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We study the smoothness properties of a complex-valued function f of several variables with absolutely convergent Fourier series. We prove sufficient conditions in terms of the Fourier coefficients of f in order that f belong to one of the multiplicative Zygmund classes Zyg(α1, α2,…,αN) and zyg(α1, α2,…,αN) for some α1, α2,…,αN > 0. These Zygmund classes of functions are defined in terms of the multiple difference operator of second order in each variable. The conditions given by us are not only sufficient, but also necessary in the case when the Fourier coefficients are nonnegative.
Key words and phrases: Multiple Fourier series; absolute convergence; multiple difference operator of second order; multiplicative Zygmund classes Zyg(α1,…,αn) and zyg(α1,…,αN)
Received: 2007-7-8
Revised: 2007-11-9
Published Online: 2009-9-25
Published in Print: 2008-7-1
© Oldenbourg Wissenschaftsverlag
