A note on the remainder in the approximation of functions by some positive linear operators

In this note we give representations for the remainder in approximations formulas generated by positive linear operators which preserve some functions including linear functions. We show that in these cases the Peano kernel from the integral representation of the remainder has the same sign on the definition domain. Applications to the iterates of some positive linear operators and quadrature formulas are given.


Introduction
The classical Bernstein operator B n : C[0, 1] → C[0, 1] is given by In [29] and [2] Stancu and L. Arama respectively obtained the following representation for the remainder term in Bernstein approximation: They also gave an integral representation of the error in the form: and showed that the (Peano) kernel has the same sign on the definition domain, i.e.
In [4] the author obtained the following formula for the error in Bernstein approximation using divided differences.
, by using the mean theorem for divided differences we get (3).
Extensions of these results for some generalized Bernstein-type operators were obtained by Leviatan in [20] (see also Arama and Arama [3] and Lupas, Muller [21] for some particular cases).
In [32] Waldron showed that an integral representation of the form (2) holds for the all operators The author used the quotient theorem to avoid the condition of interchanging between the remainder functional and the integral which is needed when we use the classical Peano theorem.On the other hand, the problem of keeping the sign for the Peano kernel in the remainder term of some approximation formulas was also considered in [13].
Throughout the paper we use the following notations: In Sect. 2 we prove Theorems 1 and 2 for all positive linear operators L : C[a, b] → C[a, b] which preserve linear functions.We show that the Peano kernel has the same sign on the definition domain.These results are extended to positive linear operators which reproduce constant functions and the function τ .Applications to the iterates of these operators and the quadrature formulas are also given in Sects.3 and 4 respectively.

Theorem 3 Let
Proof Let x ∈ [a, b] fixed and the remainder functional From [32, Theorem 3.3] we have where the Peano kernel is given by The function Theorem 2] we get i.e. the Peano kernel has the same sign for every t ∈ [a, b].Using the mean theorem from [18] it follows that there exists Taking f = e 2 we obtain From ( 7) and ( 8) we get the conclusion.
For proving the Theorem 6 we need the following two results.
Theorem 4 [14,28] Theorem 5 [14,25] Proof For every x ∈ [a, b] which is not an interpolation point of the operator L we define the functional R + x by As the operator L preserves linear functions we have x (e 2 ) = 0, then by using the Theorem 4 it follows that x is an interpolation point of the operator L. This is contradiction and therefore R + x (e 2 ) > 0. Thus the conditions from Theorem 5 are satisfied for the functional R + x .It follows that there exist three distinct points and therefore the formula (9) holds.
Next, we extend the results from Theorems 3 and 6 for the positive linear operators which preserve constant functions and the function τ .

Theorem 7 Let L : C[a, b] → C[a, b] a positive linear operator preserving the functions e
or equivalent, there exists Proof We have where The operator L preserves linear functions.Indeed, L is linear and for every x ∈ [a, b] we have and Using the Theorem 3 for the operator L given by ( 13) and taking f := f • τ −1 and x := τ (x) we get (10 ).
The proof is completed.

Remark 1
The operator L from ( 12) with L = B n , where B n is the classical Bernstein operator was considered in [6].

Applications to the iterates of positive linear operators
Throughout this section we assume that the operator Proof Using Theorem 9 with L := P τ we have that for every x ∈ [a, b] there exist the distinct points Acting k times by the operator L on (16) we get the conclusion.
The following definition can be found in [6, p. 160] (see also [19] for more about generalized convexity).
We observe that the case of the classical convexity is obtained for τ = e 1 .

Corollary 12
] is a convex functions with respect to the function τ, then we have Proof As f is a convex functions with respect to the function τ, by using [34, Theorem 2] we have that From Theorem 10 we get The conclusion follows from ( 17) and (18).
A particular case of the result from Corollary 12 was obtained in [33, Theorem 5.16].

Remark 3
From Theorem 10 we get the following condition for the convergence of the iterates of the operator L: This criterion was also presented in [12,Theorem 2] for τ = e 1 and interval [0, 1] (see also [15,16,26,27] for other criterions).

