Moduli of smoothness and best approximation of functions with singularities
type: Item
instance of: scholarly article
title: Moduli of smoothness and best approximation of functions with singularities
zbMATH Open document ID: 0963.41016
DOI: 10.1016/S0898-1221(00)00142-5
publication date: 2001-06-17T00:00:00Z
review text: Let \(\varphi_{\lambda,\alpha,0}(x)=|x|^\lambda\exp(iA|x|^{-\alpha}),\) where \(\lambda>0\), \(\alpha>0\), \(A\in R\backslash\{0\}\) and \(\varphi_{\lambda,\alpha,1}(x)= \operatorname {Re}\varphi_{\lambda,\alpha,0}(x),\) \(\varphi_{\lambda,\alpha,2}(x)= \operatorname {Im}\varphi_{\lambda,\alpha,0}(x).\) The question of the best approximation of functions \(\varphi_{\lambda,\alpha,i}\), \(i=0,1,2,\) by algebraic plynomials (resp., by entire functions of exponential type) with respect to the norm of the space \(L_p[-a,a]\), \(p\in[1,\infty]\) (resp., \(L_p(-\infty,\infty)\)) are studied. For example it is proved that the exact order of the best approximation by polynomials of order \(n\) is \(n^{-(\lambda+1/p)/(1+\alpha)}.\) Besides error estimates for the best approximation of functions \(\psi_{\lambda,\alpha,i,s}(\tau)=\varphi_{\lambda, \alpha,i}(\sin\tau)\cos^s \tau\), \(i=0,1,2,\) \(s\in N\cup\{0\}\) by means of trigonometric polynomials and periodic splines are given.
Mathematics Subject Classification ID: 41A52
zbMATH DE Number: 1471518
zbMATH Keywords: best approximation
zbMATH Keywords: moduli of smoothness
zbMATH Keywords: \(L_p\) space
zbMATH Keywords: singularity
zbMATH Keywords: polynomials
author: Michael I. Ganzburg
reviewed by: D. K. Ugulava
MaRDI profile type: Publication
cites work: On the Bernstein conjecture in approximation theory
cites work: Rational approximation to \(|x|\)
cites work: Q4081743
cites work: Best uniform rational approximation of 𝑥^{𝛼} on [0,1]
cites work: Q3948080
cites work: Decomposition theorems and approximation by a “floating" system of exponentials
cites work: Q5835496
cites work: Q5790953
cites work: Q5783394
cites work: Q4871485
cites work: Q3786878
cites work: Best approximation of functions like \(| x|^\lambda\exp (-A| x|^{-a})\)
cites work: Q5727762
cites work: Q4273944
cites work: Q5804330
cites work: Q5809332
cites work: Q3237284
cites work: On the fundamental approximation theorems of D. Jackson, S.N. Bernstein and theorems of M. Zamansky and S.B. Stechkin
cites work: Q4348446
cites work: Multidimensional Jackson theorems
cites work: Q4039793
cites work: Power Series with Bounded Coefficients
cites work: Functions of exponential type
published in: Computers & Mathematics with Applications
Recommended article: Q4355382
Recommended article: Best approximation and moduli of smoothness: Computation and equivalence theorems
Recommended article: Best approximations and moduli of smoothness of functions and their derivatives in \(L_p\), \(0 < p < 1\)
Recommended article: Q4486599
Recommended article: Q4219196
Recommended article: Moduli of smoothness of functions and their derivatives
Recommended article: Best approximations of classes of functions defined by means of the modulus of continuity
Recommended article: Q5185481
Recommended article: Q4939493
description: scientific article; zbMATH DE number 1471518