Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

Abstract

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ℤ^N, form an absolutely convergent series. Then the series

$$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$

converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $\mathbb{T}$ ^N, $\mathbb{T}$ := [−π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.