Functions of bounded \(k\)th \(p\)-variation and continuity modulus
type: Item
instance of: scholarly article
title: Functions of bounded \(k\)th \(p\)-variation and continuity modulus
zbMATH Open document ID: 1336.26008
DOI: 10.1155/2015/253636
author: Odalis Mejía
author: Pilar Silvestre
published in: Journal of Function Spaces
publication date: 2015-10-27T00:00:00Z
review text: For \(p>1\) and \(0\leq \alpha \leq 1/p\), let us consider the space \(V_{p,\alpha}^{(k)}\), \(k\in \mathbb{N}\) consisting of the 1-periodic functions on the real line whose \((p,\alpha)\) \(k\)-th variation is finite (see [\textit{S. Barza} and \textit{P. Silvestre}, Rev. Mat. Complut. 27, No. 1, 69--91 (2014; Zbl 1396.26015)] for a definition in terms of the supremum of certain partitions of an interval of length \(1\)). The spaces \(V_{p,\alpha}^{(k)}\) form a scale between the space of functions of bounded \(k\)-th variation and the Sobolev space \(W_p^k\). The main result of this paper consists of the study of sharp estimates of the \((p,\alpha)\) \(k\)-th variation of functions in the space in terms of the fractional modulus of smoothness of order \(k-1/p\) and also the study of limits in the scale of spaces \(V_{p,\alpha}^{(k)}\), \(k\in \mathbb{N}\). The work is an extension of some other previous papers such as [\textit{M. Lind}, Math. Inequal. Appl. 16, No. 1, 21--39 (2013; Zbl 1261.26008)] and Barza and P. Silvestre [loc. cit.].
reviewed by: Santiago Boza
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 6499095
zbMATH Keywords: functions of bounded variation
zbMATH Keywords: modulus of continuity
zbMATH Keywords: modulus of smoothness
zbMATH Keywords: Sobolev spaces
Wikidata QID: Q59112780
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2015/253636
OpenAlex ID: W1581625113
cites work: On fractional smoothness of functions related to p-variation
cites work: Q5848315
cites work: Q5535614
cites work: On functions of bounded (p,2)-variation
cites work: Bounded Variation and Around
cites work: Functions of bounded second \(p\)-variation
cites work: On functions of bounded \((p, k)\)-variation
cites work: Asymptotics properties of a compact set of smooth functions in the space of functions of bounded \(p\)-variation
cites work: Q4273944
cites work: Q5652921
cites work: Best approximation polynomials and a relation between continuity modules in spaces of functions of bounded \(p\)-variation
cites work: Q4039801
description: scientific article; zbMATH DE number 6499095