Generalized polynomials of minimal norm
type: Item
instance of: scholarly article
title: Generalized polynomials of minimal norm
zbMATH Open document ID: 0679.41013
DOI: 10.1016/0021-9045(89)90135-4
published in: Journal of Approximation Theory
publication date: 1989-01-01T00:00:00Z
review text: The authors consider the problem of finding optimal generalized polynomials of minimal \(L_ p\) norm, \(1\leq p<\infty\). Let \(\{u_ i\}^{N+1}_{i=1}\subset C^ N[a,b]\), [0,1]\(\subset (a,b)\), be a given set of functions with the property that \(\{u_ i\}^ K_{i=1}\) forms an ``extended Chebyshev system'' for \(K=N-1,N,N+1\). For \({\bar \tau}=(\tau_ 1,...,\tau_ N)\in R^ N\) the symbol u(\({\bar \tau}\);t) is devoted to denote the unique polynomial of the type \(u=u_{N+1}+\sum^{N}_{i=1}\alpha_ iu_ i\) with real coefficients \(\{\alpha_ i\}\) such that \(u(\tau_ k)=0\), \(k=1,...,N\). Let us consider the multiplicities \(\{\nu_ k\}^ n_{k=1}\), \(\{\mu_ k\}^ m_{k=1}\) such that \(\sum^{n}_{k=1}\nu_ k+\sum^{m}_{k=1}\mu_ k=N\) and the points \({\bar \xi}=(\xi_ 1,...,\xi_ m)\) where \(a<\xi_ 1\leq...\leq \xi_ m<b\). Set \(\Delta_ n=\{\bar x=(x_ 1,...,x_ n)\); \(0\leq x_ 1\leq x_ 2\leq...\leq x_ n\leq 1\}\) and for each \(\bar x=(x_ 1,...,x_ n)\in \Delta_ n\), let \({\bar \tau}={\bar \tau}(\bar x,{\bar \xi})\in R^ N\) be the form \[ {\bar \tau}(\bar x,{\bar \xi})=(x_ 1,...,x_ 1,x_ n,...,x_ n,\xi_ 1,...,\xi_ 1,\xi_ m,...,\xi_ m). \] If \(I(\bar x,{\bar \xi})=\int^{1}_{0}| u({\bar \tau}(\bar x,{\bar \xi});t)|^ p dt,\) \(1\leq p<\infty\), the aim of this important paper is to investigate the existence and the uniqueness of the \(\bar x^*\in \Delta_ n\) with the property \((1)\quad I(\bar x^*,{\bar \xi})=\min_{\bar x\in \Delta_ n}I(\bar x,{\bar \xi}).\) The main results are the following: ``Let \(1\leq p<\infty\). Then there is an \(\bar x^*\in \Delta_ n\) which satisfies (1). Further any such minimizing \(\bar x^*=(x^*_ 1,...,x^*_ n)\) is in interior of \(\Delta_ n\), i.e., \(0<x^*_ 1<...<x^*_ n<1\). For each n,m\(\geq 0\) so that \(n+m\leq N\), there is a unique solution \(\bar x^*\in \Delta_ n\) to the problem cited in (1).'' As an application of the above results, the authors obtain Gaussian quadrature formulas for extended Chebyshev systems.
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 41A40
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 4113208
zbMATH Keywords: application
zbMATH Keywords: Gaussian quadrature formulas
zbMATH Keywords: Chebyshev systems
author: Richard B. Barrar
author: Borislav D. Bojanov
author: Henry L. Loeb
reviewed by: Alexandru Lupaş
MaRDI profile type: Publication
full work available at URL: 0021-9045(89)90135-4
OpenAlex ID: W1982135914
cites work: Oscillating Tchebycheff systems
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cites work: Generalized Gaussian quadrature formulas
cites work: Gaussian quadrature formulae with fixed nodes
cites work: Q4114267
cites work: Q4102261
description: scientific article; zbMATH DE number 4113208