Polynomial approximation of nonlinear operators in space C
type: Item
instance of: scholarly article
title: Polynomial approximation of nonlinear operators in space C
zbMATH Open document ID: 0594.41014
DOI: 10.1007/BF00971298
author: I. K. Daugavet
author: A. A. Lenneh
published in: Siberian Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: Let F be a continuous nonlinear operator from a compact subset K of the space of continuous functions C on [0,1] into C. A polynomial operator on C is defined as \[ P(u)=\sum a_{j_ 1...j_ m}[g_ 1(u)]^{j_ 1}...[g_ m(u)]^{j_ m} \] where \(g_ k\) are bounded linear operators on C. It is shown that for any \(\epsilon >0\) there exists \(\delta >0\) and a polynomial operator P such that for all \(x\in K\) and \(\Delta\) \(x\in C\) such that \(\| \Delta x\| \leq \delta\) we have \(\| F(x)-P(x+\Delta x)\| <\epsilon.\) The proof suggests the actual construction of the operator P.
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 47H99
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 3955646
zbMATH Keywords: Banach space
zbMATH Keywords: continuous nonlinear operator
zbMATH Keywords: polynomial operator
reviewed by: Janusz Mika
MaRDI profile type: Publication
cites work: A Weierstrass theorem for real, separable Hilbert spaces
cites work: A Weierstrass theorem for normed linear spaces
cites work: A Weierstrass theorem for real Banach spaces
cites work: Q5526189
Recommended article: On polynomial approximation of operators
Recommended article: Polynomial approximation of solutions of operator-differential equations in a Hilbert space
Recommended article: Q3216922
Recommended article: Approximation of nonlinear operators by Volterra polynomials
Recommended article: On polynomial approximation of solutions of differential-operator equations
Recommended article: Q4735629
Recommended article: Approximation of solutions of operator-differential equations by operator polynomials
Recommended article: Q5476544
Recommended article: Polynomial approximation in Banach spaces
Recommended article: Q3196728
description: scientific article; zbMATH DE number 3955646