ON COMPOUND OPERATORS CONSTRUCTED WITH BINOMIAL AND SHEFFER SEQUENCES

. In this note we consider a general compound approximation operator using binomial sequences and we give a representation for its corresponding remainder term. We also introduce a more general compound approximation operator using Sheﬀer sequences. We provide convergence theorems for both studied operators.


INTRODUCTION
A sequence of polynomials (p m (x)) m≥0 is called a sequence of binomial type if deg p m = m, ∀m ∈ N, and satisfies the identities We will denote by E a the shift operator defined by (E a p) (x) = p (x + a), for every polynomial p and every real number x.
A linear operator T is said to be shift invariant if it commutes with the shift operator E a , for every real number a.
Sequences of binomial type are connected with the notion of theta operators (J.F. Steffensen [27], [28]) which were called delta operators by F. B. Hildebrand [7] and G.-C. Rota and his collaborators [15].
A delta operator Q is a shift invariant operator for which Qx = const.= 0.
Definition 1.Let Q be a delta operator.A sequence (p m (x)) m≥0 is a sequence of basic polynomials for Q (basic sequence, for short) if: For every delta operator there exists a unique basic sequence.A polynomial sequence is a sequence of binomial type if and only if it is the sequence of basic polynomials for a delta operator.
Sequences of binomial type were called poweroids by Steffensen [27], because the action of any delta operator on the binomial sequence, which is its basic sequence, is the same as the action of the derivative D on x m .
It is known [15] that if (s m (x)) m≥0 is a Sheffer sequence for a delta operator Q with the basic sequence (p m (x)) m≥0 then there exists a shift invariant and invertible operator S such that s m = S −1 p m , ∀m ∈ N, so every pair (Q, S) gives us a unique Sheffer sequence.
A Sheffer sequence satisfies the relations (1) A Sheffer sequence for the usual derivative D is an Appell sequence.
The Umbral Calculus allows a unified and simple study of binomial, Appell and Sheffer sequences.More details about Umbral Calculus can be found in [15], [5] and [6].
T. Popoviciu proposed in [14] the use of binomial sequences in order to construct a class of approximation operators of the form This kind of operators and their generalizations were intensively studied.They interpolate the function f at 0 and 1 and preserve the polynomials of degree one.The expressions for T Q m e n , n ≥ 2, were computed by C. Manole (see [10] and [11]) using the umbral calculus and later by P. Sablonnière using the generating function for the binomial sequences (see [16]).We mention that Sablonnière called them Bernstein-Sheffer operators while D. D. Stancu called them binomial operators of Tiberiu Popoviciu type.
Different results regarding the operator T Q m were obtained by several authors: D. D. Stancu and M. R. Occorsio found representations for the remainder in the approximation formula [24]; V. Miheşan proved that T Q m preserve the Lipschitz constant for a Lipschitz function [12]; D. D. Stancu and A. Vernescu studied bivariate operators of this type [26]; O. Agratini considered a generalization of T Q m in the Kantorovich sense [1]; L. Lupaş and A. Lupaş introduced and studied a modified operator of binomial type replacing x by mx and 1 by m [9], [8].More details about the role of the binomial polynomials in the Approximation Theory can be found in [2], [8] and [24].

