Approximation by Generalized Lupa¸s Operators Based on q -Integers

: The purpose of this paper is to introduce q -analogues of generalized Lupa¸s operators, whose construction depends on a continuously differentiable, increasing, and unbounded function ρ . Depending on the selection of q , these operators provide more ﬂexibility in approximation and the convergence is at least as fast as the generalized Lupa¸s operators, while retaining their approximation properties. For these operators, we give weighted approximations, Voronovskaja-type theorems, and quantitative estimates for the local approximation.


Introduction
Approximation theory rudimentarily deals with the approximation of functions by simpler functions or more easily calculated functions. Broadly, it is divided into theoretical and constructive approximation. In 1912, S.N. Bernstein [1] was the first to construct a sequence of positive linear operators to provide a constructive proof of the prominent Weierstrass approximation theorem [2] using a probabilistic approach. One can find a detailed monograph about the Bernstein polynomials in [3,4]. Cárdenas et al. [5] in 2011, defined the Bernstein type operators by B n ( f oτ −1 )oτ and showed that its Korovkin set is e 0 , τ, τ 2 instead of {e 0 , e 1 , e 2 }. These operators present an interesting byproduct sequence of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improves B n in approximating a number of increasing functions, and which, apart from the constant functions, fixes suitable polynomials of a prescribed degree. The notion of convexity with respect to τ plays an important role. Recently, Aral et al. [6] in 2014 defined a similar modification of Szász-Mirakyan type operators obtaining approximation properties of these operators on the interval [0, ∞).
Very recently motivated by the above work,İlarslan et al. [7] introduced a new modification of Lupaş operators [8] using a suitable function ρ, which satisfies the following properties: for m ≥ 1, u ≥ 0, and suitable functions f are defined on [0, ∞). If ρ(u) = u then (1) reduces to the Lupaş operators defined in [8].
Ilarslan et al. [7] gave uniform convergence results on a weighted space, where the weight function is φ(u) = 1 + ρ 2 (u) satisfying the conditions (ρ 1 ) and and (ρ 2 ) given above, in the sense of Gadjiev's results [9,10]. For the rate of convergence, the authors used a weighted modulus of continuity stated by Holhoş in [11] using the weight function. They obtained a Voronovskaya-type result and monotonicity of the sequence of operators L ρ m ( f ; u). Moreover, they obtained some quantitative type theorems on weighted spaces.
The purpose of this paper is to define the q-analogue of operators (1) which depend on ρ.
Before proceeding further, let us recall some basic definitions and notations of quantum calculus [12]. For any fixed real number q > 0, the q-integer [l] q , for l ∈ N (set of natural numbers) are defined as 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: The Gauss-formula is defined as: After development of q-calculus, Lupaş [13] introduced the q-Lupaş operator (rational) as follows: and studied its approximation properties. Similarly, Phillips [14] constructed another q-analogue of Bernstein operators (polynomials) as follows: where B m,q : C[0, 1] → C[0, 1] defined for any m ∈ N and any function f ∈ C[0, 1], where C[0, 1] denotes the set of all continuous functions on [0, 1]. The basis of these operators have been used in Computer Aided Geometric Design (CAGD) to study curves and surfaces. Then it became an active area of research in approximation theory as well as CAGD [15][16][17]. In the recent past, q-analogues of various operators were investigated by several researchers (see [18][19][20][21][22][23]).
The q-analogue is a very interesting idea. It can also be used in statistical and biological physics, multi-type directed scalefree percolation, and modeling epidemic spread with awareness and heterogeneous transmission rates in networks.

Proof. By taking into account the recurrence relation
Now by using [l + 1] q = (1 + q[l] q ) and shifting l to l + 1, we have Now, let us calculate the values of A, B, and C On adding A, B, and C we have, Now, let us calculate the values of D, E, F, and G Similarly, Also, On adding D, E, F, and G we have, , by using linearity of operators (4) and by Lemma 1 we have [m] 2 q ρ 2 (u) + (6q 6 +2q 5 +4q 4 +7q 3 +3q 2 +3q+1) [m] 3 q ρ(u).

Remark 1.
We observe from Lemma 1 and Corollary 1, that for q = 1, we get the moments and central moments of generalized Lupaş operators [7].

