On the divergence of Fourier series of functions in several variables

Abstract

The paper deals with the problems of divergence of the series from absolute values of the Fourier coefficients of functions in several variables. It is proved that as the dimension of the space increases, the absolute convergence of Fourier series with respect to any complete orthnormal system (ONS) of functions with continuous partial derivatives becomes worse. For instance, for any ɛ ∈ (0, 2) there exists a function in variables \(k > \frac{{2(2 - \varepsilon )}} {\varepsilon }\) having all the continuous partial derivatives, however the series of absolute values of its coefficients with respect to any complete orthnormal system diverges in power 2 − ɛ.