On direct and converse theorems of the approximation theory in the metric of a sign sensitive weight
type: Item
instance of: scholarly article
title: On direct and converse theorems of the approximation theory in the metric of a sign sensitive weight
zbMATH Open document ID: 0885.42002
DOI: 10.1007/BF01911125
author: A.-R. K. Ramazanov
published in: Analysis Mathematica
publication date: 1998-04-20T00:00:00Z
review text: Notations and Definitions. A pair \(p_+(x)\) and \(p_-(x)\) of continuous functions on \(\Delta= [a,b]\subset (-\infty,\infty)\) is called a continuous sign sensitive weight \((p_-(x), p_+(x))\) denoted by \(p(x)\). \(p(x)\) is regular if \(p_+(x)p_-(x)> 0\) and monotonic if \(p_+(x)\) and \(p_-(x)\) are monotonic. As usual \(x^+= x\) if \(x>0\) and \(x^+= 0\) otherwise. Similarly for \(x^-\). The following notations are also used with continuous functions \(f\) in connection with sign sensitive weights. \(|(f, p)(x)|= f^+(x) p_+(x)+ f^-(x) p_-(x)\), \(|f|_p=|f|_{p,\Delta}=|f|_{(p_-,p_+)}= \sup\{|(f,p)|: x\in\Delta\}\). For \(2\pi\)-periodic functions and weights the supremum is taken on any interval of length \(2\pi\). Also \(|f|= |f|_\Delta= \max\{|f(x)|: x\in\Delta\}\). \(v(f)= v(f,\Delta)\) is the total variation of \(f\) on \(\Delta\) and \(\Omega(f)= \Omega(f,\Delta)\) is its oscillation on \(\Delta\). The moduli of continuity used in various theorems are: \[ \omega(f,\delta)= \sup\{|(f_h- f)(x)|: x,x+ h\in\Delta, |h|\leq\delta\},\;f_h(x)= f(x+ h), \] \[ \omega(\delta, f,p)= \sup\{|(f_h- f,p)(x)|: x,x+h\in \Delta, |h|\leq\delta\}, \] \[ \omega_+(\delta, f)_p= \sup\biggl\{\{\sup|(f_h(y)- f(y),p(x))|: y,y+ h\in[x, b]\}: x\in\Delta, |h|\leq\delta\biggr\}. \] Similarly \(\omega_-(\delta, f)_p\) is defined by changing \([x,b]\) to \([a,x]\). If in the definition of \(\omega(\delta, f,p)\) instead \(|h|\leq\delta\) one takes \(0\leq h\leq\delta\), \(-\delta\leq h\leq 0\), the definitions of \(\omega_+(\delta, f,p)\) and \(\omega_-(\delta, f,p)\) are obtained. In case of \(2\pi\)-periodic functions and weights in the definition of the moduli of continuity instead of \(x,x+ h[0,2\pi]\) one can take also \(x,x+ h\in[-\infty,\infty]\). The results can be different but the notations remain the same. In the first section of the paper, the author compares the four basic known properties of the usual uniform modulus of continuity \(\omega(f,\delta)\) to the other generalize…
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 817240
zbMATH Keywords: best trigonometric approximation
zbMATH Keywords: Jackson's theorem
zbMATH Keywords: Bernstein's inequality
zbMATH Keywords: continuous sign sensitive weight
zbMATH Keywords: moduli of continuity
zbMATH Keywords: trigonometric polynomials
zbMATH Keywords: best approximation
zbMATH Keywords: best rational approximations
reviewed by: Zalman Rubinstein
MaRDI profile type: Publication
cites work: Q3506242
cites work: Some exact integral estimates of the derivative of rational and algebraic functions. Applications
cites work: The analytic order of Ш for modular elliptic curves
cites work: The Brauer-Manin Obstruction and III[2]
full work available at URL: https://doi.org/10.1007/bf01911125
OpenAlex ID: W2106558410
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description: scientific article; zbMATH DE number 817240