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


Introduction
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 N 0 = N ∪ {0}.For a given interval I, we will use the following function sets:

continuous on I}, and C B (I) = B(I) ∩ C(I).
For any x ∈ I, consider the functions ψ x : I → R defined by ψ x (t) = t − x and e i : I → R, e i (t) = t i for any t ∈ I, i ∈ {0, 1,2,3,4}.
For f ∈ C B (I), by the first-order modulus of smoothness of f is meant the function ω( f ;•) : [0,∞) → R defined for any δ ≥ 0 by (1.1) In the following, we take into account the properties of the first-order modulus of smoothness and the properties of the linear positive functional.
Lemma 1.1.Let f ∈ C B (I).Then, ω( f ;•) has the following properties: (a) ω( f ;0) = 0, (b) ω( f ;•) is an increasing function, for any x ∈ I, one has , where E(I) is a subset of the set of real functions defined on I.
In [2] we have demonstrated the following theorem.
Theorem 1.3.Let I be an interval x ∈ I, and let the function f : I → R be s times differentiable in x.According to the Taylor Expansion Theorem, one has where μ is a bounded function and lim then for any δ > 0 and x ∈ I one has

Preliminaries
In this section, we construct a general class of linear and positive operators and we demonstrate for these operators an approximation theorem and a Voronovskaja-type theorem.
Let I, J be intervals and I ∩ J is a nonempty interval.For any m ∈ N and k ∈ N 0 , consider the function ϕ m,k : J → R with the property ϕ m,k (x) ≥ 0 for any x ∈ J and the linear and positive functional A m,k : E(I) → R.
Ovidiu T. Pop 3 In the following, let E(I) and F(J) be subsets of the set of real functions defined on I, J respectively, such that the series ∞ k=0 ϕ m,k (x)A m,k ( f ) is convergent for any f ∈ E(I) and any x ∈ J.We suppose that ψ i x ∈ E(I) for any x ∈ I ∩ J and any i ∈ {0, 1,...,s + 2}.In what follows s ∈ N 0 , s is even.Definition 2.1.For m ∈ N, define the operator L m : E(I) → F(J) by for any f ∈ E(I) and x ∈ J.
Proposition 2.2.The operators (L m ) m≥1 are linear and positive on E(I ∩ J).
Proof.The proof follows immediately.
Definition 2.3.For m ∈ N and i ∈ N 0 , define T i by for any Assume that f is an s times differentiable function on I with f (s) continuous on I and an interval K ⊂ I ∩ J exists such that there exist m(s) ∈ N and the constants k j (K) ∈ R depending on K, so that for any m ∈ N, m ≥ m(s) and x ∈ K, one has where j ∈ {s, s + 2}.Then, the convergence given in (2.4) is uniform on K and for any x ∈ K and m ≥ m(s).
Proof.According to Taylor's Theorem, we have where μ is a bounded function and lim Hence, from (2.7), we have where μ x : Multiplying by ϕ m,k (x) and summing over k ∈ N 0 , we obtain (2.9) Thus, where and taking Lemma 1.2 into account, we obtain According to the relation (1.4), for any δ > 0 and t ∈ I ∩ J, we have and so Ovidiu T. Pop 5 From (2.13) and (2.15), it results that x ω f (s) ;δ . (2.16) (2.17) Taking into account that (T s L m )(x)/m αs and (T s+2 L m )(x)/m αs+2 are bounded for any m ∈ N, m ≥ m(s), and considering the fact that we have that for m ≥ m(s) and x ∈ K. Therefore, the convergence from (2.4) is uniform on K. Now, (2.10) and (2.21) yield (2.6).
In the following, we suppose that for any k ∈ N 0 and m ∈ N, we have and for any

.26)
Assume that f is continuous on I and an interval K ⊂ I ∩ J exists, such that there exist m(0) ∈ N and k 2 (K) so that for any m ∈ N, m ≥ m(0), and x ∈ K, one has

.27)
Then, the convergence given in (2.26) is uniform on K and for any x ∈ K and m ≥ m(0).
Corollary 2.10.If f ∈ E(I) is a two-times differentiable function in x ∈ I ∩ J, with f (2)  continuous in x, and if there exist α 2 , α 4 and m(2) ∈ N such that

.30)
Assume that f is a two-times differentiable function on I with f (2) continuous on I and an interval K ⊂ I ∩ J exists, such that there exist m(2) ∈ N and k j (K), so that for any m ≥ m(2) and x ∈ K, one has where j ∈ {2, 4}.Then, the convergence given in (2.30)is uniform on K.

Main results
In this section, we construct a general class of linear positive operators.Let R 0 = [0,∞) and J be an interval with J ⊂ R 0 .Let the sequence (a m ) m≥1 so that a m x ∈ J for any m ∈ N and x ∈ J.The indefinitely differentiable functions a,b : J → R have the property: for any x ∈ R 0 , and for any compact K ⊂ J the constants M 1 (K), M 2 (K) depending on K exist, such that Then, it is known that and we take it to the form where for any m ∈ N and x ∈ J.
In the following, let a fixed function w : R 0 → (0,∞), called the weight function, and the set functions (3.9) For f ∈ E(w), there exists a positive constant M such that w For m ∈ N and x ∈ J, and taking in the end (3.8)into account, we have from where it results that the series for any f ∈ E(w) and x ∈ J, where F(J) is a subset of the set of real functions defined on J.
Remark 3.4.The operators (L m ) m≥1 are linear and positive on E(w).
Proof.The relation (3.13) results from (3.8).The proof of relations (3.14) follows immediately by differentiating (3.6) with respect to u, and after that take 1 for u and a m x for x.
Then, the convergence given in (3.19) is uniform in K and for any x ∈ K and any m ≥ m(0), where M(K) is a constant depending on K.
Taking relation (3.16) into account, we apply now the Corollary 2.9.The proof is similar on a compact interval K.

.26)
Assume that f is a two-times differentiable function on R 0 with f (2) continuous on R 0 and a compact interval K ⊂ R 0 exists, such that there exist m(2) ∈ N and l i (K) so that for any m ≥ m(2) and x ∈ K, one has Now, we give some applications where a m = m for any m ∈ N. In the following, by particularization and applying Theorems 3.8 and 3.9, we can obtain approximation theorems and Voronovskaja-type theorems for some known operators.Because every application is a simple substitute in the theorems of this section, we will not replace anything.Application 3.10.If a(x) = 1 and b(x) = e x , x ∈ R 0 , we obtain the Mirakjan-Favard-Szász operators (see [3][4][5]).Application 3.11.If a(x) = g(x) = ∞ n=0 a n x n and b(x) = e x , x ∈ R 0 , we obtain the operators considered by Jakimovski and Leviatan in the paper [1].Application 3.12.If a(x) = g(x) = 1 and b(x) = cosh x = ∞ k=0 (1/(2k)!)x 2k , x ∈ R 0 , then we get the operators considered by Leśniewicz and Rempulska in the paper [6].Application 3.13.If a(x) = g(x) = 1 and b(x) = sinhx = ∞ k=0 (1/(2k + 1)!)x 2k+1 , x ∈ R 0 , we get the operators where m ∈ N and x ∈ R 0 .The operators of this type are introduced and studied by Rempulska and Skorupka in the paper [7].