Rational approximation
type: Item
instance of: scholarly article
title: Rational approximation
zbMATH Open document ID: 0334.30019
DOI: 10.1016/0001-8708(76)90117-1
author: A. R. Reddy
author: P. Erdős
published in: Advances in Mathematics
publication date: 1976-01-01T00:00:00Z
review text: Several authors have investigated the question of approximating certain functions by reciprocals of polynomials under the uniform norm on the positive real axis. In this paper, the authors survey many of the known results in this direction. They also provide simplified proofs in some cases and obtain several new results as well. Let \(f(z)= \sum ^\infty_{k=0}a_kz^k\) be an entire function with non-negative real \(a_k(a_0>0)\). If \(\pi_n\) denotes the class of all ordinary polynomials of degree \(n\) then for \(P \in \pi_n\), set \[ \psi (f,P)= \left|{1 \over f(x)}-{1 \over P(x)} \right|_{L^\infty [0, \infty]}, \lambda_{0,n}(f)= \inf_{P \in \pi_n} \psi (f,P). \] The results reviewed and obtained in this paper can be broadly classified in problems of the following type: (i) Assuming various growth restrictions on the entire function \(f\), obtain upper and lower bounds for \(\lambda_{0,n}(f)\), \(\lambda_{0,n}(f(x)/x)\), \(\liminf_{n \to infty} (\lambda_{0,n}(f))^{1/n}\), \(\limsup_{n \to \infty}(\lambda_{0,n}(f))^{1/n}\) etc, and conversely given bounds for these quantities, obtain restrictions on \(f\). (ii) Given \(f\) continuous or entire satisfying certain growth restrictions, prove the existence of polynomials \(P \in \pi_n\) for which \(\psi (f,P)\), \(\psi (f(x)/x,P)\) etc. are very small and conversely given sequence \(\{P_n \}\), \(P_n \in \pi_n\), such that these quantities are small, find the conditions which \(f\) must satisfy. (iii) Obtain sharper results corresponding to previous problems for certain special functions such as \(e^{-x}\), \((x+1)^{-n}\) etc. (iv) Under stringent conditions on the entire function \(f\), extend the above results to complex polynomials of complex variable \(z\) under the unifom norm on some subset of the complex plane. The results reviewed and generalized by the authors are mainly due to Erdős, Meinardus, Newman, Reddy, Shisha, Taylor, Varga and others. The results are too numerous to be stated in detail here (The paper co…
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 30-02
Mathematics Subject Classification ID: 41A20
zbMATH DE Number: 3522307
Wikidata QID: Q105530335
reviewed by: O. P. Juneja
MaRDI profile type: Publication
full work available at URL: 0001-8708(76)90117-1
OpenAlex ID: W4212939591
cites work: Rationale Tschebyscheff-Approximation über unbeschränkten Intervallen
cites work: Convergence of Best Rational Tchebycheff Approximations
cites work: Chebyshev rational approximations to \(e^{-x}\) in \([0,+\infty)\) and applications to heat-conduction problems
cites work: Rational approximation to certain entire functions in [0,+∞)
cites work: A note on rational approximation
cites work: Rational Approximation on the Positive Real Axis
cites work: A Note on Rational Approximation
cites work: RATIONAL APPROXIMATION TO e-⊧x⊧ ON THE WHOLE REAL LINE
cites work: Q5562737
cites work: Properties of functions related to their rate of approximability by rational fractions
cites work: Q3226016
cites work: Chebyshev rational approximations to certain entire functions in \([0,+\infty)\)
cites work: Converse Theorems and Extensions in Chebyshev Rational Approximation to Certain Entire Functions in [ 0, + ∞)
cites work: Rational approximation to \(e^{-x}\)
cites work: Addendum to "Converse Theorems and Extensions in Chebyshev Rational Approximation to Certain Entire Functions in [ 0, + ∞)"
cites work: A contribution to rational Chebyshev approximation to certain entire functions in \([0, \infty)\)
cites work: A note on rational Chebyshev approximation on the positive real axis
cites work: A note on rational approximation on \([0,\infty)\)
cites work: Rational Chebyshev Approximation to certain Entire Functions of Zero Order on the Positive Real Axis. II
cites work: Rational Chebyshev approximation to certain entire functions of zero order on the positive real axis
cites work: Some results in Chebyshev rational approximation
cites work: Recent advances in Chebyshev rational approximation on finite and infinite intervals
cites work: On approximation by rational functions
cites work: A class of rational approximations of the positive real axis - a survey
cites work: A characterization of entire functions \(\sum^\infty_{k=0} a_kz^k\) with all \(a_k\geq 0\)
cites work: Rational Chebyshev approximation on \([0,+\infty)\)
cites work: Q4126831
cites work: Zur rationalen Approximierbarkeit von \(e^{-x}\) über \([0, \infty)\)
cites work: Q5727762
cites work: Q5633606
cites work: Rational approximation. III
cites work: Rational Approximation to xn II
cites work: Rational approximation of \(e^{-x}\) on the positive real axis
description: scientific article; zbMATH DE number 3522307