On uniqueness and integrability of multiple trigonometric series (Q1813238)

The author considers \(m\)-dimensional trigonometric series of the form \((*)\) \(\sum_ n a_ n e^{2\pi in\cdot x}\). In the main two theorems he finds (by application of his previous results [Mat. Sb., Nov. Ser. 132(174), No. 1, 104-130 (1987; Zbl 0639.42009)]) conditions which imply that the series \((*)\) is the Fourier series of an integrable function. The first result (Theorem 1) involves conditions on the Abelian means of the series, the second (Theorem 3) conditions on integral means of the partial sums of \((*)\). There are applications to the theory of \(m\)- harmonic real functions, and to localization properties of Fourier series.