Polynomials with bounds and numerical approximation
type: Item
instance of: scholarly article
title: Polynomials with bounds and numerical approximation
zbMATH Open document ID: 1379.65010
author: Bruno Després
published in: Numerical Algorithms
publication date: 2017-11-24T00:00:00Z
review text: Let \(P_n\) be the set of polynomials of degree at most \(n\) and \(U_n=\{p\in P_n: x\in I \Rightarrow p(x)\in I\}\) where \(I=[0,1]\). A large part of the paper is devoted to finding a good parametrization for such bounded polynomials. This is first obtained by using the Lukács theorem for nonnegative polynomials and Euler's four squares identity which gives a \(2\pi\)-periodic solutions depending on three angles for \(n=1\) and by iteration for \(n>1\). This is used to prove that \(\inf_{p\in U_n}\|f-p\|\leq 2\inf_{p\in P_n}\|f-p\|\). Some ideas are given on how this can be generalized to bivariate polynomials. This requires to replace the four squares property by the eight squares identity of Degen. Some implementation details are given and illustrated with some numerical applications.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65D15
zbMATH DE Number: 6812494
zbMATH Keywords: polynomial with bounds
zbMATH Keywords: numerical approximation
zbMATH Keywords: quaternion algebras
zbMATH Keywords: numerical examples
zbMATH Keywords: Euler's four squares identity
describes a project that uses: fminunc
describes a project that uses: AMPL
describes a project that uses: Matlab
describes a project that uses: Chebfun
describes a project that uses: HE-E1GODF
describes a project that uses: Sostools
describes a project that uses: Sollya
describes a project that uses: Chebfun2
MaRDI profile type: Publication
full work available at URL: s11075-017-0286-0
OpenAlex ID: W2345300376
cites work: Q4210476
cites work: On polynomials of best one sided approximation
cites work: A computational journey into the mind
cites work: Efficient and accurate computation of upper bounds of approximation errors
cites work: Sollya: An Environment for the Development of Numerical Codes
cites work: A generalization of an inequality of V Markov to multivariate polynomials
cites work: Q4706139
cites work: Uncertainty Propagation; Intrusive Kinetic Formulations of Scalar Conservation Laws
cites work: A mathematical introduction to compressive sensing
cites work: A Modeling Language for Mathematical Programming
cites work: Uncertainty quantification for systems of conservation laws
cites work: Markov and Bernstein type inequalities for polynomials
cites work: Optimal scaling parameters for sparse grid discretizations
cites work: Q3395491
cites work: Q4889859
cites work: A Generalized Empirical Interpolation Method: Application of Reduced Basis Techniques to Data Assimilation
cites work: Formal Proofs for Nonlinear Optimization
cites work: Q5605151
cites work: Q4861743
cites work: Analytic Inequalities
cites work: Q5835296
cites work: Compositions of quadratic forms
cites work: Q4354425
cites work: An Extension of Chebfun to Two Dimensions
cites work: The kernel polynomial method
DOI: 10.1007/S11075-017-0286-0
description: scientific article; zbMATH DE number 6812494