Generalized q-Bernstein-Schurer Operators and Some Approximation Theorems

In 1987, Lupaş [1] introduced the first q-analogue of Bernstein operator and investigated its approximating and shapepreserving properties. Another q-generalization of the classical Bernstein polynomials is due to Phillips [2]. After that many generalizations ofwell-knownpositive linear operators, based on q-integers, were introduced and studied by several authors. Recently the statistical approximation properties have also been investigated for q-analogue polynomials. For instance, in [1] q-analogues of Bernstein-Kantorovich operators; in [3] q-Baskakov-Kantorovich operators; in [4] q-Szász-Mirakjan operators; in [5, 6] q-Bleimann, Butzer and Hahn operators; in [7] q-analogue of Baskakov and Baskakov-Kantorovich operators; in [8] q-analogue of Szász Kantorovich operators; in [9, 10] q-analogue of Stancu-Beta operators; and in [11] q-Lagrange polynomials were defined and their classical approximation or statistical approximation properties were investigated. Schurer [12] introduced the following operators L m,p :


Introduction and Preliminaries
In 1987, Lupas ¸ [1] introduced the first -analogue of Bernstein operator and investigated its approximating and shapepreserving properties.Another -generalization of the classical Bernstein polynomials is due to Phillips [2].After that many generalizations of well-known positive linear operators, based on -integers, were introduced and studied by several authors.Recently the statistical approximation properties have also been investigated for q-analogue polynomials.For instance, in [1] q-analogues of Bernstein-Kantorovich operators; in [3] -Baskakov-Kantorovich operators; in [4] -Szász-Mirakjan operators; in [5,6] -Bleimann, Butzer and Hahn operators; in [7] -analogue of Baskakov and Baskakov-Kantorovich operators; in [8] -analogue of Szász Kantorovich operators; in [9,10] -analogue of Stancu-Beta operators; and in [11] -Lagrange polynomials were defined and their classical approximation or statistical approximation properties were investigated.
Schurer [12] introduced the following operators  , : Recently, Muraru [13] introduced the -analogue of these operators and investigated their approximation properties and rate of convergence using modulus of continuity.Note that Radu [14] has also used q-intgers to define and study the approximation properties of the q-analogue of Kantorovich operators.
In this paper, we study the statistical approximation properties by -Bernstein-Schurer operators.We also give some direct theorems.
We recall certain notations of -calculus.Let  > 0. For any  ∈ N 0 := {0} ∪ N, the -integer []  is defined by and the -factorial []  ! by Also, the -binomial coefficients are defined by Details on -integers can be found in [14].

Statistical Approximation
In this section we obtain the Korovkin type weighted statistical approximation properties for these operators defined in (5).Korovkin type approximation theory [15] has also many useful connections, other than classical approximation theory, in other branches of mathematics (see Altomare and Campiti [16]).First we recall the concept of statistical convergence for sequences of real numbers which was introduced by Fast [ In this case, we write  − lim    = .Note that every convergent sequence is statistically convergent but not conversely, even unbounded sequence may be statistically convergent.For example, let  = (  ) be defined by Then  − lim   = 0, but  is not convergent.
Recently the idea of statistical convergence has been used in proving some approximation theorems, in particular, Korovkin type approximation theorems by various authors, and it was found that the statistical versions are stronger than the classical ones.Authors have used many types of classical operators and test functions to study the Korovkin type approximation theorems which further motivate continuation of this study.After the paper of Gadjiev and Orhan [18], different types of summability methods have been deployed in approximation process, for example, [19][20][21][22][23]. Recently, statistical convergence has been used to the summability of Walsh-Fourier series [24].
Let   [0,  + 1] be the space of all bounded and continuous functions on [0,  + 1].Then   [0,  + 1] is a normed linear space with ‖‖ = sup ≥0 |()|.Let  be a function of the type of modulus of continuity.The principal properties of the function are the following: Let   [0,  + 1] be the space of all real valued functions  defined on [0,  + 1] satisfying the following condition: for any ,  ∈ [0,  + 1].
Theorem 6.Let ( * , ) be the sequence of the operators (5), and the sequence  = (  ) satisfies (14) For a given  > 0, let us define the following sets: It is obvious that  ⊆   ; it can be written as By using (14), we get Therefore  ({ ≤  : Now, if we choose then, by ( 14), we can write Now for given  > 0, we define the following four sets:
Let  *  [0,  + 1] be the space of all bounded functions for which lim  → ∞ () is finite.