COMPOUND POWEROID OPERATORS
Let Q be a delta operator with the basic sequence (p m (x)) m≥0 .If p m (1) = 0, ∀m ∈ N, for every function f ∈ C [0, 1] we consider the general approximation operator defined by , while s and r are two nonnegative integers satisfying the condition 2sr ≤ m. If Different instances of this compound poweroid operator were previously studied by D. D. Stancu and his collaborators as follows: was introduced and studied by D. D. Stancu (see [18]); if s is arbitrary, the operator S D m,r,s is a special case of the operator L α,β m,r 1 ,...,rs , considered by D. D. Stancu in [19]  α] was studied by D. D. Stancu and J. W. Drane in [23]; 3. D. D. Stancu and A. C. Simoncelli studied in [25] the compound poweroid operator for Q m,r,s f ) converges uniformly to f on the interval [0, 1] .Using the Peano theorem, the authors also gave a representation of the remainder R α,β m,r,s f for the approximation formula Stancu considered also a class of linear positive compound operators S α,β,γ,δ m,r,s f (see [22]) with modified knots defined by the following relation If s = 0 or r = 0 then S Q m,r,0 and S Q m,0,s reduce to the binomial operator of T. Popoviciu T Q m , defined by (2).From the definition of a basic sequence it results that Using these relations we obtain that the polynomial Lemma 3. The values of the operator S Q m,r,s for the test functions are where pm (1) .
Proof.From the definition of a sequence of binomial type we have that Hence the operator S Q m,r,s preserves the polynomials of degree one.Analogously, we obtain that Using the expression found by C. Manole in [11] for , we obtain .
From the relations (4) it results that (S Q m,r,s e 2 )(x) converges to e 2 (x) if s,r → 0, so using the Bohman-Korokvin criterion of convergence we have the following result.Taking into account an inequality proved by O. Shisha and B. Mond (see [17]), we can write Using the expressions obtained for S Q m,r,s e i , for i = 0, 1, 2, we obtain that Example 1.If we consider the delta operator T = ln (I + D), its basic sequence is the sequence of exponential polynomials: where S (m, k) = [0, 1, . . ., k; e m ] are the Stirling numbers of second kind.In this case, Manole obtained ϕm( 1) , and he proved that there exist two positive constants c 1 and c 2 such that Hence, ϕ m−1 (1)  ϕm(1) → 0, as m → ∞, which implies d T m → 0. Consequently, S T m,r,s f defined by f k+jr m converges to the function f , uniformly on the interval [0, 1] .

EVALUATION OF THE REMAINDER
Using Peano's theorem, the remainder of the approximation formula ( 5) where G Q m,r,s (t; x) is the Peano kernel defined by is negative, one can apply the mean value theorem to the integral from ( 6) and we obtain that there exists ξ ∈ [0, 1] such that Taking f (x) = x 2 in the previous relation, we obtain that , so it follows that, for every function f ∈ C 2 [0, 1], the remainder in formula ( 5) is of the following form:

COMPOUND SHEFFER OPERATORS
Let Q be a delta operator with the basic sequence (p m (x)), S a shift invariant and invertible operator and s m = S −1 p m a Sheffer sequence.We can also generalize the operator defined in (3), by considering another compound approximation operator containing a Sheffer sequence (additionally to the basic sequence p m ): sn (1) .When S = I this operator reduces to the operator defined by (3).
For s = 0, S Q,S m,r,0 is in fact the operator that we studied in our paper [3], We remind that for this operator we have obtained the following expressions for the test functions , where (1)   sm (1) , .
If p n (0) ≥ 0 and s n (0) ≥ 0, ∀n ∈ N, then w Q,S n,k (x) ≥ 0, ∀x ∈ [0, 1] (see [3]) and so the operator S Q,S m,r,s f is a positive approximation operator.For the operator (7) we have S Q,S m,r,s f (0) = f (0).In the following we compute the values of the operator S Q,S m,r,s for the test functions.
From the convolution-type relation (1) satisfied by a Sheffer sequence, it is obvious that S Q,S m,r,s e 0 = e 0 .For e 1 we have m−sr e 0 (x), and using the relations (8) we obtain that S Q,S m,r,s e 1 (x) = x (m−sr)a m−sr +rsas m .
Finally, for e 2 we have Using again the relations (8) we can rewrite the last expression as In [3] we proved that if L Q,S m is a positive operator then So, if lim m→∞ a m = lim m→∞ b m = 1, then lim m→∞ c m = 0. Taking into account the previous relations and applying the Bohman-Korokvin criterion convergence we can state the following result.It can be easily proved that, in the same conditions, the operator S G,S m,r,s converges also uniformly to f on [0, 1] .

Theorem 4 .
Let Q be a delta operator with the basic sequence (p m (x)) m≥0 , p m (1) = 0 and p m (0) ≥ 0, for every positive integer m.If d Q m → 0 then the sequence of linear and positive operators S Q m,r,s f converges to the function f, uniformly on the interval [0, 1].Now we establish an estimate for the order of approximation of a function f ∈ C [0, 1] by means of the operator S Q m,r,s using the first modulus of continuity.