The Ward property for a \(\mathcal P\)-adic basis and the \(\mathcal P\)-adic integral.
type: Item
instance of: scholarly article
title: The Ward property for a \(\mathcal P\)-adic basis and the \(\mathcal P\)-adic integral.
zbMATH Open document ID: 1034.26004
DOI: 10.1016/S0022-247X(03)00426-8
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-10-14T00:00:00Z
review text: Let \({\mathcal P}= \{p_j\}^\infty_{j=0}\) be a fixed sequence of integers, with \(p_j> 1\) for \(j\in\mathbb{N}\cup \{0\}\). We set \(m_0= 1\) and \(m_k= p_0p_1\cdots p_{k-1}\) for \(k\geq 1\). For fixed \(k\in\mathbb{N}\cup \{0\}\), the closed intervals \[ \Biggl[{r\over m_k},\, {r+1\over m_k}\Biggr]= I^{(k)}_r,\qquad r=0,1,\dots, m_k- 1, \] are called the \({\mathcal P}\)-adic intervals. The set of all such intervals is denoted by \({\mathcal I}_{{\mathcal P}}\). For a given positive function \(\delta\) on \([0, 1]\), we set \[ {\mathcal P}_\delta= \{(I, x): I\in{\mathcal I}_{{\mathcal P}},\, x\in I\subset(x- \delta(x), x+\delta(x)),\, x\in E\}. \] The family \(\{{\mathcal P}_\delta\}_\delta\), where \(\delta\) runs over the set of all positive functions on \([0,1]\), is called the \({\mathcal P}\)-adic basis on \([0, 1]\). It is a differentiation basis. In this paper, it is shown that a \({\mathcal P}\)-adic basis possesses the Ward property if and only if the sequence by which it is defined is bounded. Recall that a basis \(\beta\) satisfies the Ward property whenever each additive interval function is \(\beta\)-differentiable almost everywhere on the set of all points at which at least one of its extreme \(\beta\)-derivatives is finite. Properties (for examples, full descriptive characterizations, VBG and the Lusin condition) of Henstock-Kurzweil type integrals with respect to a \({\mathcal P}\)-adic basis are discussed.
Mathematics Subject Classification ID: 26A39
Mathematics Subject Classification ID: 26A42
zbMATH DE Number: 1992567
zbMATH Keywords: \(\mathcal P\)-adic basis
zbMATH Keywords: \(\mathcal P\)-derivative
zbMATH Keywords: Ward property
zbMATH Keywords: variational measure
zbMATH Keywords: Henstock-Kurzweil integral
zbMATH Keywords: VBG function
author: Benedetto Bongiorno
author: Luisa Di Piazza
author: Valentin Skvortsov
reviewed by: Tuan Seng Chew
MaRDI profile type: Publication
cites work: Q3715607
cites work: Infinite variation and derivatives in \(R^m\)
cites work: A new full descriptive characterization of Denjoy-Perron integral
cites work: On variational measures related to some bases
cites work: A Full Descriptive Definition of the Gage Integral
cites work: On absolute continuity
cites work: Variational Measures in the Theory of the Integration in ℝ m
cites work: The \(S\)-Henstock integration and the approximately strong Lusin condition
cites work: Q3999467
cites work: The inversion of approximate and dyadic derivatives using an extension of the Henstock integral
cites work: Q4307619
cites work: Une théorie de Denjoy des martingales dyadiques. (A Denjoy theory of dyadic martingales)
cites work: Q4258069
cites work: Henstock integration in the plane
cites work: Q4106410
cites work: Q3216832
cites work: Theory of Ward for cell functions. I: Wards theory in an abstract setting. II: Structure theorem for (finitely) additive pointwise Lipschitzian cell functions without Vitali hypothesis
cites work: Q5574796
cites work: Q3209350
cites work: Variations and variational measures in integration theory and some applications
cites work: \(A\)-integrable martingale sequences and application to Haar series
cites work: Series in multiplicative systems convergent to Denjoy-integrable functions
cites work: Derivation bases on the real line. I
cites work: Derivates of interval functions
Recommended article: On the Ward Theorem for $\mathcal{P}$-adic-path bases associated with a bounded sequence
Recommended article: Variational measures related to local systems and the Ward property of $$\mathcal{P}$$ -adic path bases
Recommended article: Ein p-adisches Integral und seine Anwendungen. II
Recommended article: On integrality of \(p\)-adic iterated integrals
Recommended article: Q3695357
Recommended article: INTEGRAL BASES OVER p-ADIC FIELDS
Recommended article: A \(p\)-adic Waldspurger formula
Recommended article: A partial analog of the integrability theorem for distributions on \(p\)-adic spaces and applications
Recommended article: An invariant \(p\)-adic \(q\)-integral on \(\mathbb Z _p\)
Recommended article: A generalization of Coleman's \(p\)-adic integration theory
description: scientific article; zbMATH DE number 1992567