# Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

Article

First Online:

Received:

Revised:

Accepted:

- 62 Downloads
- 2 Citations

## Abstract

We consider converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum

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

*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 (α_{1},..., α_{ N }) and lip (α_{1},..., α_{ N }) for some α_{1},..., 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.## Key words and phrases

multiple Fourier series absolute convergence multiple difference operator of first order in each variable multiplicative Lipschitz classes Lip (*α*

_{1}, ...,

*α*

_{N}) and lip (

*α*

_{1}, ...,

*α*

_{N})

## 2000 Mathematics Subject Classification

42B99 42A32## References

- [1]R. P. Boas Jr., Fourier series with positive coefficients,
*J. Math. Anal. Appl.*,**17**(1967), 463–483.MATHCrossRefMathSciNetGoogle Scholar - [2]R. DeVore and G.G. Lorentz,
*Constructive Approximation*, Springer (Berlin, 1993).MATHGoogle Scholar - [3]V. Fülöp, Double sine series with nonnegative coefficients and Lipschitz classes,
*Colloq. Math.*,**105**(2006), 25–34.MATHCrossRefMathSciNetGoogle Scholar - [4]F. Móricz, Absolutely convergent Fourier series and function classes,
*J. Math. Anal. Appl.***324**(2006), 1168–1177.MATHCrossRefMathSciNetGoogle Scholar - [5]J. Németh, Fourier series with positive coefficients and generalized Lipschitz classes,
*Acta Sci. Math. (Szeged)*,**54**(1990), 291–304.MATHMathSciNetGoogle Scholar - [6]A. Zygmund,
*Trigonometric Series*, Cambridge Univ. Press (Cambridge, 1959).MATHGoogle Scholar

## Copyright information

© Springer Science+Business Media B.V. 2008