On continuity of functions of several variables of the class of generalized bounded variation (Q1095333)

Let \(a_ k\), \(b_ k\) be the Fourier coefficients of the \(2\pi\)-periodic function f. If f is of bounded variation over the period and \(\min \{f(x- 0),f(x+0)\}\leq f(x)\leq \max \{f(x-0),f(x+0)\},\) N. Wiener proved in 1924 that \(n^{-1}\sum^{n}_{k=1}\sqrt{a^ 2_ k+b^ 2_ k}=o(1)\) is a necessary and sufficient condition for f to be continuous. Other necessary and sufficient conditions were given by S. M. Lozinskij. In recent time \textit{B. I. Golubov} [Soobshch. Akad. Nauk Gruz. SSR 74, 297- 300 (1974; Zbl 0302.42006)] considered the analogous problem for functions of several variables and of bounded p-variation in the sense of Hardy. In the present paper the author introduces the notion of generalized bounded variation for functions of several variables and proves some properties of such functions, including conditions for their continuity.