ABOUT A GENERAL PROPERTY FOR A CLASS OF LINEAR POSITIVE OPERATORS AND APPLICATIONS

. In this paper we demonstrate a general property for a class of linear positive operators. By particularization, we obtain the convergence and the evaluation for the rate of convergence in term of the ﬁrst modulus of smoothness for the Bernstein operators, Durrmeyer operators, Kantorovich operators and Bleimann, Butzer and Hahn operators.


INTRODUCTION
In this section, we recall some notions and results which we will use in this article (see [10]).
We For m a non zero natural number, define the operator for any f ∈ E([a, b]), for any x ∈ I and for a natural number i, define T * m,i by (2) In the following, let s be a fixed natural number, s even and we suppose that the operators L * m m≥1 verify the conditions: there exists the smallest If I ⊂ R is a given interval and f ∈ C B (I), then the first order modulus of smoothness of f is the function ω 1 : [0, ∞) → R defined for any δ ≥ 0 by In [10] are the results contained in the following.
where j ∈ {s, s + 2}, then for any natural number m, m ≥ m(s).

PRELIMINARIES
In this section, we recall some operator definitions which we will use in this article.In the following, let m be a non zero natural number. Let where p m,k (x) are the fundamental polynomials of Bernstein defined as follows ( 9) The operators for any x ∈ [0, 1], are named Durrmeyer operators, introduced in 1967 by J. L. Durrmeyer in [8] and were studied in 1981 by M. M. Derriennic in [6].
In 1980, G. Bleimann, P. L. Butzer and L. Hahn introduced in [5] a sequence of linear positive operators for any x ∈ [0, ∞), for any non zero natural number m.These operators are named Bleimann, Butzer and Hahn operators.

MAIN RESULTS
In the following, in Theorem 2 we consider s = 0.For the operators recall in the preliminaries we have α 0 = 0 and α 2 = 1.Then Theorem 2 becomes: and there exists a non zero natural number m(0) and a real numbers k 0 , k 2 so that for any natural number m, m ≥ m(0) and for any

. , m} and for any f ∈ C([0, 1]).
In this application, we obtain the Bernstein operators and for any non zero natural number m, so k 0 = 1, k 2 = 1 4 (see [9] or [13]).Because the condition ( 13) and ( 14) take place, Theorem 3 is announced thus: for any x ∈ [0, 1], the convergence given in (17) is uniform on [0, 1] and for any non zero natural number m. Application 6.Let m be a non zero natural number, the functionals A m,k : In this case, we obtain the Durrmeyer operators.We have , for any x ∈ [0, 1], for any non zero natural number m, so k 0 = 1, k 2 = 1 4 for any m ∈ N, m ≥ 3 (see [10]).Then, we have: for any x ∈ [0, 1], the convergence given in (19) is uniform on [0, 1] and for any natural number m, m ≥ 3.
for any x ∈ [0, 1], the convergence given in (21) is uniform on [0, 1] and consider I ⊂ R, I an interval and we shall use the function sets: E(I), F (I) which are subsets of the set of real functions defined on I, B(I) = {f |f : I → R, f bounded on I}, C(I) = {f |f : I → R, f continuous on I} and C B (I) = B(I) ∩ C(I).For x ∈ I, consider the function ψ x : I → R, ψ x (t) = t − x, for any t ∈ R. Let a, b, a , b be real numbers, I ⊂ R interval, a < b, a < b , [a, b] ⊂ I, [a , b ] ⊂ I and [a, b] ∩ [a , b ] = ∅.For any non zero natural number m, consider the functions ϕ m,k : I → R with the property that ϕ m,k (x) ≥ 0 for any x ∈ [a , b ], for any k ∈ {0, 1, . . ., m} and the linear positive functionals A m,k : E([a, b]) → R for any k ∈ {0, 1, . . ., m}.

Proposition 1 .Theorem 2 .
For m a non zero natural number, the L * m operators are linear and positive.Let f : [a, b] → R be a function.If f is a s times derivable on [a, b], the function f (s) is continuous on [a, b] and there exists a non zero natural number m(s) and a real numbers k s , k s+2 so that for any natural number m, m ≥ m(s) and for any x ∈ [a, b] ∩ [a , b ], we have Proof.Theorem 3 is a consequence of Theorem 2.In Applications 4, 6, 8 and 10 we consider a = a = 0, b = b = 1 and ϕ m,k = p m,k , for any m, k natural numbers, m = 0 and k ∈ {0, 1, . . ., m}.