THE GENERALIZATION OF VORONOVSKAJA’S THEOREM FOR A CLASS OF LINEAR AND POSITIVE OPERATORS

. In this paper we generalize Voronovskaja’s theorem for a class of linear and positive operators, and then, through particular cases, we obtain statements veriﬁed by the Bernstein, Schurer, Stancu, Kantorovich and Durrmeyer operators.


INTRODUCTION
In this section, we recall some notions and results which we will use in this article.
In 1932, E. Voronovskaja, proved the result contained in the following theorem.
In 2002, D. Bȃrbosu proved the result contained in the following theorem.
Let m be a nonzero natural number and the operators for any x ∈ [0, 1].These operators were introduced in 1967 by J.L. Durrmeyer in [5] and were studied in 1981 by M.M. Derriennic in [4], where the following theorem can be found.
If f is a two times derivable function on [0, 1] and the function f is continuous on [0, 1], then the convergence from For m be a nonzero natural number, let the operators The operators K m , where m is a nonzero natural number, are named Kantorovich operators, introduced and studied in 1930 by L.V. Kantorovich (see [10]).
For 0 ≤ α ≤ β and m a nonzero natural number, define , where m is a nonzero natural number, are named Bernstein-Stancu operators, introduced and studied in 1969 by D.D. Stancu in the paper [12].
In [12] is the result contained in the following theorem.
) be a two times derivable function at the point x ∈ [0, 1].Then the equality We 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 : Definition 1.5.If I ⊂ R is a given interval and f ∈ B(I), then the first order modulus of smoothness of f is the function ω 1 : [0, ∞) → R defined for any δ ≥ 0 by (1.12) In the following, we take into account the properties of the first order modulus of smoothness and the properties of the linear positive functional.

PRELIMINARIES
Theorem 2.1.Let I ⊂ R be an interval, a ∈ I, n ∈ N and the function f : I → R, f is n times derivable at a.According to Taylor's expansion theorem for the function f around a, we have where µ is a bounded function and lim Proof.If n = 0, the proof is immediately.Let n be a nonzero natural number.According to Taylor's expansion with the Lagrange's remainder, we have where ξ is between a and x.From (2.1) and (2.4), we obtain µ(x − a) =

MAIN RESULTS
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 ] and f is a s times derivable function at x, the function f (s) is continuous at x, then If f is a s times derivable function on [a, b], the function f (s) is continuous on [a, b] and there exists m(s) ∈ N and k j ∈ R so that for any natural number m, m ≥ m(s) and for any where j ∈ {s, s + 2}, then the convergence given in for any natural number m , m ≥ m(s).
Proof.Let m be a nonzero natural number.According to Taylor's theorem for the function f around x, we have where µ is a bounded function and lim Taking that A m,k is the linear positive functional into account, from (3.6) we have where Multiplying by ϕ m,k (x) and summing after k, where k ∈ {0, 1, . . ., m}, we obtain where Then and taking Lemma 1.7 into account, we obtain According to the relation (2.3), for any δ > 0 and for any t we have and so From (3.9) and (3.10), it results that , the inequality above becomes m , for any k ∈ {0, 1, . . ., m} and for any f ∈ C([0, 1]).In this application, we obtain the Bernstein operators and if i is a natural number, then for any x ∈ [0, 1] (see [6] or [10]).
In [6] are the results contained in the following theorem.
Theorem 3.4.If i is a natural number, then for any x ∈ [0, 1], where If s is a natural number, s even and j ∈ {s, s + 2}, then a j = 0, and then from (3.17) it results that there exists a natural number m (s) so that (3.20) for any x ∈ [0, 1] and for any natural number m, m ≥ m(s), where j ∈ {s, s + 2}.Because the conditions (3.2) and (3.4) take place, where α j = j 2 , j ∈ {s, s + 2}, Theorem 3.1 and Corollary 3.2 are enounced thus: and f is a s times derivable function at x and the function f (s) is continuous at x, then if s is a natural number and for any k ∈ {0, 1, . . ., m} and for any f ∈ L 1 ([0, 1]).In this case, we obtain the Durrmeyer operators.With calculus, for m a nonzero natural number, we have and for any m ∈ N, m ≥ 3, any x ∈ [0, 1] and Then, according to Corollary 3.2, Theorem 1.3 takes place.If m, i are natural numbers, let for any x ∈ [0, 1] (see [4]).
for any m, i ∈ N and any x ∈ [0, 1].
In [4] the result contained in the following corollary can be found.
Corollary 3.7.For any natural number j, j even, there exists k j ∈ R so that According to the Corollary 3.2, we have: f is s times derivable at x and the function f (s) is continuous at x, then if s is a natural number and If f is a s times derivable function on [0, 1] and the function f for any k ∈ {0, 1, . . ., m} and for any f ∈ L 1 ([0, 1]).In this case, we obtain the Kantorovich operators.If i is a natural number and x ∈ [0, 1], then and taking into account that (1 − x) i+1−j − (−x) i+1−j < 2 for any x ∈ [0, 1], α 2 = 1 and α 4 = 2, we have

Lemma 1 . 7 .
Let A : E(I) → R be a linear positive functional.Then a) for any f, g ∈ E(I) with f (x) ≤ g(x), for any x ∈ I, we have A(f ) ≤ A(g) and b) |A(f )| ≤ A(|f |), for any f ∈ E(I).
for any f ∈ E([a, b]) and for any x ∈ I. Proposition 2.3.For m be a nonzero natural number, the L m operators are linear and positive on E([a, b] ∩ [a , b ]).Proof.The proof follows immediately.Definition 2.4.Let m be a nonzero natural number and L m : E([a, b]) → F (I) be an operator defined in (2.5).For a natural number i, define T * m,i

,Corollary 3 . 2 .
for any natural number m, m ≥ m(s) and for any x ∈ [a, b] ∩ [a , b ], from which, the convergence from (3.3) is uniform on [a, b] ∩ [a , b ].From (3.8) and (3.13), (3.6) follows.Let f : [a, b] → R be a function.If x ∈ [a, b] ∩ [a , b ] and f is s times derivable and the function f (s) is continuous at x, then (3.14) lim m→∞