ON COMPOUND OPERATORS DEPENDING ON s PARAMETERS

. In this note we introduce a compound operator depending on s parameters using binomial sequences. We compute the values of this operator on the test functions, we give a convergence theorem and a representation of the remainder in the corresponding approximation formula. We also mention some special cases of this operator.


INTRODUCTION
In this note we introduce a compound operator using polynomial sequences of binomial type.We begin by defining these sequences and their link with delta operators.for every real numbers x and y and every positive integer m.
In the following we will consider linear operators defined on the algebra of polynomials.
A linear operator T is a shift invariant operator if E a T = T E a , for every a, where E a is the shift operator defined by E a p (x) = p (x + a) .
A linear operator Q is called a delta operator if Q is shift invariant and Qx = const.= 0. Some examples of delta operators are: the derivative D, the forward and backward difference operators ∇ α = E α − I and ∆ α = I − E −α , the Touchard operator T = ln ( We say that a sequence of polynomials (p m (x)) m≥0 is the basic sequence for the delta operator It is known that every delta operator has a unique basic sequence (see [19]).
Proposition 3. [19].If (p m (x)) m≥0 is a basic sequence for a delta operator then it is a sequence of binomial type; if (p m (x)) m≥0 is a sequence of binomial type then there exists a delta operator for which (p m (x)) m≥0 is the basic sequence.Definition 4. If T is a linear operator, then its Pincherle derivative T is defined by T = T X −XT, where the linear operator X is defined by (Xp) (x) = xp (x) for all x and all polynomials p.

COMPOUND OPERATORS DEPENDING ON S PARAMETERS
Let Q be a delta operator with the basic sequence (p k (x)) k≥0 , which satisfy p m (1) = 0 and p m (0) ≥ 0 for every positive integer m.For every function f ∈ C [0, 1] we introduce the compound operator (2) F r 1 ,...,rs m,k,j (f ) , If p m (0) ≥ 0 for every positive integer m then p m (x) ≥ 0, ∀x ∈ [0, 1] so this condition assures the positivity of the operator L Q m,r 1 ,...,rs f (x) .
From Definition 2 ii), it results that ..,rs f (0) contains only a nonzero term, for k = j = 0, while the only nonzero term in L Q m,r 1 ,...,rs f (1) appears for k = m−r 1 −...−r s and j = s.Consequently, it is easy to see that this approximation operator interpolates the function f at both ends of the interval [0, 1] , that is We remark that for s = 0 the operator L Q m,r 1 ,...,rs reduces to the binomial operator of T. Popoviciu T Q m .In the following we will compute the values of this operator for the test functions e n (x) = x n , for n = 0, 1, 2. For this we need Manole's results contained in the next Proposition 5. [13], [14].The values of the binomial operators of T. Popoviciu type on the test functions are: for i = 0, 1 and (3) where and Q is the Pincherle derivative of delta operator Q.
..,rs is the approximation operator defined by (2) then we have the following relations L Q m,r 1 ,...,rs e i = e i , for i = 0, 1 and where Proof.First we make the convention that s j = 0, if s < 0 or j < 0.
Because (p m (x)) is a basic sequence for the delta operator Q according to Proposition 3 it is a polynomial sequence of binomial type and using Definition 2 we have m k=0 p Q m,k (x) = 1 so we can write In the case of the next test function e 1 we have Finally, for e 2 we can write If we use the relations (3) we can rewrite the last expression as After some simple computations we obtain the expression from the conclusion of lemma.
Using the well known theorem of Bohman-Korovkin and the expressions obtained in the above lemma for L Q m,r 1 ,...,rs e i , i = 0, 1, 2, we can state the following convergence theorem Let Q be a delta operator having the basic sequence p m (x) with p m (1) = 0 and p m (0) ≥ 0 for every positive integer m.

SPECIAL CASES
1.If r 1 = ... = r s = r the compound operator defined by (2) reduces to the operator which we have studied in [7] (6) Here we have r j (r j − 1) .
2.1.For s = 1 the above operator reduces to the following operator which was constructed by D.D. Stancu in [26] using a probabilistic approach.
The above mentioned author have found the eigenvalues for this operator We mention also that D. D. Stancu in [25] obtained a quadrature formula using this operator where, if we suppose that f ∈ C 2 [0, 1] , the remainder has the following simple form For f ∈ C (s+1) [0, 1] O. Agratini gave an estimate for the difference in which appears the first modulus of continuity ω 1 for the derivatives of order s and s + 1 of f (see [1]).The bivariate analogue of the operator defined by (7), having as domain the square [0, 1] × [0, 1] was studied by D.D. Stancu in [28].In the same paper a cubature formula (using this operator) was constructed.2.2.The operator obtained for s = 1 and r = 2, L D m,2 has been studied by H. Brass [4].x Taking into account that d α α m = 1+αm (1+α)m , we obtain the following expression for this operator on e 2 , L α α m,r 1 ,...,rs e 2 (x) = .4. For Q arbitrary and s = 1 the operator defined by (2) reduces to the operator

AN INTEGRAL REPRESENTATION FOR THE REMAINDER
We consider the following approximation formula ( 9) From Lemma 6 it results that the degree of exactness of this formula is 1.
Because for a fixed value of x, G Q m,r 1 ,...,rs (t; x) is negative we can apply the mean value theorem and we obtain that it exists ξ ∈ [0, 1] such that R Q m,r 1 ,...,rs f (x) = f (ξ) G Q m,r 1 ,...,rs (t; x) dt.

Definition 1 .
A sequence of polynomials (p m (x)) m≥0 is called a sequence of binomial type if deg p m = m, ∀m ∈ N and it satisfies the relations p m (x + y) = m k=0 m k p k (x) p m−k (y) one obtains the operator introduced and studied by D.D. Stancu in[27]