Statistical Approximation of q-Bernstein-Schurer-Stancu-Kantorovich Operators

In 1987, Lupaş [1] introduced a q-type of the Bernstein operators and in 1997 another generalization of these operators based on q-integers was introduced by Phillips [2].Thereafter, an intensive research has been done on the q-parametric operators. Recently the statistical approximation properties have also been investigated for q-analogue polynomials. For instance, in [3] q-Bleimann, Butzer, and Hahn operators; in [4] Kantorovich-type q-Bernstein operators; in [5] qanalogue of MKZ operators; in [6] Kantorovich-type qSzász-Mirakjan operators; in [7] Kantorovich-type discrete qBeta operators; in [8] Kantorovich-type q-Bernstein-Stancu operators were introduced and their statistical approximation properties were studied. The main aim of this paper is to introduce two kinds of Kantorovich-type q-Bernstein-Stancu operators and study the statistical approximation properties of these operators with the help of the Korovkin-type approximation theorem. We also estimate the rate of statistical convergence by means of modulus of continuity and with the help of the elements of the Lipschits classes. Before proceeding further, let us give some basic definitions and notations. Throughout the present paper, we consider 0 < q < 1. For any n = 0, 1, 2, . . ., the q-integer [n]q is defined as (see [2])


Introduction
In 1987, Lupas ¸ [1] introduced a -type of the Bernstein operators and in 1997 another generalization of these operators based on -integers was introduced by Phillips [2].Thereafter, an intensive research has been done on the -parametric operators.Recently the statistical approximation properties have also been investigated for -analogue polynomials.For instance, in [3] -Bleimann, Butzer, and Hahn operators; in [4] Kantorovich-type -Bernstein operators; in [5] analogue of MKZ operators; in [6] Kantorovich-type -Szász-Mirakjan operators; in [7] Kantorovich-type discrete -Beta operators; in [8] Kantorovich-type -Bernstein-Stancu operators were introduced and their statistical approximation properties were studied.
The main aim of this paper is to introduce two kinds of Kantorovich-type -Bernstein-Stancu operators and study the statistical approximation properties of these operators with the help of the Korovkin-type approximation theorem.We also estimate the rate of statistical convergence by means of modulus of continuity and with the help of the elements of the Lipschits classes.
We now redefine  (,) , (; ) as Let us give some lemmas as follows.
The proof of the above theorem follows along Theorem 3; thus we omit the details.

Statistical Approximation of Korovkin Type
Further on, let us recall the concept of statistical convergence which was introduced by Fast [12].
A sequence  =   is called statistically convergent to a number  if, for every  > 0, { ∈  : |  − | ≥ } = 0.This convergence is denoted as st − lim    = .It is known that any convergent sequence is statistically convergent, but its converse is not true.Details can be found in [14].
In approximation theory by linear positive operators, the concept of statistical convergence was used by Gadjiev and Orhan [15].They proved the following Bohman-Korovkintype approximation theorem for statistical convergence.
Now for a given  > 0, let us define the following sets: From (41), one can see that  ⊆  1 ∪  2 , so we have By ( 22) and (38) it is clear that So we have Finally, in view of (10), one can write Using (22), we can write       (,) ,  ( In view of (54), (59), and (67), the proof is complete.