Exact value of widths of a certain class of solutions of linear differential equations
type: Item
instance of: scholarly article
title: Exact value of widths of a certain class of solutions of linear differential equations
zbMATH Open document ID: 0645.41020
DOI: 10.1007/BF02633194
author: I. N. Volodina
published in: Analysis Mathematica
publication date: 1985-01-01T00:00:00Z
review text: The distance of a subset A of a normed vector space R from a subspace V, i.e. the expression \(E(A,x):=\sup_{x\in A}\inf_{y\in V}\| x- y\|\) is called the width of A with respect to V. This concept, introduced by A. N. Kolmogorov in 1963, is of increasing interest. - In the above paper the author uses the vector space \(C_{2\pi}\) of all real \(2\pi\)-periodic functions. The subset \(A=K(P)\) is defined by all functions of \(C_{2\pi}\) satisfying a certain differential inequality: Let \(P=P_ r(z)\) be a monic polynomial of degree \(r\in {\mathbb{N}}\) with real coefficients. Then K(P) consists of al \(2\pi\)-periodic functions x(t) which are absolutely continuous with their derivatives up to the order r-1 and satisfy \[ \| P(\frac{d}{dt})x\|_{\infty}= \sup_{t}| P(\frac{d}{dt})x(t)| \leq 1. \] The subspace is the common space \(T_{2n-1}\) of all real trigonometric sums of order at most n-1. The author gives under a suitable restriction of n for the corresponding width \[ \epsilon_ n:=\sup_{x\in K(P)}\inf_{y\in T_{2n-1}}\| x-y\| \] in the uniform norm an exact expression which is based on a special periodic function \(B_ r\), defined by a Fourier series in which the values of P at the points \(i\nu\), \(\nu\in {\mathbb{Z}}\) occur.
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 42A10
zbMATH DE Number: 4052219
zbMATH Keywords: width
zbMATH Keywords: Fourier series
MaRDI profile type: Publication
cites work: Q3506242
cites work: Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse
cites work: Q4050542
full work available at URL: https://doi.org/10.1007/bf02633194
OpenAlex ID: W75339013
Recommended article: Exact values of relative widths of classes of differentiable functions
Recommended article: Q3159713
Recommended article: Q3779174
Recommended article: Exact solutions of a linear functional differential equation
Recommended article: Estimates of bounded solutions of linear differential equations
Recommended article: On solutions of linear differential equations with entire coefficients
Recommended article: On particular solutions of linear partial differential equations
Recommended article: Exceptional values of solutions of linear differential equations
Recommended article: The nonzero solutions to a class of linear differential equations
Recommended article: Q3362384
description: scientific article; zbMATH DE number 4052219