Equiconvergence of Expansions in Multiple Fourier Series and in Fourier Integrals with 'Lacunary Sequences of Partial Sums'

摘要

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 ∈ R~1) containing N - 1 components which are elements of "lacunary sequences."