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

Abstract

Let a sequence of d-dimensional vectors n_k = (n_k¹, n_k²,..., n_k^d) with positive integer coordinates satisfy the condition n_k^j = α_jm_k +O(1), k ∈ ℕ, 1 ≤ j ≤ d, where α₁ > 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.