* Approximation properties of Kantorovich type q-Balázs-Szabados operators

Approximation theory is an important area of research. Recently, several interesting studies have been conducted (see [1–7]). The statistical approximation properties of the some operators have also been recently investigated by several authors. For example, in [8] Meyer-König and Zeller operators based on q-integers; in [9] q-analogues of Bernstein-Kantrovich operators; in [10] q-Bleimann, Butzer and Hahn operators; in [11] q-Baskokov-Kantrovich operators; in [12] Kantrovich type q-Bernstein operators; in [13] q-Stancu-Beta operators; in [14] Kantorovich-type q-Bernstein-Stancu operators were defined and their statistical approximation properties were investigated. Firstly, we recall some basic definitions used in q-calculus. Details can be found in [15–17]. For any nonnegative integer r, the q-integer of the number r is defined by

Firstly, we recall some basic definitions used in q-calculus. Details can be found in [15][16][17]. For any nonnegative integer r, the q-integer of the number r is defined by where q is a fixed positive real number. The q-factorial is defined by For integers n, r with 0 ≤ r ≤ n, the q-binomial coefficients are defined by For fixed 0 < q < 1, we denote the q-derivative of a function f with respect to x Bernstein-type rational functions are defined by Balázs [18]. Balázs and Szabados modified and studied the approximation properties of these operators [19]. The q-analogue of the Balázs-Szabados operators is defined by Doğru [20] as follows β q for all n ∈ N and 0 < β ≤ 2 3 . Also, Doğru gave the following equalities (1 + an x) (1 + an qx) . (1.5) We will use (1.5) instead of (1.4) throughout this paper. In [21], a kind of real and complex q-Balázs-Szabados-Kantorovich operators were defined, and it was given an upper estimate on compact disks. Now, we give the following new kinds of q-Balázs-Szabados-Kantorovich operators: β q for all n ∈ N, q ∈ (0, 1) and 0 < β ≤ 2 3 .

Definition 1. A new kind of q-Balázs-Szabados-Kantorovich operators is defined as follows:
q-Balázs-Szabados-Kantorovich operators can be called as q-BSK operators for convenience. Since f is nondecreasing and from the definition of q-integral, q-BSK operator is a positive operator. And also, q-BSK operator is linear, so q-BSK operator is a linear and positive operator. We have the following lemma for the q-BSK operators.

Lemma 2. The following equalities hold for the q-BSK operators
(1.15) Proof. Using Lemma 2, after a simple calculation, the proof can be obtained easily, so we omit it.

Weighted statistical approximation properties
The concept of the statistical convergence was introduced by H. Fast [22]. We recall some definitions about the statistical convergence. The density of a set K ⊂ N is defined by The natural density, δ, of a set K ⊂ N is defined by provided the limits exist [23].
Any convergent sequence is statistically convergent but not conversely (see [20]). A real function ρ is called a weight function if it is continuous on R and lim Using (1.13) in Lemma 2, we get For a given ε > 0, let us define the following sets: }︁ , }︁ , Under the conditions given in ( Again for a given ε > 0, let us define the following sets: }︁ , Under the conditions given in ( In this part, we give the rates of convergence of the q-BSK operators by means of the weighted modulus of smoothness The weighted modulus of smoothness for the functions f in Bρ 0 (R+) is defined as for δ > 0. It is clear that for each f in Bρ 0 (R+) , Ωρ 0 (f ; .) is well-defined and satisfies the following properties (see [24]) Ωρ 0 (f ; λδ) ≤ (λ + 1) Ωρ 0 (f ; δ) , δ > 0, (2.7) Ωρ 0 (f ; nδ) ≤ nΩρ 0 (f ; δ) , δ > 0, n ∈ N, lim δ→0 + Ωρ 0 (f ; δ) = 0, We give the following rate of convergence for the q-BSK operators.
Theorem 2. Let q = (qn) be a sequence satisfying the conditions given as in (2.1). For all nondecreasing functions f in Bρ 0 (R+) , we have Proof. Let n ∈ N and f ∈ Bρ 0 (R+) . From (2.6) and (2.7), we can write From the linearity and positivity of the q-BSK operators and using Cauchy-Schwarz inequality, we obtain Finally, choosing δ = µn (x), the proof is completed.

Local approximation
Let C B (R+) be the space of all real valued continuous bounded functions defined on R+. The norm on the space C B (R+) is the supremum norm ‖f ‖ = sup {|f (x)| : x ∈ R+}. Also, Peetre's K-functional is defined By [25] (p.117), there exists a positive constant C > 0 such that where is the second order modulus of continuity of functions f in C B [0, ∞). Further, the usual modulus of continuity is defined by Now, we give local results for the q-BSK operators.
Theorem 3. Let q = (qn) be a sequence satisfying the conditions given as in (2.1) and f ∈ C B (R+). Then for all n ∈ N , there exists a positive constant C > 0 such that are given as in and (1.13) and (1.15).
Proof. For x ∈ R+, we introduce the auxiliary operator as follows: Let x ∈ R+ and g ∈ W 2 . Using the Taylor formula ApplyingȒn to both sides of the above equation, we obtain On the other hand, so the results in weighted space give us the weighted approximation degree of the q-BSK operators to f ,and also the results of local approximation give us the approximation degree of the q-BSK operators to f . We give an illustrative example which shows the rate of convergence of the operators q-BSK to certain functions in the following example: Example 1. In case of β = 0.5, the convergence of the q-BSK operators to f (x) = x + sin(3x) is illustrated in Figure 1 and Figure 2 acording to increasing values of n and q, respectively.
The figures clearly show that, for increasing values of n and q, the degree of approximation becomes better.