Theorem 13 For every f ∈ C[a, b] we have the estimation
where m, M are given by Theorem 10.
Proof From Definition 1 it follows that τ 2 is convex with respect to τ.
] a positive linear operator which preserves constants and the function τ.From [34, Theorem 2] we have that τ 2 ≤ Q(τ 2 ).Using Theorem 9 it follows that Taking Q := L k in (19) we get the conclusion.
For simplicity the next result is proved on the interval [0, 1].

Theorem 14
If there exists a constant c ∈ (0, 1] such that the operator L satisfies then for every f ∈ C[0, 1] we get the following sharp estimations: where m, M are given by Theorem 10. Proof From (20) it follows by induction that Using Theorems 10 and 13 we get ( 21) and ( 22) respectively.

Remark 5
The constants c k and 1−c k in ( 21) and ( 22) respectively are the best possible.Indeed, taking f = τ 2 we get m = M = 1.
We have We obtain the sharp inequalities ( 21) and (22) with

The remainder term of quadratures generates by positive linear operators
Let the positive linear operator where where the Peano kernel is given by By integration of (26) on the interval [a, b] we get the quadrature formula where For the remainder term we have As K n (x, t) ≤ 0, x, t ∈ [a, b] then, by applying the mean theorem from [18] it follows that there exists ξ ∈ (a, b) such that Using (15) we get Remark 6 The Peano kernel from the remainder term of the quadrature keeps the same sign on the interval [a,b]: If we integrate an approximation formula generated by a positive linear operator which preserves only constant functions we can obtain a quadrature formula with the degree of exactness greater than zero (see for example the Fejer-Hermite quadrature formula from [31, p. 78]).We show that by using a positive linear operator which fixes linear functions the degree of exactness of the quadrature formula (27) can not be increased.[a, b] given by (25) preserves the linear function, then the degree of exactness of quadrature ( 27) is one.

Theorem 15 If the positive linear operator L
Proof From (28) we have that Using [34,Theorem 2] we have e 2 ≤ L n (e 2 ).As e 2 = L n (e 2 ) and the function e 2 − L n (e 2 ) is continuous we get R n (e 2 ) < 0. The proof is ended.
Next, we give two examples of quadratures formula with degree of exactness one.

Example 3
We consider the quadrature formula generated by the q-Stancu operator given by ( 23) with τ = e 1 .We have with where w q,α n,i , i = 0, . . ., n are given by ( 24) for α ≥ 0 and q ∈ (0, 1].If f ∈ C 2 [0, 1], then using (28) we get The formula (29) has the degree of exactness one.For n = 2 the quadrature formula is If α = 0 and q = 1, then we get the Bernstein quadrature formula (see [31]) (for i = 0 and i = n the first and the second equation respectively are missed).We also have, for every i = 0, 1, . . ., n − 1, The linear spline operator is linear, positive and it reproduces the affine functions.The quadrature generated by this operator is The quadrature formula (32) with remainder term given by (33) on the interval [0, 1] was also obtained in [9] using a different method.
Author contributions Not applicable.
Funding Not applicable.

Theorem 8
Let L : C[a, b] → C[a, b] a positive linear operator preserving the functions e 0 and τ.If f , τ ∈ C 2 [a, b], then we have

Remark 2 Theorem 9
From (14) we have that the Peano kernel has the same sign on the definition domain.Let L : C[a, b] → C[a, b] a positive linear operator preserving the functions e 0 and τ.If f ∈ C[a, b], then for every x ∈ [a, b] there exist three distinct points ξ 1 is a positive linear operator which preserves constants and the function τ .Theorem 10 For every f ∈ C[a, b] we have the estimation m

) Example 4
Let the distinct nodes belonging to the interval[a, b], a = x 0 < x 1 < ... < x n = b.The linear spline operators S n : C[a, b] → C[a, b] is defined by S n ( f ) = n i=0 b i f (x i ),where b i , i = 0, 1, . . ., n are the B-spline functions b
n. are distinct nodes.We assume that the operator L n preserves the constant functions and the function τ .If f , τ ∈ C 2 [a, b], then from Theorem 8 we have the formula