On the convergence and the Gibbs phenomenon of multiple Fourier series for functions of bounded harmonic variation (Q1896867)

The author defines the class of functions of bounded harmonic variation (HBV) for functions of \(n (\geq 2)\) variables by induction on \(n\). The definition of HBV in the case \(n= 1\) is due to \textit{D. Waterman} [Stud. Math. 44, 107-117 (1972; Zbl 0237.42001)]. The author proves theorems on the convergence of multiple Fourier series of a function \(f\in \text{HBV}\). The function \(f\) is assumed either to be continuous on a compact set or to have a discontinuity of the first kind at a point. These results sharpen and extend for \(n\geq 3\) those proved for \(n= 2\) by \textit{A. A. Saakyan} [Izv. Akad. Nauk Arm. SSR, Mat. 21, No. 6, 517-529 (1986; Zbl 0614.42009)]. The Gibbs phenomenon for multiple Fourier series is also considered.