tailieunhanh - Approximation by Chlodowsky type of Szász operators based on Boas-Buck-type polynomials

A Chlodowsky variant of generalized Szász-type operators involving Boas–Buck-type polynomials is considered and some convergence properties of these operators by using a weighted Korovkin-type theorem are given. A Voronoskaja-type theorem is proved. | Turk J Math (2018) 42: 2243 – 2259 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Approximation by Chlodowsky type of Szász operators based on Boas–Buck-type polynomials Mohammad MURSALEEN1 ,, Ahmed Hussin AL-ABIED1 ,, Ana Maria ACU2,∗, 1 Department of Mathematics, Aligarh Muslim University, Aligarh, India 2 Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Sibiu, Romania Received: • Accepted/Published Online: • Final Version: Abstract: A Chlodowsky variant of generalized Szász-type operators involving Boas–Buck-type polynomials is considered and some convergence properties of these operators by using a weighted Korovkin-type theorem are given. A Voronoskaja-type theorem is proved. The convergence properties of these operators in a weighted space of functions defined on [0, ∞) are studied. The theoretical results are exemplified choosing the special cases of Boas–Buck polynomials, namely Appell-type polynomials, Laguerre polynomials, and Charlier polynomials. Key words: Szász operators, modulus of continuity, rate of convergence, weighted space, Boas–Buck-type polynomials 1. Introduction and preliminaries In recent years, there is an increasing interest to study linear positive operators based on certain polynomials, such as Appell polynomials, Laguerre polynomials, Charlier polynomials, Sheffer polynomials, and Hermite polynomials. In 1969, Jakimovski and Leviatan [16] introduced Szász-type operators by using Appell polynomials, as follows: Pn (f ; x) = ( ) ∞ e−nx ∑ k pk (nx)f , g(1) n () k=0 ∑∞ k where pk (x) , k ≥ 0, are the Appell polynomials defined by g(u)eux = and k=0 pk (x)u ∑∞ k g(u) = k=0 ak u is an analytic function in the disk | u | 1 and g(1) ̸= 0 . If g(u) = 1 , then pk (x) = xk (see [9]) and we obtain Szász–Mirakjan operators: k! Sn (f ; x) = e −nx ( ) ∞ ∑ (nx)k k f . k! n k=0 Very recently, the .