On Sets of Unbounded Divergence at Each Point of Multiple Fourier Series of a Function Equal Zero on a Closed Set

Abstract

The problem investigated is to characterize sets E, the sets of unbounded divergence (at each point) of single and multiple Fourier series under condition of convergence of these series to zero at each point of the complement of E.

For any nonempty open set B ⊂ T ^N = [0, 2π]^N, N ≥ 1, a Lebesgue integrable function f ₀ is constructed which equals zero on the set U = T ^N \ B whose multiple trigonometric Fourier series diverges unboundedly (in the case of summation over squares) at each point of the set

$$\varepsilon (\overline {\mathfrak{B}} ) = \bigcup\limits_{j = 1}^N {({\text{pr}}_{{\text{(}}j{\text{)}}} \{ \overline {\mathfrak{B}} \} \times T^{N - 1} )} ,$$

, where $\overline {\mathfrak{B}}$ is the closure of the set ${\mathfrak{B}}$, pr_(j) ${\{ \overline {\mathfrak{B}} \} }$ is the orthogonal projection of the set ${\overline {\mathfrak{B}} }$ on the axis Ox _j , j = 1,...,N. It is also proved that if $\varepsilon (\overline {\mathfrak{B}} ) \ne T^N$, then for any function f equal zero on the set U the multiple trigonometric Fourier series of the function f (in the case of summation over rectangles) converges at each point of the set T ^N \ $\varepsilon (\overline {\mathfrak{B}} )$.