Linear operators that preserve some test functions

Abstract

The paper centers around a pair of sequences of linear positive operators. The former has the degree of exactness one and the latter has its degree of exactness null, but, as a novelty, it reproduces the third test function of Korovkin theorem. Quantitative estimates of the rate of convergence are given in different function spaces traveling from classical approximation to approximation in abstract spaces. Particular classes are also studied.

Local Approximation Properties of Modified Baskakov Operators

“…The idea was first introduced by King [9] for Bernstein polynomials. Similar problems were accomplished for Szász-Mirakjan operators [7], Szász-Mirakjan-Beta operators [8], Meyer-König and Zeller operators [12], Bernstein-Chlodovsky operators [1], q-Bernstein operators [10] and some other kinds of summation-type positive linear operators [2]. Very recently, local and global approximation properties of modified Szász-Mirakjan operators have been investigated in [11] and [6], respectively.…”

Modified Kantorovich Operators Providing a Better Error Estimation

“…King-type approximation operators [1][2][3][4][5][6][7] [0 ) x ≥ and n N ∈ , then we get the following modified positive linear operators:…”

A class of Markov type operators which preserveej,j⩾1

“…The Markov operators which preserve the monomial e 2 are named King type operators. They have been approached in recent years, starting with the paper of King [14] (see also [1,10,3,8,4]). …”