About Some Linear and Positive Operators Defined by Infinite Sum

Abstract

Abstract. In [13], we study a class of linear and positive operators defined by finite sum. In this paper we demonstrate general properties for a class of linear positive operators defined by infinite sum. By particularization, we obtain statements, the convergence and the evaluation for the rate of convergence in therm of the first modulus of smoothness for the Mirakjan-Favard-Szasz operators, Baskakov operators and Mayer-Konig and Zeller operators. We don't study the convergence of these operators with the well known theorem of Bohman-Korowkin. IntroductionIn this section, we recall some notions and results which we will use in this axticle.Let

About Some Linear Operators

“…For x ∈ J and m ∈ N, the following hold (2) (1) + 4a (3) (1) a(1) x + 3m 2 x 2 + a(1) + 10a (1) (1) + 6a (2) (1) a(1) mx + a (1) (1) + 7a (2) (1) + 6a (3) (1) + a (4) (1) a (1) .…”

Some general Baskakov type operators

“…The paper is organized as follows. In Section 2 we recall some results obtained by O.T.Pop in [7] which are essentially used for obtaining the main results of the paper. Section 3 is devoted to the construction of the general class of linear and positive operators defined by infinite sum, which we announced in the start.…”

“…In this section we recall some results from [7], which we shall use in the present paper. Let I; J be real intervals with the property I \ J is a nonempty interval.…”

The Voronovskaja type theorem for a general class of Szász-Mirakjan operators

“…Let N be the set of positive integers and N 0 D N [ f0g: In this section we recall some results from [12], which we shall use in the present paper. Let I; J be real intervals and I \ J ¤ ¿: For any n; k 2 N 0 , n ¤ 0 consider the functions ' n;k W J !…”