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

We investigate the equiconvergence on $\mathbb T^N=[-\pi,\pi)^N$ of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions $f\in L_p({\mathbb T}^N)$ and $g\in L_p({\mathbb R}^N)$, $p>1$, $N\ge 3$, $g(x)=f(x)$ on $\mathbb T^N$, in the case where the “partial sums” of these expansions, i.e., $S_n(x;f)$ and $J_\alpha(x;g)$, respectively, have “numbers” $n\in {\mathbb Z}^N$ and $\alpha\in {\mathbb R}^N$ ($n_j=[\alpha_j]$, $j=1,\dots,N$, $[t]$ is the integral part of $t\in \mathbb R^1$) containing $N-1$ components which are elements of “lacunary sequences.”