Genuine q -Stancu-Bernstein–Durrmeyer Operators

: In the present paper, we introduce the genuine q -Stancu-Bernstein–Durrmeyer operators Z q , α n ( f ; x ) . We calculate the moments of these operators, Z q , α n ( t j ; x ) for j = 0,1,2, which follows a symmetric pattern. We also calculate the second order central moment Z q , α n (( t − x ) 2 ; x ) . We give a Korovkin-type theorem; we estimate the rate of convergence for continuous functions. Furthermore, we prove a local approximation theorem in terms of second modulus of continuity; we obtain a local direct estimate for the genuine q -Stancu-Bernstein–Durrmeyer operators in terms of Lipschitz-type maximal function of order β and we prove a direct global approximation theorem by using the Ditzian-Totik modulus of second order.


Introduction
A sequence of positive linear operators S (α) n : C[0, 1] → C[0, 1] was introduced and studied by a Romanian mathematician D.D. Stancu [1] in 1968.The operators depend on a non-negative parameter α and are defined as follows: n,k (x) is the Polya distribution with density function given by p (α) If α = 0, the operators S (α) n reduce to the well-known classical Bernstein polynomials.A. Lupaş and L. Lupaş [2] studied on a particular case for Stancu's operators by setting α = 1 n and obtained where (y) n = y(y + 1)(y + 2) . . .(y + n − 1) is the rising factorial with (y) 0 = 1.
For any function f ∈ C[0, 1], α ≥ 0, q > 0 and each n ∈ N, modification of the Stancu's operators based on the q-integers is introduced by G. Nowak [3] in 2009 as follows: where G. Nowak studied the Korovkin type approximation properties for the q-analogue of the Stancu's operators B q,α n ( f , x).If α = 0, B q,α n ( f , x) reduces to the q-Bernstein polynomials defined by GM.Phillips in [4] as [k] q [n] q p nk (q; x), where p n,k (q, x) (1 − q s x).
For q → 1−, they reduce to Bernstein-Stancu operators.For α = 0 and q → 1−, they reduce to the classical Bernstein polynomials.
On the other hand, integral modification of classical Bernstein polynomials was introduced by J.L. Durrmeyer ([5]) in 1967.This type of modification was studied in several forms by different authors (see [6][7][8][9] and etc.).Classical genuine Bernstein-Durrmeyer operators were independently introduced by W. Chen ([10]) in 1987, and by T. N. T. Goodman & A. Sharma ( [11]) later in 1991 and these operators were investigated by many authors, see for example [12,13].They possess many interesting properties, in particular they reproduce linear functions and thus interpolate every function f ∈ C[0, 1] at x = 0 and x = 1.V. Gupta ([14]) introduced the q-analogue of the Bernstein-Durrmeyer operators and studied some approximation properties for these operators.
In this paper, we introduce genuine q-Stancu-Bernstein-Durrmeyer Operators Z q,α n in detail.We calculate the moments of these operators Z q,α n (t j ; x) for j = 0, 1, 2 and observe that they follow a symmetrical pattern.We calculate the second order central moment Z q,α n ((t − x) 2 ; x).We examine convergence properties of the operators, estimate the rate of convergence and we prove a Korovkin-type theorem.We also present a local approximation theorem by using the second order modulus of smoothness, we provide a local direct estimate in terms of Lipschitz type maximal function of order β and we offer a global approximation theorem by using Ditzian-Totik modulus of second order.Compared to the previous generalizations of this type of operator, the advantage of genuine q-Stancu-Bernstein-Durrmeyer operators is that they reproduce linear functions and they interpolate every function First of all, we provide some notations and definitions of q-integers: Let 0 < q < 1.For any n ∈ N ∪ {0}, the q-integer [n] q is defined by [n] q := 1 + q + . . .+ q n−1 , [0] q := 0; and the q-factorial [n] q ! by For integers 0 ≤ k ≤ n, the q-binomial is defined by The q-analogue of integration in the interval [0, A] (see [28]) is defined by For more information on q-calculus one can see [28].
The paper is organized as follows: In Section 2, we calculate the moments of these operators Z q,α n (t j ; x) for j = 0, 1, 2 and the second order central moment Z q,α n ((t − x) 2 ; x), we show that Z q,α n (t m ; x) is a polynomial of degree less than or equal to min(m, n) and we prove that for f ∈ C[0, 1] and 0 < q < 1, Z q,α n ( f ; x) = f (x) for all x ∈ [0, 1] if and only if f is linear.In Section 3, we examine convergence properties of the operators.In Section 4, we give an estimation for the rate of convergence, we prove a local approximation theorem by using the second order modulus of smoothness, we obtain a local direct estimate in terms of Lipschitz type maximal function of order β and we prove a global approximation theorem by using second order Ditzian-Totik modulus.In Section 5, we present conclusions and suggest further studies.

