ON THE APPROXIMATION OF FUNCTIONS BY MEANS OF THE OPERATORS OF BINOMIAL TYPE OF TIBERIU POPOVICIU

. In 1931, Tiberiu Popoviciu has initiated a procedure for the construction of sequences of linear positive operators of approximation. By using the theory of polynomials of binomial type ( p m ) he has associated to a function f ∈ C [0 , 1] a linear operator deﬁned by the formula

Examples of such operators were considered in several subsequent papers.
In this paper we present a convergence theorem corresponding to the sequence (Tmf ) and we also present a more general sequence of operators of approximation Sm,r,s, where r and s are nonnegative integers such that 2sr ≤ m.
We give an integral expression for the remainders, as well as a representation by using divided differences of second order.MSC 2000.41A36, 41A25, 41A80.

POPOVICIU
It is known that a sequence of polynomials (q m ), where m is a nonnegative integer, is said to be of binomial type if deg q m = m, p 0 ≡ 1 and it obeys the following identities for any nonnegative integer m.
The corresponding sister sequence (p m ), where q m = p m /m!, obeys the identities By starting from the class of polynomials of binomial type (q m ), the great Romanian mathematician Tiberiu Popoviciu had already in 1931 [9] the wonderful idea to indicate a general method for construction linear positive operators, useful in the constructive theory of functions.
It is known that the sequence (q m ) is of binomial type if and only if it is defined by a generating relation x suggested to Tiberiu Popoviciu to introduce an operator of binomial type, which associates to a function f ∈ C[0, 1] the polynomial which can be used for the approximation of the function f .It is easy to see that, in fact, we have a m = q m (1), where we suppose that q m (1) = 0.
Taking this into account and the identity for the sister polynomials (p m ): we can write We call it the operator T m of binomial type of Tiberiu Popoviciu.
If we consider a result of T. Popoviciu [9], rediscovered later by P. Sablonnière [11], we have p ν (x) ≥ 0 on [0, 1] (ν ∈ N), if and only if the coefficients c k from (1.3) are non-negative.In this case the operator T m is of positive type.
B) If we use the binomial polynomials represented by the factorial powers p m (x) = x [m,−α] (α ∈ R + ), then we get the operator S α m , defined by where The operator S α m was introduced in 1968 in our paper [13].It was later investigated in several other papers [1], [2], [4], [7].

CONVERGENCE OF THE SEQUENCE (T m f )
Assuming that all the coefficients c k from (1.3) are non-negative, for the convergence of (T m f ), where f ∈ C[0, 1], we can use the convergence criterion of Bohman-Popoviciu-Korovkin.
According to the identities satisfied by the binomial polynomials, we can see at once that we have: T m e 0 = e 0 .
In the case of the test function f = e 1 , we get If we take into account that It is easy to see that employing the change of index of summation k − 1 = ν and then denoting again the summation variable by k, we get Because the binomial sequence (p m ) is generated by the following expansion by differentiation we obtain It follows that we are able to write We can now use the connection between sequences of polynomials of binomial type and sequences of Sheffer polynomials.
One says that a sequence of polynomials (s m ) m≥0 is a Sheffer sequence for a theta operator θ if we have: a) s 0 (x) = c = 0; b) θs m (x) = ms m−1 (x).Now we notice that a sequence (s m ) is a Sheffer sequence relative to a sequence of binomial type (p m ) if it satisfies the functional equation A sequence of polynomials of Sheffer type is generated by an expansion similar with that connected with the binomial sequence [11], namely Because at (2.2 ) we have an expansion of the form (2.3), corresponding to the functions ϕ and ψ = ϕ , in our case we have the Sheffer sequence s m (x) = p m+1 (x)/x.Imposing to this sequence to satisfy the equation (2.2), we get If we decrease m by one and we choose u = x and v = 1 − x, we find Replacing it in (2.1) we find that T m e 1 = e 1 .This result was found earlier by C. Manole [6] and then by P. Sablonnière [11].
Going on to the test function e 2 , we mention two results.
1) The first one was found by C. Manole [6]: where θ being the Pincherle derivative of the theta operator θ for which (p m ) is a basic sequence.
2) The second result belongs to P. Sablonnière [11]: where p m = p m (1) and r m = r m (1), the sequence (r m ) being generated by the expansion Now we are able to state the following If we consider the approximation formula we can say that this formula has the degree of exactness equal to one.On the other hand, it is easily to check that x = 0 and x = 1 are interpolation points of this polynomial, since (T m f )(0) = f (0), (T m f )(1) = f (1).

