Uniform estimates of monotone and convex approximation of smooth functions
type: Item
instance of: scholarly article
title: Uniform estimates of monotone and convex approximation of smooth functions
zbMATH Open document ID: 0818.41015
published in: Journal of Approximation Theory
publication date: 1995-03-20T00:00:00Z
review text: The paper estimates the degree of approximation of monotone and of convex smooth functions by means of monotone and convex polynomials, respectively. A function \(f\in C[-1, 1]\cap C^ r(- 1,1)\) is said to be in the class \(B^ r\), if \(\sup_{0< x< 1} |(1- x^ 2)^{r/2} f^{(r)}(x)|< \infty\). Let \(\varphi(x):= \sqrt{1- x^ 2}\) and define for \(k\geq 1\), the weighted Ditzian-Totik modulus of smoothness of \(f\) by \[ \omega^ k_{\varphi, r}(f, t):= \sup_{0< h\leq r} \|(1- kh\varphi(x)/2- x)^{r/2} (1- kh\varphi(x)/2- x)^{r/2} \Delta^ k_{h\varphi(z)}(f, \cdot)\|, \] where \(\Delta^ k_{h\varphi(x)}\) is the familiar non-uniform difference. Denote by \(E^{(1)}_ n (f)\), and \(E^{(2)}_ n (f)\), the degrees of monotone and convex approximation by polynomials, respectively. Then the paper is devoted into proving the following significant results: Theorem 1. For \(k\geq 1\) and \(r\geq 3\), there exists a constant \(C= C(r, k)\), such that if \(f\in B^ r\) is nondecreasing, then for every \(n\geq r+ k-1\), \[ E^{(1)}_ n (f)\leq Cn^{-r} \omega^ k_{\varphi, r}(f^{(r)}, 1/n). \] Theorem 2. For \(k\geq 1\) and \(r\geq 5\), there exists a constant \(C= C(r, k)\), such that if \(f\in B^ r\) is convex, then for every \(n\geq r+ k- 1\), \[ E^{(2)}_ n (f)\leq Cn^{-r} \omega^ k_{\varphi, r} (f^{(r)}, 1/n). \] It is known that when \(r< 3\), in the monotone case, and when \(r< 5\) in the convex case, estimates like the above are usually impossible to obtain. In fact in the monotone case they hold only when \(r+ k\leq 2\), and in the convex case only when \(r+ k\leq 3\).
Mathematics Subject Classification ID: 41A29
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 720667
zbMATH Keywords: Jackson estimates
zbMATH Keywords: modulus of smoothness
zbMATH Keywords: monotone and convex approximation
author: Kirill A. Kopotun
reviewed by: D. Leviatan
MaRDI profile type: Publication
full work available at URL: jath.1995.1005
OpenAlex ID: W2033710365
DOI: 10.1006/JATH.1995.1005
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description: scientific article; zbMATH DE number 720667