Operators and Estimation of Their Moments
Definition 1.For f ∈ C[0, 1], α ≥ 0 and 0 < q < 1, we define the following genuine q-Stancu-Bernstein-Durrmeyer Operators: where for n = 1 the sum is empty, i.e., equal to 0, Moments and central moments play an important role in approximation theory.In the following lemma, we obtain explicit formulas for Z q,α n (t m ; x) for m = 0, 1, 2 and Z q,α n Proof.By using the definition of q-Beta function (see [28]), we have the following equalities for r = 0, 1, . . .
We are going to use the following identities for the proof of the theorem (see [3]): Using Definition 1 and equality (1) and by using the above identities, we have Now, for the proof of Z q,α n (t − x) 2 ; x , we use Z q,α n t 2 ; x and the linearity of the operators as follows: Lemma is proved.
Lemma 2. Z q,α n (t m ; x) is a polynomial of degree less than or equal to min(m, n).
Proof.By using simple calculations, we obtain Now using the identity that where c j (m) > 0, j = 1, 2, . . ., m, are the constants independent of k, we get From [3], we know that B α n,q t j ; x is a polynomial of degree less than or equal to min(j, n) and c j (m) > 0, j = 1, 2, . . ., m.Thus it follows that Z q,α n (t m ; x) is a polynomial of degree less than or equal to min(m, n).Theorem 1.Let f ∈ C[0, 1] and 0 < q < 1.Then Z q,α n ( f ; x) = f (x) for all x ∈ [0, 1] if and only if f is linear.
Proof.From Theorem 9 of [29], we know that for a positive linear operator D on C[0, 1] which reproduces linear functions, if D(t 2 , x) > x 2 for all x ∈ (0, 1), then D( f ) = f if and only if f is linear.Now, since Z q,α n is a positive linear operator on C[0, 1] which reproduces linear functions, it sufficies to show that Z q,α n t 2 ; x > x 2 for all x ∈ (0, 1).Let h(x) = Z q,α n t 2 ; x − x 2 .One can easily see that h(0) = h(1) = 0. On the other hand we have, Now since the function h is concave down on (0, 1) and h(0) = h(1) = 0, we conclude that h(x) > 0 for all x ∈ (0, 1) which implies that Z q,α n t 2 ; x > x 2 for all x ∈ (0, 1).This completes the proof of the theorem.

Convergence of Genuine q-Stancu-Bernstein-Durrmeyer Operators
Theorem 2. Let (q n ) and (α n ) be two sequences such that 0 < q n ≤ 1 and α n ≥ 0. Then the sequence Z q n ,α n n ( f ) converges to f uniformly on [0, 1] for each f ∈ C[0, 1] if and only if lim n→∞ q n = 1 and lim n→∞ α n = 0.
Proof.From the definition of Z q,α n ( f ) and Lemma 1 it follows that the operators Z q n ,α n n are positive linear operators on C[0, 1] and reproduce linear functions.The well-known Korovkin theorem implies that Z q n ,α n n On the other hand, if we assume that for any Consequently Since α n and [n] q n are both nonnegative we get α n → 0 and [n] q n → ∞ (q n → 1), which completes the proof of the theorem.
Proof.From Definition 1 and from Lemma 1, we have

