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.”