Approximate Taylor polynomials and differentiation of functions
type: Item
instance of: scholarly article
title: Approximate Taylor polynomials and differentiation of functions
zbMATH Open document ID: 0811.26005
DOI: 10.12775/TMNA.1994.009
author: Fon-Che Liu
author: Wei-Shyan Tai
published in: Topological Methods in Nonlinear Analysis
publication date: 1995-04-05T00:00:00Z
review text: The authors have proved that for a measurable function \(u\) defined on a measurable subset \(D\) of the \(n\)-dimensional Euclidean space the following conditions are equivalent: \(u\) has the Lusin property of order \(k\) on \(D\); \(u\) has an approximate \((k-1)\)-Taylor polynomial at almost every point of \(D\); \(u\) is approximately differentiable of order \(k\) at almost every point of \(D\). Here the Lusin property of order \(k\) means that for each \(\varepsilon> 0\) there exists a \(C^ k\)-function \(g\) defined on \(\mathbb{R}^ n\) such that \(m(\{x\in D: u(x)\neq g(x)\})< \varepsilon\), where \(m\) is a Lebesgue measure, \(p\) is an approximate \((k- 1)\)-Taylor polynomial if \(p(x,y)\) is a polynomial centered at \(x\) of degree at most \((k-1)\) such that \[ \text{ap} \limsup_{y\to x} | u(y)- p(x,y)|\cdot | y- x|^{-k}< +\infty \] and \(u\) is approximately differentiable of order \(k\) if there is a polynomial \(p(x,y)\) centered at \(x\) and of degree at most \(k\) such that \[ \text{ap} \lim_{y\to x} | u(y)- p(x,y)|\cdot| y- x|^{-k}= 0. \] {}.
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 26B35
Mathematics Subject Classification ID: 41A58
zbMATH DE Number: 640275
zbMATH Keywords: approximate Taylor polynomial
zbMATH Keywords: approximate differentiability
zbMATH Keywords: Lusin property
reviewed by: Władysław Wilczyński
MaRDI profile type: Publication
full work available at URL: tmna.1994.009
OpenAlex ID: W2587663619
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description: scientific article; zbMATH DE number 640275