Shape preserving approximation on the real line with exponential weights
type: Item
instance of: scholarly article
title: Shape preserving approximation on the real line with exponential weights
zbMATH Open document ID: 1184.41006
author: Oleksandr Maizlish
published in: Journal of Approximation Theory
publication date: 2009-05-28T00:00:00Z
review text: The paper deals with weighted polynomial approximation on the real line. In the 1950's, Achieser, Mergelyan and Pollard solved the well-known `Bernstein's weighted problem' posed by \textit{S. Bernstein} in 1924 [Bull. Math. Soc. France 52, 399--410 (1924; JFM 50.0195.01)]. As stated by \textit{H. Pollard} [Proc. Am. Math. Soc. 4, 869--875 (1953; Zbl 0057.29802)] it was written as follows. Here for \(W: \mathbb{R}\rightarrow (0,1]\), \(C_W\) denotes the space of all \(f\in C(\mathbb{R})\) such that \(\lim _{x\rightarrow \pm \infty}f(x)W(x)=0\) and \(\|f\|_W:=\sup _{x\in \mathbb{R}}|f(x)W(x)|\). Theorem. Suppose \(x^n\in C_{W}\), for an non-negative integer \(n\). Then, in favor that \(\{x^n\}_{n=0}^{\infty}\) is fundamental in \(C_{W}\) it is necessary and sufficient that {\parindent=8mm \begin{itemize}\item[(i)] \(\int _{-\infty}^{+\infty}\frac{\log (1/W(t))}{1+t^2}\,dt=\infty\); \item[(ii)] there exists a sequence of polynomials \(\{P_n\}\) such that for each \(x\): \[ \lim _{n\rightarrow \infty}P_n(x)W(x)=1\quad\text{ while }\quad\sup _{n\geq 1}\|P_n\|_W<\infty. \] \end{itemize}} In particular, the so-called Freud's weight \(W_{\alpha }(x):=e^{-|x|^{\alpha }}\) satisfies conditions (i) and (ii) if \(\alpha \geq 1\), and therefore the linear span of \(x^n\), \(n\in \mathbb{Z}_+\) is dense in \(C_{W_{\alpha }}\) if \(\alpha \geq 1\). The main goal of the paper is to investigate `Bernstein's weighted problem' in Shape Preserving Approximation using Freud's weight. For \(f\in C_W\), let \[ E_n(f,W):=\inf _{P\in \Pi_n}\|f-P\|_W \quad \quad E_n^k(f,W):=\inf _{P\in \Pi_n^k}\|f-P\|_W, \] where \(\Pi _n\) and \(\Pi _n^k\) denote the sets of all algebraic and all \(k\)-monotone algebraic polynomials of degree \(\leq n\), respectively (\(f:\mathbb{R}\rightarrow \mathbb{R}\) is said to be \(k\)-monotone if \([x_0,x_1,\dots x_k,f]\geq 0\) for any collection of \(k+1\) distinct points \(x_0,x_1,\dots , x_k\)). The main result of the paper reads as follows: Theorem 1. Let \(W_{…
reviewed by: Daniel Cárdenas Morales
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 5558974
zbMATH Keywords: weighted approximation
zbMATH Keywords: Freud's weights
zbMATH Keywords: \(k\)-monotone approximation
MaRDI profile type: Publication
full work available at URL: j.jat.2008.07.005
OpenAlex ID: W2035942352
cites work: A criterion for \(n\)-convexity
cites work: Q3994552
cites work: Shape-preserving approximation by polynomials
cites work: Q3406470
cites work: Solution of Bernstein's Approximation Problem
cites work: Another Proof that Convex Functions are Locally Lipschitz
DOI: 10.1016/J.JAT.2008.07.005
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description: scientific article; zbMATH DE number 5558974