Statistical approximation of modified Schurer-type q-Bernstein Kantorovich operators
© Lin; licensee Springer. 2014
Received: 16 July 2014
Accepted: 12 November 2014
Published: 26 November 2014
New modified Schurer-type q-Bernstein Kantorovich operators are introduced. The local theorem and statistical Korovkin-type approximation properties of these operators are investigated. Furthermore, the rate of approximation is examined in terms of the modulus of continuity and the elements of Lipschitz class functions.
MSC:41A10, 41A25, 41A36.
In 1987, Lupaş  introduced a q-analogue of Bernstein operators, and in 1997 another q-generalization of the Bernstein polynomials was introduced by Phillips . After that generalizations of the Bernstein polynomials based on the q-integers attracted a lot of interest and were studied widely by a number of authors. Some new generalizations of well-known positive linear operators based on q-integers were introduced and studied by several authors (e.g., see [3–6]). On the other hand, the study of the statistical convergence for sequences of positive operators was attempted by Gadjiev and Orhan . Very recently, the statistical approximation properties have also been investigated for q-analogue polynomials. For instance, in q-Bleimann, Butzer and Hahn operators; in  Kantorovich-type q-Bernstein operators; in  a q-analogue of MKZ operators; in  Kantorovich-type q-Szász-Mirakjan operators; in  Kantorovich-type q-Bernstein-Stancu operators were introduced and their statistical approximation properties were studied.
The paper is organized as follows. In Section 2, we introduce a new modification of Schurer-type q-Bernstein Kantorovich operators and evaluate the moments of these operators. In Section 3 we study local convergence properties in terms of the first and the second modulus of continuity. In Section 4, we obtain their statistical approximation properties with the help of the Korovkin-type theorem proved by Gadjiev and Orhan. Furthermore, in Section 5, we compute the degree of convergence of the approximation process in terms of the modulus of continuity and the Lipschitz class functions.
2 Construction of the operators
In order to investigate the approximation properties of , we need the following lemmas.
Lemma 2.1 ()
For , since , so (2.5) holds.
Using (2.3) and (2.4), by a simple calculation we can get the stated result (2.7). □
3 Local approximation
By the Korovkin theorem, we can state the following theorem.
as . Because of the linearity and positivity of , the proof is complete by the classical Korovkin theorem. □
is the second modulus of smoothness of .
Let , for any , the usual modulus of continuity for f is defined as .
We next present the following local theorem of the operators in terms of the first and the second modulus of continuity of the function .
which together with (3.3) gives the proof of the theorem. □
4 Korovkin-type statistical approximation properties
Further on, let us recall the concept of statistical convergence which was introduced by Fast .
Let the set and , the natural density of K is defined by if the limit exists (see ), where denotes the cardinality of the set .
A sequence is called statistically convergent to a number L if for every , . This convergence is denoted as . It is known that any convergent sequence is statistically convergent, but not conversely. Details can be found in .
In approximation theory by linear positive operators, the concept of statistical convergence was used by Gadjiev and Orhan . They proved the following Bohman-Korovkin-type approximation theorem for statistical convergence.
Theorem 4.1 ()
In this section, we establish the following Korovkin-type statistical approximation theorems.
Proof From Theorem 4.1, it is enough to prove that for , .
In view of (4.2), (4.4) and (4.6), the proof is complete. □
5 Rates of convergence
for any , and .
Now, we give the convergence rate of to the function in terms of the modulus of continuity.
where is given by (2.9).
Take , , be a sequence satisfying condition (4.1) and choose in (5.2), the desired result follows immediately. □
This research is supported by the Fundamental Research Funds for the Central Universities (Nos. N110323010, N130323015), Science and Technology Research Founds for Colleges and Universities in Hebei Province (No. Z2014040), and the Research Fund for Northeastern University at Qinhuangdao (No. XNB201429).
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