ON THE COMPOSITE BERNSTEIN TYPE QUADRATURE FORMULA

. Considering a given function f ∈ C [0 , 1], the interval [0 , 1] is divided in m equally spaced subintervals (cid:2) k − 1 m , km (cid:3) , k = 1 , m . On each of such type of interval the Bernstein approximation formula is applied and a corresponding Bernstein type quadrature formula is obtained. Making the sum of mentioned quadrature formulas, the composite Bernstein type quadrature formula is obtained.

For any f ∈ C[0, 1], any x ∈ [0, 1] and any n ∈ N, the following equality is called the Bernstein approximation formula, where R n is the remainder operator associated to the Bernstein operator B n , i.e.R n f is the remainder term of the approximation formula (1.3).Regarding the remainder term of (1.3), Tiberiu Popoviciu [4] established the following: Theorem 1.1.For any f ∈ C[0, 1] there exist the distinct points ξ 1 , ξ 2 , ξ 3 ∈ [0, 1] such that, for any x ∈ [0, 1], the remainder term of (1.3) can be represented under the form In (1.4) the brackets denote the divided difference of function f with respect the distinct knots ξ 1 , ξ 2 , ξ 3 .It is well known the following estimation of the remainder term of (1.3), (see [7]).
The inequality (1.5) follows directly from (1.4), applying the mean value theorem for divided differences and it is attributed to D. D. Stancu.
In the following we suppose that f ∈ C 2 [0, 1].Starting with the Bernstein approximation formula (1.3), in [7] the following Bernstein quadrature formula (1.7) is obtained, where The focus of the present paper is to construct the composite Bernstein type quadrature formula.For this aim, the interval [0, 1] will be divided in m equally spaced subintervals k−1 m , k m , k = 1, m.On each of such type of interval, the Bernstein quadrature formula (1.7) will be applied.Next, adding the mentioned quadrature formulas, the desired Bernstein type quadrature formula on [0, 1] will be obtained.

MAIN RESULTS
We start with two auxiliary results.
Proof.One applies in a way similar to the case of relation (1.5), taking the transformation t → x−a b−a into account.
In what follows, let us to consider the interval [0, 1] divided in the equally spaced subintervals k−1 m , k m , k = 1, m.In each interval k−1 m , k m , k = 1, m, one considers the distinct knots x i = kn−n+i mn , i = 0, n.Applying Lemma 2.1.,yields the following Bernstein type polynomial The corresponding Bernstein type approximation formula on the interval , the remainder term of (2.4) verifies the inequality , the following Bernstein type quadrature formula holds, for any k = 1, m, where Proof.Integrating (2.4) on k−1 m , k m , k = 1, m, and taking (2.3) into account, yields The last integral is the Euler function of first kind B(i + 1, n − i + 1).Using the well known properties of Euler function of first kind, it follows For the remainder term, taking (2.5) into account, we get (2.9) and from (2.9) one arrives to the desired inequality (2.8).
Theorem 2.4.For any f ∈ C 2 [0, 1], the following composite Bernstein type quadrature formula (2.10) holds, where Adding the Bernstein type quadrature formulas (2.6) for k = 1, m, we get the following composite Bernstein type quadrature formula (2.12) Remark 2.5.It is easy to see that we get the same result for the remainder term of the composite Bernstein type quadrature formula as the result obtained by D. D. Stancu in [7], for the Bernstein quadrature formula.

Lemma 2 . 1 . 1 (Lemma 2 . 2 .
Suppose that a, b ∈ R, a < b and f ∈ C[a, b].Then, the Bernstein polynomial associated to the function f is defined by(2.1)(B n f )(x) = − a) j (b − x) n−j f a + j b−a n .Proof.It is easy to observe that the correspondence t → x−a b−a transform the interval [a, b] in the interval [0, 1].Taking (1.1), (1.2) and the above remark into account, yields (2.1).Suppose that a, b ∈ R, a < b and f ∈ C 2 [a, b].Then, the remainder term of the Bernstein approximation formula on [a, b] verifies the inequality