On the degree of piecewise shape-preserving approximation by polynomials
type: Item
instance of: scholarly article
title: On the degree of piecewise shape-preserving approximation by polynomials
zbMATH Open document ID: 1305.41011
published in: Journal of Approximation Theory
publication date: 2014-11-28T00:00:00Z
review text: Let \(\mathbb{P}_{n}\) be the space of all algebraic polynomials of degree \(< n,\) and \(C[-1,1]\) the space of continuous functions on \([-1,1]\) equipped with the norm \(\| f \| = \max_{x \in [-1,1]} | f(x) |.\) Given \(s \geq 1,\) denote by \(\mathbb{Y}_{s}\) the set of all collections \(Y_{s} = \{ y_{i} \}_{i=1}^{s}\) of points \(y_{i}\) such that \(-1 = y_0 < y_1 < \dots < y_s < y_{s+1} = 1.\) We define the set of pointwise \(q\)-convex functions as \(\Delta^{(q)}(Y_s) = \{ f \in C[-1,1] \cap C^{q-2}(-1,1) \mid (-1)^{i-s} f^{(q-2)} \text{ is non-strictly convex on }[y_i,y_{i+1}], 0 \leq i \leq s \}. \) Further, we say that two functions \(f,\) \(g\) are \(q\)-coconvex if \(f, g \in \Delta^{(q)}(Y_s)\) for some \(q \geq 2,\) \( Y_s \in \mathbb{Y}_{s}.\) For \(f \in \Delta^{(q)}(Y_s)\) we denote by \(E_{n}^{(q)}(f;Y_s) = \inf _{P_n \in \mathbb{P}_{n} \cap \Delta^{(q)}(Y_s)} \| f -P_n \|\) the degree of \(q\)-coconvex approximation by algebraic polynomials of \(f\) relative to \(Y_s.\) The author proves that the degree of piecewise \(q\)-coconvex approximation \(E_{n}^{(q)}(f;Y_s)\) of a \(q\)-convex function by algebraic polynomials of degree \(< n\) asymptotically depends only on \(\alpha,\) \(Y_s\) and \(q,\) \[ n^{\alpha} E_{n}^{(q)}(f;Y_s) \leq c(\alpha, s, q), \;n \geq N^{*}(\alpha, Y_s, q), \] assuming this inequality holds for an \(N^{*}\) dependent on \(f\) and that the degree of unconstrained approximation has the same decrease rate: \(n^{\alpha} \inf_{P_n \in \mathbb{P}_{n}} \| f - P_n \| \leq 1,\) \(n \geq N\) and \(N \leq s+q+1.\)
reviewed by: Zoltán Finta
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 6374934
zbMATH Keywords: comonotone polynomial approximation
zbMATH Keywords: coconvex polynomial approximation
zbMATH Keywords: degree of approximation
zbMATH Keywords: uniform norm
author: Oleksandr Vlasiuk
MaRDI profile type: Publication
full work available at URL: j.jat.2014.10.005
OpenAlex ID: W2088109138
cites work: Q3817079
cites work: Q3522935
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cites work: Positive results and counterexamples in comonotone approximation
cites work: Some positive results and counterexamples in comonotone approximation. II
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DOI: 10.1016/J.JAT.2014.10.005
description: scientific article; zbMATH DE number 6374934