Degree of simultaneous coconvex polynomial approximation
type: Item
instance of: scholarly article
title: Degree of simultaneous coconvex polynomial approximation
zbMATH Open document ID: 0926.41014
DOI: 10.1007/BF03322045
author: D. Leviatan
author: Kirill A. Kopotun
published in: Results in Mathematics
publication date: 1998-10-13T00:00:00Z
review text: Let \(f\in C^1[-1,1]\) change its convexity \(s\)-times at the points \(y_j\in (-1,1)\) \((j-1, \dots,s)\). Then \(f\) is approximated by polynomials \(p_n\), which are coconvex with \(f\), i.e., \(p_n\) changes its convexity exactly at the same points \(y_j\) \((j=1, \dots,s)\). It is proved that for each sufficiently large \(n\), there exists a polynomial \(p_n\) of degree \(n\), which is coconvex with \(f\) and satisfies \[ \begin{aligned} \| f-p_n \|_\infty & \leq C(s)n^{-1} \omega_2 (f^1, n^{-1}),\\ \| f'-p_n' \|_\infty & \leq C(s) \omega_2 (f',n^{-1}). \end{aligned} \] The constant \(C(s)\) is independent of \(f\), \(n\), and the location of the points \(y_j\) \((j=1, \dots, s)\). Here \(\omega_2\) denotes the second Ditzian-Totik modulus of smoothness.
Mathematics Subject Classification ID: 41A29
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 1209937
zbMATH Keywords: coconvex polynomial approximation
zbMATH Keywords: Jackson-type estimates
zbMATH Keywords: simultaneous approximation
zbMATH Keywords: Ditzian-Totik modules of smoothness
reviewed by: Manfred Tasche
MaRDI profile type: Publication
cites work: On Comonotone Approximation
cites work: Coconvex polynomial approximation of twice differentiable functions
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cites work: Comonotone polynomial approximation in \(L_p[-1,1], 0<p\leq\infty\)
cites work: The degree of coconvex polynomial approximation
cites work: ON COMONOTONE APPROXIMATION BY POLYNOMIALS IN Lp SPACE
description: scientific article; zbMATH DE number 1209937