Relations between rational and polynomial approximations in Banach spaces
type: Item
instance of: scholarly article
title: Relations between rational and polynomial approximations in Banach spaces
zbMATH Open document ID: 0859.41012
DOI: 10.1007/BF02342338
author: M. A. Nazarenko
published in: Analysis Mathematica
publication date: 1997-03-25T00:00:00Z
review text: Notations. \((\mathbb{B},|\cdot |)\) is a Banach space over the field of real or complex numbers having the below described properties. There exists a linearly independent sequence \(\{g_n\}^\infty_{n=0}\) of elements which have a structure of a commutative algebra with unit element \(g_0\). Let \(\mathbb{L}\) be the linear envelope of \(\{g_n\}\) and let \(\mathbb{P}_n = L(g_0, \dots, g_n)\) be the linear envelope of the set of polynomials of degree not exceeding \(n\), whereby the degree \(\partial P\) of a polynomial \(P\) is the maximal index of vectors \(g_k\) with nonzero coefficients representing \(P\), \(\partial (\alpha g_0)=0\), \(\partial (PQ)= \partial P+ \partial Q\). It is possible to divide two polynomials \(H,P\in \mathbb{L}\) with a remainder as follows: There exists \(Q,S\in \mathbb{L}\) such that \(H=P Q+S\), \(\partial S\leq \partial P\), \(\partial P+ \partial Q= \partial H\). For every \(Q\in \mathbb{L}\) there exists \(T=T (Q)>0\) such that for every \(t\in (0,t)\), \((g_0- tQ)^{-1} = g_0+tQ+ \widehat o(t) \in \mathbb{B}\) where \(|\widehat o(t) |= o(t)\). Rational functions \(\mathbb{R}_{n,m}\) are defined by the product of a polynomial of degree not exceeding \(n\) by a reciprocal of a polynomial of degree not exceeding \(m\). The terms ``numerator'' and ``denominator'' used through out with double quotes in the paper, shall be used without quotes in this review. The least deviation of an element \(f\in \mathbb{B}\) from the subspaces \(\mathbb{P}_n\) and \(\mathbb{R}_{n,m}\) is defined as usual by \({\mathbf e}_n (f)= \inf_{P_n\in \mathbb{P}_n} |f_- P_n|\), \({\mathbf r}_{n,m} (f)= \inf_{R_{n,m} \in\mathbb{R}_{n,m}} |f-R_{n,m} |\). It is known that the best approximating element exists in any finite dimensional Banach space. \(\mathbb{Z}_+\) denotes the set of nonnegative integers. In the first section the author proves two theorems, generalizing some results of \textit{A. L. Levin} and \textit{V. M. Tikhomirov} [Dokl. Akad. Nauk SSSR 174, …
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 913006
reviewed by: Zalman Rubinstein
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02342338
OpenAlex ID: W1539665871
description: scientific article; zbMATH DE number 913006