Research Article About Some Linear Operators

The aim of this paper is to construct a class of linear operators in more general conditions. The method was inspired by Jakimovski and Leviatan (see [1]). We do not study the convergence of these operators with the well-known theorem of Bohman-Korovkin. The evaluation theorems for the rate of convergence are different from the well-known theorem of Shisha-Mond. We prove the Voronovskaja-type theorem for these operators. In the end, we give particularizations of these operators. We recall some notions and results which we will use in this paper. Let N be the set of positive integer numbers and N0 = N∪ {0}. For a given interval I , we will use the following function sets: B(I)= { f | f : I →R, f bounded on I}, C(I)= { f | f : I →R, f continuous on I}, and CB(I)= B(I)∩C(I). For any x ∈ I , consider the functions ψx : I →R defined by ψx(t)= t− x and ei : I →R, ei(t)= ti for any t ∈ I , i∈ {0,1,2,3,4}. For f ∈ CB(I), by the first-order modulus of smoothness of f is meant the function ω( f ;·) : [0,∞)→R defined for any δ ≥ 0 by