Construction of dual bases
type: Item
instance of: scholarly article
title: Construction of dual bases
zbMATH Open document ID: 1272.65016
author: Paweł Woźny
published in: Journal of Computational and Applied Mathematics
publication date: 2013-04-22T00:00:00Z
review text: Given a set \(\{b_i\}_{i=0}^N\) of linearly independent functions and the space \({\mathcal B}_{N}=\text{span}\{b_i,i=0,\ldots,N\}\), the author proposes a simple recurrent way of constructing a dual basis \(\{d_i^{(N)}\}_{i=0}^N\) of \({\mathcal B}_{N}\) in the time \(O(N^3)\). Such a method and the corresponding algorithm is an improvement of those already known in literature because of their disadvantage of either inverting a full matrix or precomputing an orthonormal basis of the considered space. The new efficient algorithm leads to a system of linear equations whose coefficient matrix has a simple LU decomposition, it is non singular and it can be easily inverted. An application to least squares approximation is presented. Finally, four examples on monomial basis, Bernstein basis, B-spline functions and best polynomial approximation are shown.
reviewed by: Paola Lamberti
Mathematics Subject Classification ID: 65D15
Mathematics Subject Classification ID: 65D07
zbMATH DE Number: 6156307
zbMATH Keywords: dual bases
zbMATH Keywords: approximation
zbMATH Keywords: Bernstein basis polynomials
zbMATH Keywords: B-spline functions
zbMATH Keywords: algorithm
zbMATH Keywords: orthonormal basis
zbMATH Keywords: LU decomposition
zbMATH Keywords: least squares approximation
MaRDI profile type: Publication
full work available at URL: j.cam.2012.12.007
OpenAlex ID: W2144147547
DOI: 10.1016/J.CAM.2012.12.007
description: scientific article; zbMATH DE number 6156307