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

Abstract

Let a sequence of d-dimensional vectors n k = (n 1 , n 2 ,..., n ) with positive integer coordinates satisfy the condition n = α j m k +O(1), k ∈ ℕ, 1 ≤ j ≤ d, where α 1 > 0,..., α d > 0 and {m 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.