{{m_k}}}$ (g, x) converges almost everywhere for any function g ∈ φ(L)([0, 2π)), then, for any d ∈ ℕ and f ∈ φ(L)(ln⁺ L)^d−1([0, 2π)^d), the sequence ${S_{{n_k}}}$ (f, x) of rectangular partial sums of the multiple trigonometric Fourier series of the function f and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.