Structural and geometric characteristics of sets of convergence and divergence of multiple Fourier series with Jn n kn -lacunary sequence of rectangular partial sums

摘要

We study multiple trigonometric Fourier series of functions f in the classes (L_p left( {mathbb{T}^N } right)), p > 1, which equal zero on some set (mathfrak{A}, mathfrak{A} subset mathbb{T}^N , mu mathfrak{A} > 0) (µ is the Lebesgue measure), (mathbb{T}^N = left[ { - pi ,pi } right]^N), N ≥ 3. We consider the case when rectangular partial sums of the indicated Fourier series S n (x; f) have index n = (n 1, ..., n N ) ∈ ℤ N , in which k (k ≥ 1) components on the places {j 1, ..., j k } = J k ⊂ {1, ..., N} are elements of (single) lacunary sequences (i.e., we consider multiple Fourier series with J k -lacunary sequence of partial sums). A correlation is found of the number k and location (the “sample” J k ) of lacunary sequences in the index n with the structural and geometric characteristics of the set (mathfrak{A}), which determines possibility of convergence almost everywhere of the considered series on some subset of positive measure (mathfrak{A}_1) of the set (mathfrak{A}).