A Note on the Definition of Bounded Variation of Higher Order for Double Sequences

Bhikha Lila Ghodadra,Vanda Fülöp

doi:10.46793/kgjmat2004.563g

Abstract

In this study the deﬁnition of bounded variation of order p (p ∈ ℕ) for double sequences is considered. Some inclusion relations are proved and counter examples are provided for ensuring proper inclusions.

Highlights

While studying convergence properties of double trigonometric and Walsh series, many authors have considered double sequences which are of bounded variation or more generally of bounded variation of order (p, 0), (0, p), and (p, p)
Many results regarding the convergence of trigonometric and Walsh series with coefficients of bounded variation of higher order are proved. It seems that showing the inclusion relations for such classes of sequences and constructing counter examples for showing proper inclusions have not yet been done
By a sequence, we mean a function from Z to C, and by a double sequence, we mean a function from Z × Z to C

Summary

Introduction

While studying convergence properties of double trigonometric and Walsh series, many authors have considered double sequences which are of bounded variation or more generally of bounded variation of order (p, 0), (0, p), and (p, p) (see, e.g., [1, 3]). It seems that showing the inclusion relations for such classes of sequences and constructing counter examples for showing proper inclusions have not yet been done. 2. One Dimensional Case We recall the definition of bounded variation of order p for a single sequence (see [2, Defintion 1.4]). }, i.e., {ak} such that ak → 0 as |k| → ∞, is said to be of bounded variation of order p (p ∈ N) if

Results

Conclusion