DURRMEYER-STANCU TYPE OPERATORS

. Considering two given real parameters α, β satisfying the conditions 0 ≤ α ≤ β . D. D. Stancu [7] constructed and studied the linear positive operators P ( α,β ) m : C ([0 , 1]) → C ([0 , 1]), deﬁned for any f ∈ C ([0 , 1]) and any positive inte-ger m by (1). In this paper we are dealing with the Durrmeyer form of Stancu’s operators. Some approximation properties of these Durrmeyer-Stancu operators are established. As a particular case, we retrieve approximation properties for the classical Durrmeyer operators [5].

The aim of the present paper is to construct the Durrmeyer type operators associated to the operators (1), denoted Durrmeyer-Stancu operators and to establish some of them approximation properties.
In Section 2 we recall the well-known theorem due to O. Shisha and B. Mond [6] and some properties of Durrmeyer operators [5].We also introduce here the Durrmeyer-Stancu operators, denoted D (α,β) m and we compute the images of test functions by these operators.
Section 3 contains the main results of the paper which are: a convergence theorem for the sequence D (α,β) m f m∈N ; the approximation order of f by f , under different assumption on the approximated function f .As particular case, we get the classical Durrmeyer operator (3).

AUXILIARY RESULTS
As usually, ω 1 (f ; δ) denotes the first order modulus of smoothness for the function f and e j (x) = x j (j = 0, 1, . . . ) are the test functions.
First, we recall the following result, due to O. Shisha and B. Mond [6].
Proof.It is easy to observe the following identities: Next, one applies Lemma 2.
Lemma 6.For any positive integer m and any x ∈ [0, 1] the following m holds, where and for any positive integer m.
For any positive integer m, let us to introduce the following notations: ; (m+β) 2 (m+2) .Let f m [0, 1] → R be defined for any x ∈ [0, 1] by: If β = 0, follows α = 0 and f m (x) = 0, for any x ∈ [0, 1].If β = 0 the function f is a polynomial function of second degree and because a m > 0 follows that f m has a maximum value in the point .
Proof.One applies Theorem 1 and Lemma 5.

Theorem 1 .
Let I be a non-empty interval of real axis, let L : C B (I) → B(I) be a linear and positive operator and, for x ∈ I, let ϕ x : I → R be defined by ϕ x (t) = |t − x|, for any t ∈ I. a) Let f ∈ C B (I) be given; for any δ > 0 and any x ∈ I the following