On the simultaneous approximation of functions and their derivatives by the Szász-Mirakyan operator
type: Item
instance of: scholarly article
title: On the simultaneous approximation of functions and their derivatives by the Szász-Mirakyan operator
zbMATH Open document ID: 0697.41009
DOI: 10.1016/0021-9045(88)90094-9
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: The Szász-Mirakyan operator \(S_ n\) is defined by \(S_ n(f,x):=\sum^{\infty}_{k=0}f(k/n)p_ k(nx)\), \(p_ k(t)=e^{-t}t^ k/k!\). The class \(B_ r^{(\alpha)}:=\{f|\) \(f^{(r-1)}\in C[0,\infty)\), \(f_{\pm}^{(r)}(x)\) exist everywhere and are bounded on every finite subinterval of [0,\(\infty)\) and \(f_{\pm}^{(r)}(t)=O(t^{\alpha t})\) (t\(\to \infty)\) for some \(\alpha >0\}\), \(r=0,1,...\); \(f_{\pm}^{(0)}(x)\) means \(f(x\pm))\) is larger than the class of functions of generalized bounded variation. The author gives a precise estimate of the rate of convergence of the operator \(S_ n(f,x)\) to \((f_+^{(r)}(x)+f_-^{(r)}(x))\) for \(f\in B_ r^{(\alpha)}\).
reviewed by: S.Aljančić
Mathematics Subject Classification ID: 41A28
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 4142715
zbMATH Keywords: simultaneous approximation
zbMATH Keywords: Szász-Mirakyan operator
zbMATH Keywords: functions of generalized bounded variation
author: Xiehua Sun
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90094-9
OpenAlex ID: W2023778784
cites work: Generalization of Bernstein's polynomials to the infinite interval
cites work: Q5624137
cites work: Approximation of unbounded functions on unbounded interval
cites work: On the rate of convergence of the Szász-Mirakyan operator for functions of bounded variation
cites work: On the rate of convergence of Bernstein polynomials of functions of bounded variation
cites work: Q3761032
cites work: On the rate of convergence of Fourier series for functions of HBMV
description: scientific article; zbMATH DE number 4142715