AN INTEGRAL REPRESENTATION OF THE REMAINDER
We can establish an integral form for the remainder of the approximation formula (2.6).
and x is a fixed point of the interval [0, 1] then the remainder of the Tiberiu Popoviciu approximation formula (2.6) can be represented under the following integral form where the Peano kernel G m (t; x) is given by the formula Proof.The representation (3.1) can be obtained at once if we apply the well known theorem of Peano.
If we introduce the notation we obtain Concerning the Peano kernel, we can state Theorem 3.2.If we assume that x ∈ s−1 m , s m (1 ≤ s ≤ m), the equation (3.2) permits to write the explicit formula From these relations it is easy to see that on the square Consequently, the equation y = G m (t) = G m (t; x) represents a spline function of degree one, having the knots k m (k = 0, 1, . . ., m).It may actually be shown that G m (t; x) represents the solution of a secondorder differential system, under certain boundary conditions, so that it is the corresponding Green's function.
, then the remainder of the T. Popoviciu approximation formula (2.6) can be represented by the following Cauchy type formula where ξ ∈ (0, 1).
Applying the first law of the mean to the integral (3.1) and replacing in the formula (2.6), we obtain If we substitute here f (x) = e 2 (x) = x 2 , we find that and we obtain formula (3.4).

REPRESENTATION OF THE REMAINDERS BY CONVEX COMBINATION OF SECOND-ORDER DIVIDED DIFFERENCES
Let L m : C[0, 1] → C[0, 1] be a linear positive operator, defined by a formula of the following form where on the interval [0, 1] we have q m,k (x) ≥ 0.
We assume that we have and the formula has the degree of exactness N = 1.
As was shown in 1958 by I.J.Schoenberg, if L m is not the identity operator then we have L m e 2 = e 2 , that is R m e 2 = 0.
In many cases the remainder of the approximation formula (4.2) can be represented under the following form where D m f is a linear functional representing a convex combination of secondorder divided differences of the function f on the point x and two consecutive nodes.

ILLUSTRATIONS
I. In 1964 we have proved that in the case of the Bernstein operator B m we have II.In the case of the linear positive operator S α m , introduced in 1968 in our paper [13], α being a parameter which might depend on m, the remainder of the corresponding approximation formula is Here we have used the notation III.In the case of the operators S m,r,s , defined by the formula One observes that the polynomial S m,r,s f is interpolatory at both sides of the interval [0, 1], that is (S m,r,s f )(0) = f (0), (S m,r,s f )(1) = f (1).
For the remainder of the approximation formula (4.6) one can find (see [15]) the following representation:

1 pm
ON THE APPROXIMATION OF FUNCTIONS BY MEANS OF THE OPERATORS OF BINOMIAL TYPE OF TIBERIU POPOVICIU * DIMITRIE D. STANCU † Dedicated to the memory of Acad.Tiberiu Popoviciu Abstract.In 1931, Tiberiu Popoviciu has initiated a procedure for the construction of sequences of linear positive operators of approximation.By using the theory of polynomials of binomial type (pm) he has associated to a function f ∈ C[0, 1] a linear operator defined by the formula (Tmf ) (x) = k (x)p m−k (1 − x)f k m .

Theorem 2 . 1 .
If f ∈ C[0, 1] and the operator T m of Tiberiu Popoviciu is of positive type, then the sequence of polynomials (T m f ) converges uniformly to the function f on the interval [0, 1] if we have: k (x + α, 1 − x + α) x, k m , k+1 m ; f .

(
S m,r,s f )(x) = m−sr k=0 p m−sr,k (x) s j=0 p s,j (x)f k+jr m , where r and s are nonnegative integer parameters satisfying the condition: 2sr ≤ m, we have the approximation formula (4.6) f (x) = (S m,r,s f )(x) + (T m,r,s f )(x).