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Integral representation of functions of bounded second Φ-variation in the sense of Schramm

Gimenez J., Merentes N., Rivas S.

Opuscula Mathematica

2012

Vol. 32, no. 1

137-151

In this article we introduce the concept of second Φ--variation in the sense of Schramm for normed-space valued functions defined on an interval [a; b] ⊂ R. To that end we combine the notion of second variation due to de la Vallée Poussin and the concept of φ-variation in the sense of Schramm for real valued functions. In particular, when the normed space is complete we present a characterization of the functions of the introduced class by means of an integral representation. Indeed, we show that a function [formula] (where X is a reflexive Banach space) is of bounded second Φ-variation in the sense of Schramm if and only if it can be expressed as the Bochner integral of a function of (first) bounded variation in the sense of Schramm.

On a property of phi-variational modular spaces

Wang J., Wu C.

Opuscula Mathematica

2010

Vol. 30, no. 2

209-215

Maligranda pointed out whether condition (B.1) is satisfied in the variational modular space X*ρ is an open problem. We will answer this open problem in X*ρ', a subspace of X*ρ. As a consequence this modular space X*ρ' can be F-normed.

On the order of BVφ - approximation of convolution integrals over the line group

Bardaro C., Vinti G.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr spec.

47-63

Here we study the rate of BVφ - approximation of higher order for a class of linear integral operators of convolution type over the line group.