Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

Abstract

We investigate the equiconvergence on T N = [−π, π) N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L p (T N ) and g ∈ L p (R N ), p > 1, N ≥ 3, g(x) = f(x) on T N , in the case where the “partial sums” of these expansions, i.e., S n (x; f) and J α(x; g), respectively, have “numbers” n ∈ Z N and α ∈ R N (n j = [α j ], j = 1,..., N, [t] is the integral part of t ∈ R1) containing N − 1 components which are elements of “lacunary sequences.”