Approximation Properties of q-Stancu-Bernstein-Durrmeyer Operators
We consider the following K-functional: where the space C 2 [0, 1] is defined as Then, from the well known result in [30], there exists an absolute constant where In the following theorem, we state our first main result for this section.We prove a local approximation theorem for the genuine q-Stancu-Bernstein-Durrmeyer Operators Z q,α n .Theorem 3.There exists an absolute constant L > 0 such that Proof.Using the Taylor formula we obtain that It is obvious that Taking the infimum on the right hand side over all h ∈ C 2 [0, 1], we obtain Now the desired inequality follows from ( 3) and ( 4).
We know that a function f ∈ C[0, 1] belongs to the Lipschitz type space Lip M (β), (M > 0 and 0 < β ≤ 1), provided that Then we have the following result.Then, for all f ∈ Lip M (β), n ∈ N, α ≥ 0 and 0 < q < 1, we have where M is a constant depending on β and f .
In the next theorem, we give the direct global approximation theorem for the genuine q-Stancu-Bernstein-Durrmeyer Operators Z q,α n .Before stating the theorem we recall the weighted K-functional of second order for f ∈ C[0, 1], it is defined by and g ∈ AC loc [0, 1] means that g is differentiable and g is absolutely continuous in every closed interval [a, b] ⊂ [0, 1].Furthermore, the Ditzian-Totik modulus of second order is given by It is well known that K-functional K 2,φ f , δ 2 and Ditzian-Totik modulus ω φ 2 ( f , δ) are equivalent, see [30].
Proof.From the Taylor expansion, we can write s)dsdp.
Now, applying Z q,α n to both sides of the last equation we get .
Now, if we take the infimum on the right hand side over all h with h ∈ AC loc [0, 1] we obtain Finally, from the fact that K 2,φ ( f , δ 2 ) and ω 2 φ ( f , δ) are equivalent we obtain the de- sired result.

Conclusions
Inspired from the previous studies, our goal was to construct a generalization of Stancu-Bernstein-Durrmeyer type operators based on the q-integers which reproduces linear functions and interpolates every function f ∈ C[0, 1] at x = 0 and x = 1.
In this paper, by using the q-analogue of integers, we defined the genuine q-Stancu-Bernstein-Durrmeyer Operators Z q,α n ( f ; x).We gave explicit formulas for the moments of these operators Z q,α n (t j ; x) for j = 0, 1, 2 and for the second order central moment Z q,α n ((t − x) 2 ; x).We proved a Korovkin-type theorem and we provided an estimation of the rate of convergence for these operators.We proved a local approximation theorem by using the second order modulus of smoothness, we obtained a local direct estimate in terms of Lipschitz-type maximal function of order β and we proved a global approximation theorem by using Ditzian-Totik modulus of second order.The newly defined genuine q-Stancu-Bernstein-Durrmeyer operators have some advantages compared to the similar generalizations of this type of operator which was defined before.The advantage of genuine q-Stancu-Bernstein-Durrmeyer operators is that they preserve linear functions, i.e., Z q,α n (e 0 ; x) = e 0 (x) and Z q,α n (e 1 ; x) = e 1 (x) for e i (x) = x i (i = 0, 1) and furthermore, they interpolate every function f ∈ C[0, 1] at x = 0 and x = 1, i.e., Z q,α n ( f ; 0) = f (0) and Z q,α n ( f ; 1) = f (1).As a future work we would like to construct genuine (p, q)-Stancu-Bernstein-Durrmeyer operators Z p,q,α n ( f ; x) by using the theory of post quantum calculus and examine their approximation properties.