On a new mean and functions of bounded deviation
type: Item
instance of: scholarly article
title: On a new mean and functions of bounded deviation
zbMATH Open document ID: 0937.26015
DOI: 10.1007/BF02841534
author: V. Pereyra
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1999-10-05T00:00:00Z
review text: Let \(L_{2\pi}\) be the space of all \(2\pi\)-periodic and Lebesgue integrable functions over \([0,2\pi]\). Then for \(f\in L_{2\pi}\) and real \(x\), write \[ \begin{aligned} \varphi(t) &= \tfrac 12 \{f(x+t)+ f(x-t)\}= \varphi_0(t),\\ \varphi_\alpha(t) &= \alpha t^{-\alpha} \int_0^t (t-u)^{\alpha-1} \varphi(u) du \quad(\alpha>0),\\ P(t) &= \varphi(t)- \varphi_1(t). \end{aligned} \] Chandra's mean \(P(t)\), as pronounced by the authors, was first studied by the present reviewer [Period. Math. Hung. 5, 121-130 (1974; Zbl 0286.42010)]. In the present paper, generalizing this mean in the form: \[ \begin{aligned} p(k,t) &= \sum_{m=0}^k (-1)^m \binom km \varphi_m(t) \quad (k=1,2,\dots),\\ p(0,t) &= \varphi(t), \end{aligned} \] the authors have proved the following: Theorem 1. For each positive integer \(k\) \[ \begin{aligned} \varphi(t)\in \text{BV}(0,\pi) &\Leftrightarrow p(k,t)\in \text{BV} (0,\pi) \quad\text{and}\quad \frac{p(k,t)}{t}\in L(0,\pi), \tag{a}\\ p(k,t)\in \text{BV}(0,\pi) &\Rightarrow p(k+1,t)\in \text{BV} (0,\pi), \tag{b}\\ t^{-1} p(k,t)\in L(0,\pi)&\Rightarrow t^{-1} p(k+1,t)\in L(0,\pi). \tag{c}\end{aligned} \] The case \(k=1\) of part (a) was earlier known to the present reviewer [Jñānābha, Sect. A 8, 17-24 (1978; Zbl 0478.26003)]. In Theorem 2, the authors have demonstrated, by using Chandra's mean, that the class of functions of bounded deviation (BD) strictly includes the class of functions of bounded variation (BV).
Mathematics Subject Classification ID: 26E60
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 1348067
zbMATH Keywords: fractional integral mean
zbMATH Keywords: Chandra's mean
zbMATH Keywords: functions of bounded deviation
zbMATH Keywords: functions of bounded variation
reviewed by: Prem Chandra
MaRDI profile type: Publication
cites work: The Absolute Summability (A) of Fourier Series
cites work: On Integral Lipschitz Conditions and Integral Bounded Variation
cites work: Necessary and sufficient Tauberian conditions for the absolute convergence of Fourier series and conjugate series
cites work: Necessary and sufficient conditions for the absolute Euler summability
cites work: On the convergence criteria of a Fourier series
cites work: Q3254471
full work available at URL: https://doi.org/10.1007/bf02841534
OpenAlex ID: W2049232686
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description: scientific article; zbMATH DE number 1348067