On functions of bounded \((p, k)\)-variation
type: Item
instance of: scholarly article
title: On functions of bounded \((p, k)\)-variation
zbMATH Open document ID: 1241.26007
DOI: 10.1155/2012/202987
published in: Journal of Function Spaces and Applications
publication date: 2012-05-10T00:00:00Z
review text: Summary: We introduce and study the concept of \((p, k)\)-variation \((1 < p < \infty, ~k \in \mathbb N)\) of a real function on a compact interval. In particular, we prove that a function \(u : [a, b] \rightarrow \mathbb R\) has bounded \((p, k)\)-variation if and only if \(u^{(k-1)}\) is absolutely continuous on \([a, b]\) and \(u^{(k)}\) belongs to \(L_p[a, b]\). Moreover, an explicit connection between the \((p, k)\)-variation of \(u\) and the \(L_p\)-norm of \(u^{(k)}\) is given which is parallel to the classical Riesz formula characterizing functions in the spaces \(RV_p[a, b]\) and \(A_p[a, b]\). This may also be considered as an alternative characterization of the one variable Sobolev space \(W^k_p[a, b]\).
Mathematics Subject Classification ID: 26A45
zbMATH DE Number: 6032351
zbMATH Keywords: \((p, k)\)-variation
zbMATH Keywords: absolutely continuous
zbMATH Keywords: classical Riesz formula
author: N. Merentes
author: Sergio Rivas
author: J. Sanchez
Wikidata QID: Q58908151
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/202987
OpenAlex ID: W2079312563
cites work: Q4756366
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cites work: Functions of Bounded k th Variation
cites work: Further results on an integral representation of functions of generalised variation
cites work: Q5553606
description: scientific article; zbMATH DE number 6032351