Weighted Approximation
We start by noting that ρ not only defines a Korovkin-type set {1, ρ, ρ 2 } but also characterizes growth of the functions that are approximated.
Also, we define the following subspaces of B φ [0, ∞) as Obviously, For the weighted uniform approximation by linear positive operators acting from C φ [0, ∞) to B φ [0, ∞), we state the following results due to Gadjiev in [9,10].
Then we have

Remark 2.
It is clear from Lemma 1 and Lemma 2 that the operators L ρ m,q act from C φ [0, ∞) to B φ [0, ∞). Also the convergence of these operators are applicable in studying switched linear systems, see: Subspace confinement for switched linear systems, also see in [24,25].

Theorem 2.
Let q m be a sequence in (0, 1), such that q m → 1 as m → ∞. Then for each function Proof. By Lemma 1 (i) and (ii), it is clear that and by Lemma 1 (iii), we have Then from Lemma 1 and (5) we get lim Hence, the proof is completed.

Rate of Convergence or Order of Approximation
In this section, we determine the rate of convergence for L ρ m,q by weighted modulus of continuity ω ρ ( f ; δ) which was recently considered by Holhoş [11] as follows: where f ∈ C φ [0, ∞), with the following properties:

Theorem 3 ([11]). Let
where the sequences a m , b m , c m , and d m converge to zero as m → ∞. Then

Theorem 4.
Let for each f ∈ C φ [0, ∞) with 0 < q < 1. Then we have where ω ρ is the weighted modulus of continuity defined in (6) and Proof. By using Lemma 1, we have Finally, Thus, the sequences a m,q , b m,q , c m,q , and d m,q are calculated. The sequences a m , b m , c m , and d m converge to zero as m → ∞. Then for all f ∈ C φ [0, ∞), where δ m,q = 2 (a m,q + 2b m,q + c m,q )(1 + a m,q ) + a m,q + 3b m,q + 3c m,q + d m,q .
Hence, by substituting the values of a m,q , b m,q , c m,q and d m,q we obtain the desired result.

Voronovskaya-Type Theorem
In this section, using a technique developed in [5] by Cardenas-Morales, Garrancho and Raşa, we prove pointwise convergence of L ρ m,q by obtaining Voronovskaya-type theorems.
Proof. By using the q-Taylor expansion of f oρ −1 at ρ(u) ∈ [0, ∞), there exist a point w lying between u and z, then we have Therefore, by (14) together with the assumption on f ensures that and is convergent to zero as w → u. Now applying the operators (4) to the equality (13), we obtain By Lemma 1 and Corollary 1, we get and lim By estimating the last term on the right hand side of equality (15), we will get the proof. Since from (14), for every > 0, lim w→u λ q m u (w) = 0. Let δ > 0 such that |λ q m u (w)| < for every w ≥ 0. Using a Cauchy-Schwartz inequality, we have we obtain lim Thus, by using Equations (16), (17) and (19) to Equation (15) the proof is completed.

Local Approximation
In this section, we present local approximation theorems for the operators L ρ m,q . By C B [0, ∞), we denote the space of real-valued continuous and bounded functions f defined on the interval [0, ∞). The norm · on the space C B [0, ∞) is given by The second order modulus of smoothness is as follows, Let ρ be a function satisfying the conditions (ρ 1 ), (ρ 2 ), and ||ρ || is finite. Then, there exists an absolute constant C > 0 such that Proof. Let s ∈ W 2 and u, z ∈ [0, ∞). Using Taylor's formula we have Using the equality Now, put v = ρ(y) in the last term in equality (21), we get By using Lemma 1 and (23) and applying the operator (4) to the both sides of equality (21), we deduce As we know ρ is strictly increasing on [0, ∞) and with condition (ρ 2 ), we have For Hence we have Taking infimum over all s ∈ W 2 we obtain Now, we recall local approximation in terms of α order Lipschitz-type maximal functions given in [27]. Let ρ be a function satisfying the conditions (ρ 1 ), (ρ 2 ), 0 < α ≤ 1 and Lip M (ρ(u); α), M ≥ 0 is the set of functions f satisfying the inequality Moreover, for a bounded subset E ⊂ [0, ∞), we say that the function f ∈ C B [0, ∞) belongs to where M α, f is a constant depending on α and f .

Conclusions
Here, the q-analogue of the generalized Lupaş operators are constructed. We have investigated convergence properties, order of approximation, Voronovskaja-type results and also quantitative estimates for the local approximation. The constructed operators provide better flexibility in approximating functions and rate of convergence which are dependent on the selection of the function ρ and extra parameter q. These operators also possess interesting properties and depending on the selection of q, can obtain better approximation while q = 1. The basis of these operators can be used to draw curves and surfaces in Computer Aided Geometric Design (CAGD).