Best constants in preservation inequalities concerning the first modulus and Lipschitz classes for Bernstein-type operators
type: Item
instance of: scholarly article
title: Best constants in preservation inequalities concerning the first modulus and Lipschitz classes for Bernstein-type operators
zbMATH Open document ID: 0923.41015
author: José A. Adell
author: Ana Pérez-Palomares
published in: Journal of Approximation Theory
publication date: 1998-10-27T00:00:00Z
review text: Many families \(\{L_t:t\in T\}\) of positive linear operators considered in the literature on approximation theory admit a probabilistic representation of the form \[ L_tf(x)= Ef\bigl(Z_t(x)\bigr) \quad (x\in I,\;t\in T),\tag \] where \(I=[0,1]\) or \(I=[0,\infty)\), \(T=\mathbb{N}\) or \(T=(0,\infty)\), \(E\) denotes mathematical expectation, \(\{Z_t(x):x\in I,\;t\in T\}\) is a doubly-indexed stochastic process of integrable random variables taking values in \(I\), and \(f\) is any real-valued measurable function on \(I\) for which the right-hand side of (*) is well defined. For such families the authors give probabilistic characterisations of the best possible constants both in preservation inequalities concerning the first modulus and in preservation of Lipschitz classes of first order. Such constants for the Bernstein, Szász, gamma, Baskakov and beta operators are computed. This continues a study begun in \textit{J. A. Adell} and \textit{J. de la Cal}, in Approximation, probability and related fields Plenum, New York, 1994, 1-18 (1994; Zbl 0863.41007)].
Mathematics Subject Classification ID: 41A44
zbMATH DE Number: 1197075
reviewed by: H. R. Dowson
MaRDI profile type: Publication
full work available at URL: jath.1998.3164
OpenAlex ID: W2034759823
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DOI: 10.1006/JATH.1998.3164
description: scientific article; zbMATH DE number 1197075