Absolute convergence of multiple Fourier series revisited

摘要

In a recent paper [4], Gogoladze and Meskhia generalized the classical results of Bernstein, Szasz, Zygmund and others related tq absolute convergence of single trigonometric Fourier series. Our aim is to extend these results from single to multiple Fourier series. To this effect, we introduce the notions of multiplicative moduli of continuity and that of smoothness. Multiplicative Lipschitz classes of functions in several variables, and functions of bounded s-variation in the sense of Vitali are also considered.