Means and averages of Taylor polynomials
type: Item
instance of: scholarly article
title: Means and averages of Taylor polynomials
zbMATH Open document ID: 0807.41017
published in: Journal of Mathematical Analysis and Applications
publication date: 1993-09-05T00:00:00Z
review text: Let \(f\) have a continuous \((r+ 1)\)st derivative, nowhere 0 on \([a,b]\) and \(P_ c\) its Taylor polynomial at \(c\). Noting that \(P_ a(M)= P_ b(M)\), if \(r\) is odd, and \(2f(M)= P_ a(M)+ P_ b(M)\), if \(r\) is even, have unique solutions, the author denotes these by \(M^ r_ p\) if \(f(x)= x^ p\) (creating some confusion with his previous notation \(M^ r_ f\)). He observes that \(M^ r_{r+1}\), \(M^ r_{r/2}\), and \(M^ r_{-1}\) are the arithmetic, geometric, and harmonic means, respectively, of \(a\) and \(b\). However, an argument is offered showing that the generalization, via general probability measures, of this definition to mean values of \(n\) numbers never yields the arithmetic, geometric or harmonic means. Finally, the author makes a correction to his 1990 paper in the same Journal 149, No. 1, 220-235 (1990; Zbl 0706.26019).
reviewed by: János Aczél
Mathematics Subject Classification ID: 41A58
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 399076
zbMATH Keywords: probability measures
author: Alan Horwitz
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1223
OpenAlex ID: W1984775028
DOI: 10.1006/JMAA.1993.1223
description: scientific article; zbMATH DE number 399076