Stancu type q-Bernstein operators with shifted knots

In the present paper, Stancu type generalizations of the q-analog of Lupaş Bernstein operators with shifted knots are introduced. Some approximation results and rate of convergence for these operators are investigated. A Voronovskaja type theorem and local approximation results for the mentioned operators are studied. The extra parameters γ, δ, q, a and b provide more flexibility for approximation.


Introduction
Approximation theory basically deals with approximation of functions by simpler functions or more easily calculated functions. Broadly it is divided into theoretical and constructive approximation. In 1912 Bernstein [5] was the first to construct a sequence of positive linear operators as follows: where u ∈ [0, 1], f is bounded on [0, 1]. A constructive proof of the well-known Weierstrass approximation theorem using a probabilistic approach was provided. Here C[0, 1] denotes the set of all continuous functions on [0, 1] which is equipped with the sup-norm · . He showed that if f ∈ C[0, 1], then B m (f ; u) converges to f (u) uniformly on [0, 1]. One can find a detailed monograph about the Bernstein polynomials in [12,13].
Before proceeding, let us recall some basic definitions and notations of quantum calculus [9]. For any fixed real number q > 0 satisfying the conditions 0 < q ≤ 1, the q-integer [k] q , for k ∈ N is defined as (1-q) , q = 1, k, q = 1 © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. and the q-factorial by The q-binomial expansion is (u + y) m q := (u + y)(u + qy) u + q 2 y · · · u + q m-1 y , and the q-binomial coefficients are as follows: From the above Gauss-formula is defined as After development of q-calculus, Lupaş [14] in 1987 introduced the q-Lupaş operator (rational) as follows: and studied its approximation properties. Similarly, Phillips [23] in 1996 constructed another q-analog of Bernstein operators (polynomials) as follows: Bases of these operators have been used in computer aided geometric design (CAGD) to study curves and surfaces. From then onward it became an active area of research in approximation theory as well as CAGD. In the recent past, q-analogs of various operators were investigated by several researchers (see [6,15,19,22,24]). Also see [1, 2, 4, 10, 11, 16-18, 21, 25] for other modifications.
Motivated by the above work, in the next section we present a Stancu type modification of Lupaş q-Bernstein shifted operators and will study its approximation properties.

Definitions and auxiliary results
(f ; u) be given by (2.1). Then the following properties hold: Proof (1) By using definition of q-binomial coefficients and Gauss-formula, we have after calculating the values of A, B and C we get which is the required result.
(4) For f (t) = t 3 , we have After calculating the values of D, E, F and G, we get (5) For f (t) = t 4 , we have After calculating the values of H, I, J, K and L, we obtain

Lemma 3.2 By using the linearity of operators S
[m] q +b + u 4 . We can easily see that S

Main results
Firstly, we prove some theorems on the convergence of S  Consider the sequence of operators [m] qm +δ , 1].
Then obviously and using (4.1) we obtain Now, by applying the Korovkin theorem [12] (see also [3]) to the sequence of positive linear operators S * m , we obtain for every continuous function f . Therefore Proof For any u, y ∈ [a, b], it is well known that Therefore, we get Thus, we obtain the desired result. Proof By the Taylor formula we may write where r(t, u) is the remainder term and lim t→u r(t, u) = 0. Applying S By the Cauchy-Schwartz inequality, we have Observe that r 2 (u, u) = 0 and r 2 (·, u) ∈ C This completes the proof of the theorem.

Local approximation
If σ > 0 and W 2 = {s ∈ C[0, 1]; s , s ∈ C[0, 1]}, then the K -functional is defined as where the second order modulus of smoothness for f ∈ C[0, 1] is defined as The usual modulus of continuity for f ∈ C[0, 1] is defined as Our next main result is the following local approximation theorem.
Proof We definē which implies that