Polynomial approximation on infinite intervals with weights having inner zeros
type: Item
instance of: scholarly article
title: Polynomial approximation on infinite intervals with weights having inner zeros
zbMATH Open document ID: 1012.41008
DOI: 10.1023/A:1019769202986
author: Giuseppe Mastroianni
author: Jószef Szabados
published in: Acta Mathematica Hungarica
publication date: 2003-05-18T00:00:00Z
review text: The authors generalize Languerre weights on \(\mathbb R^+\) by multiplying them by translations of finitely many Freud type weights which have singularities, and prove polynomial approximation theorems in the corresponding weighted spaces. Let \(\mathcal V\) denote the set of continuous functions \(v(x)\) which are even on \(\mathbb R=(-\infty,\infty)\), \(v(0)=0\), \(v'(x)>0\) on \(\mathbb R^+:=(0,\infty)\), and \[ A(v):=\lim_{x\to 0} \frac{xv'(x)}{v(x)} <\infty . \] Further let (1) \(u(x)=w_\alpha(x)V(x)\), \(V(x)=\prod^s_{k=1}v_k(x-t_k)\) with \(w_\alpha(x)=x^\alpha e^{-x}\), \(\alpha>-1\), \(x>0\), \(0<t_1<t_2<\dots <t_s<\infty\), \(v_k\in \mathcal V\), \(k=1,\dots,s\). Let \(L^p(A)\), \(A\subset \mathbb R\), \(1\leq p<\infty\), denote the set of all measurable functions \(f\) with the usual norm \(\|f\|_p(\int_A|f(x)|^pdx)^{1/p}\). If \(\sigma\) is a weight function then \(f\in L^p_\sigma(A)\) if \(f\sigma\in L^p(A)\). In order to give the central result of the paper to all functions that have singular points at the zeros \(t_i\) of the weight \(u\), the authors introduce some new classes of functions. For \(1\leq p<\infty\), \(L^p_u:=L^p_u(\mathbb R^+)\) and \(L^\infty_u\) denote the set of all continuous functions in \(\mathbb R^+\setminus \{t_1,t_2,\dots,t_s\}=:\tilde{\mathbb R}^+\) such that \(\lim_{x\to t_i}(fu)(x)=0\), \(i=1,\dots,s\) and \(\lim_{x\to 0}(fu)(x)=\lim_{x\to \infty}(fu)(x)=0\). Finally is defined \(W^p_r(u)=\{f\in L^p_u:f^{(r-1)}\in AC(\tilde{\mathbb R}^+)\), \(\|f^{(r)}\varphi^r u\|_p<\infty\}\) where \(r\geq 1\), \(1\leq p\leq \infty\), \(\varphi(x)=\sqrt x\) and \(AC(\tilde{\mathbb R}^+)\) is the set of all absolutely continuous functions in \(\tilde{\mathbb R}^+\). Theorem. Assume that for \(v_k\in \mathcal V\) we have \(0<A(v_k)\neq j-\frac 1p\), \(k=1,\dots,s\); \(j=1,\dots,r\) with \(1\leq p\leq \infty\), or \(A(v_k)=0\) for some \(1\leq k\leq s\) with \(p=\infty\). Then for each function \(f\in W^p_r(u)\), we have \[ E_n(f)_{u,p}\leq…
reviewed by: Zoltán Finta
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 1912066
zbMATH Keywords: weights with singularities
zbMATH Keywords: \(K\)-functionals
zbMATH Keywords: moduli of continuity
MaRDI profile type: Publication
full work available at URL: a:1019769202986
OpenAlex ID: W1791268161
description: scientific article; zbMATH DE number 1912066