Equiconvergence of expansions in multiple trigonometric Fourier series and integrals in the case of a lacunary sequence of partial sums

Abstract

296 We investigate the equiconvergence onN = (-π, π) N of expansions in multiple trigonometric Fourier series and Fourier integrals of functions f ∈ Lp(� N ) and g ∈ L p (� N ), p > 1, N ≥ 2, g(x )= f (x) onN , in the case when the "partial sums" of these expansions, i.e., Sn(x; f ) and Jα(x; g), respectively, have "indices" n =( n1, n2, …, nN) ∈ � N and α = (α1, α2, …, αN) ∈ � N with components nj and αj satisfying the relation |α j - n j | ≤ C, j = 1, 2, …, N, where C is a constant inde� pendent of n or α. In particular, we consider the case when some of the components nj are elements of lacunary sequences.