On continuity of functions of several variables of the class of generalized bounded variation
type: Item
instance of: scholarly article
title: On continuity of functions of several variables of the class of generalized bounded variation
zbMATH Open document ID: 0632.42016
DOI: 10.1007/BF01951358
author: T. I. Akhobadze
published in: Acta Mathematica Hungarica
publication date: 1986-01-01T00:00:00Z
review text: 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.
reviewed by: S.Aljančić
Mathematics Subject Classification ID: 42B99
Mathematics Subject Classification ID: 42A16
zbMATH DE Number: 4028094
zbMATH Keywords: generalized bounded variation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4028094