Uniform convergence of expansions into a multiple trigonometric Fourier series or a Fourier integral

Questions of convergence almost everywhere of expansions into a multiple trigonometric Fourier series or a Fourier integral are studied for functions from $L_p$, $p\ge1$, with summation over rectangles. Moreover, a ldquogeneralized localization principle,rdquo understood in the sense of convergence almost everywhere, is considered in the paper.