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

Abstract

We introduce two kinds of Kantorovich-type q-Bernstein-Schurer-Stancu operators. We first estimate moments of q-Bernstein-Schurer-Stancu-Kantorovich operators. We also establish the statistical approximation properties of these operators. Furthermore, we study the rates of statistical convergence of these operators by means of modulus of continuity and the functions of Lipschitz class.

1. Introduction

In 1987, Lupaş [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.

Before proceeding further, let us give some basic definitions and notations. Throughout the present paper, we consider . For any , the -integer is defined as (see [2]) and the -factorial as

For the integers , , , the -binomial or the Gaussian coefficient is defined as

For an arbitrary function , the -differential is given by

The -Jackson integral in the interval is defined as (see [9]) provided that sums converge absolutely.

Suppose . The -Jackson integral in a general generic interval is defined as

2. Construction of the Operators

For any , a fixed nonnegative number and , a real parameters satisfying the conditions , we introduce the Kantorovich-type -Bernstein-Schurer-Stancu operators as follows: where .

The moments of these operators are obtained as follows.

Lemma 1. For , , one has

Proof. It is obvious that
For , since , (8) holds.
For , Using the properties of the generalized -Schurer-Stancu operators ([10, Lemma 2]) we have
For , we obtain In view of (13) and [10, Lemma 2] by simple calculation we can get the stated result (10).

Lemma 2. For , one has

Proof. In view of Lemma 1, we have

Now, we consider a sequence satisfying the following two expressions: By Korovkin’s theorem, we can state the following theorem.

Theorem 3. Let be a sequence satisfying (20) for . Then for any function , the following equality is satisfied.

Proof. We know that is linear positive. By Lemma 1, if we choose the sequence satisfying (20) and using the equality we have Because of the linearity and positivity of , the proof is complete by the classical Korovkin theorem.

We now redefine as Let us give some lemmas as follows.

Lemma 4. For , , one has

Proof. It is obvious that (25) holds.
For , Taking into account [11, Lemma 1] we have
For , we obtain From (29) and [11, Lemma 1] by simple calculation we arrive at the desired result (27).

Lemma 5. For , one has

Proof. From Lemma 4, it is immediately seen that

We can give the following result, a theorem of Korovkin type.

Theorem 6. Let be a sequence satisfying (20) for . Then for any function , the sequence converges to uniformly on .

The proof of the above theorem follows along Theorem 3; thus we omit the details.

3. Statistical Approximation of Korovkin Type

Further on, let us recall the concept of statistical convergence which was introduced by Fast [12].

Let us set and ; the natural density of is defined by if the limit exists (see [13]), where denotes the cardinality of the set .

A sequence is called statistically convergent to a number if, for every , . This convergence is denoted as . 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-Korovkin-type approximation theorem for statistical convergence.

Theorem 7 (see [15]). If the sequence of linear positive operators satisfies the conditions for , , then, for any ,

In this section, we establish the following Korovkin-type statistical approximation theorems.

Theorem 8. Let , , be a sequence satisfying the following conditions: then for , one has

Proof. From Theorem 7, it is enough to prove that for , .
By (8), we can easily get From equality (9) we have
Now for a given , let us define the following sets: From (41), one can see that , 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 Then, from (22) and (38), we have Here for a given , let us define the following sets: It is clear that . So we get By (49), we have which implies that In view of (40), (45), and (53), the proof is complete.

Theorem 9. Let , , be a sequence satisfying (38); then for , one has .

Proof. From Theorem 7, it is enough to prove that for , .
Using (25), we can easily get From equality (26) we have
Now for a given , let us define the following sets: From (55), one can see that , so we have By (38) it is clear that So we have
Finally, in view of (27), one can write Using , then we can write So, Then, from (38), we have Here for a given , let us define the following sets: It is clear that . So we get By (63), we have which implies that In view of (54), (59), and (67), the proof is complete.

4. Rates of Statistical Convergence

Let for any and . Then we have , so for any , we get

Owing to for , it is obvious that we have for any , and .

Next we will give the rates of convergence of both and in terms of the modulus of continuity.

Theorem 10. Let , be a sequence satisfying (38); then for any function , , one has where

Proof. Using the linearity and positivity of the operator and inequality (69), for any and , we get In view of Lemma 2, take , as a sequence satisfying (38) and choose in (72); the desired result follows immediately.

Theorem 11. Let , be a sequence satisfying (38); then for any function , , one has where

Proof. Using the linearity and positivity of the operator and inequality (69), for any and , we get In view of Lemma 5, take , as a sequence satisfying (38) and choose in (75); the desired result follows immediately.

Finally, we give the rates of statistical convergence of both and with the help of functions of the Lipschitz class. We recall a function on , if the inequality holds.

Theorem 12. Let on , . Let be a sequence satisfying the condition given in (38). If we take as in (71), then one has

Proof. Let on , . Since is linear and positive, by using (76), we have If we take , and apply the Hölder inequality and Lemma 2, then we obtain

Theorem 13. Let on , . Let , be a sequence satisfying the condition given in (38). If we take as in (74), then one has

Proof. Let on , . Since is linear and positive, by using (76), we have If we take , and apply the Hölder inequality and Lemma 5, then we obtain

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (no. N110323010).