Approximation properties of a class of linear operators

Abstract

This work focuses on a class of linear positive operators of discrete type. We present the relationship between the local smoothness of functions and the local approximation. Also, the degree of approximation in terms of the moduli of smoothness is established, and the statistical convergence of the sequence is studied.

On new class of linear and positive operators

“…Due to intersecting properties of the Szász-Mirakyan operators, these operators have been intensively studied by many mathematicians [8][9][10][11][12]. Furthermore, its modification the Jain operators and its different generalizations were also discussed by many researchers in recent years, see [13][14][15][16][17][18]. This motivated us to generalize Jain operators as variant of the Lupaş operators defined by (1.1).…”

Asymptotic behaviour of Jain operators

“…One special case of it is Phillips operator [16]. Recently, one of the authors [5] presented the relationship between the local smoothness of functions and the local approximation by the operators (1) and studied the statistical convergence of the sequence.…”

Some approximation properties of a new class of linear operators

“…The rate of convergence of these operators was developed by Rempulska and Walczak, and the asymptotic expansion introduced by Abel et al In 1972, Jain introduced another generalization of the Szász‐Mirakyan operators . The relation between the local smoothness of function and local approximation, the degree of approximation, and the statistical convergence of the Jain operators were studied by Agratini . The Durrmeyer‐type generalizations of the Jain operators and its approximation properties were elaborated by Tarabie, Mishra and Patel, and Agratini .…”

“…9 The relation between the local smoothness of function and local approximation, the degree of approximation, and the statistical convergence of the Jain operators were studied by Agratini. 10 The Durrmeyer-type generalizations of the Jain operators and its approximation properties were elaborated by Tarabie, 11 Mishra and Patel,12,13 and Agratini. 14 The generalized Jain operators as variant of the Lupaş operators were studied by Patel and Mishra.…”