Operator Methods in Approximation TheoryView this Special Issue
Bivariate Chlodowsky-Stancu Variant of (p,q)-Bernstein-Schurer Operators
In this study, it is proposed to define bivariate Chlodowsky variant of -Bernstein-Stancu-Schurer operators. Therefore, Korovkin-type approximation theorems and the error of approximation by using full modulus of continuity are presented. Beside this, we introduce a generalization of the bivariate Chlodowsky variant of -Bernstein-Stancu-Schurer operators and investigate its approximation in more general weighted space. Moreover, the numerical results are discussed in order to validate the accuracy of the bivariate Chlodowsky variant of -Bernstein-Schurer operators.
In this area, many operators about -integer and -integer [1–25] were studied. It may not be possible to find the exact solution for the models developed for the engineering fields because of its mathematical intractability. Therefore, these operators can be used effectively to find an algorithm for approximating solutions .
In 2014, Vedi and Ozarslan introduced the Chlodowsky variant of -Bernstein-Schurer-Stancu operators in  as where is increasing sequence of real numbers satisfying , , , , and and investigated its approximation properties.
In 2017, Vedi and Ozarslan defined the two dimensional Chlodowsky variant of -Bernstein-Schurer-Stancu operators in  bywhere and are increasing sequences of real numbers satisfying and , , , and investigated its approximation properties on the rectangular unbounded domain.
Moreover, Gemikonakli and Vedi-Dilek  introduced the Chlodowsky variant of Bernstein-Schurer operators based on -integers as where , , and .
Let us discuss some well-known basic definitions of -calculus. For , the -numbers are given as  For each the -factorial is represented by and -binomial coefficients are defined aswhere
Then, Chlodowsky variant of -Bernstein-Stancu-Schurer operators was constructed in  aswhere , , with , , and has the same properties as the operator .
Lemma 1. Let be defined in .
Then the first few moments of the operators are
The bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on -integers is defined and then the first few moments of the operator are provided in Section 2. Next, in Section 3, some Korovkin-type theorems are studied. And following this, the order of convergence of the bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on -integers by means of the first modulus of continuity is obtained in Section 4. Moreover, in Section 5, we study the generalization of the bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on -integers and seek its approximation properties in more general weighted space. Finally, in Section 6, numerical results for the operators constructed are provided in detail.
2. Construction of the Operators
Let denoteand and are increasing sequences of real numbers satisfyingFor , we construct the bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on -integers aswhere , , and , .
Lemma 2. Let be given in (11).
Then, the first few moments of the operators are
3. Korovkin-Type Approximation Theorems
In this section, Korovkin-type approximation theorems are given for the bivariate Chlodowsky variant of Bernstein–Stancu-Schurer operators based on -integers. For fixed consider the space , which consists of all continuous functions satisfying the conditionObviously, is a linear normed space with the following norm:
Theorem 3. Let the numbers and be any fixed positive real numbers.
Let with and and be increasing sequences of positive real numbers that satisfy the following properties: For all , we obtain
Proof. Using Lemma 2, we haveAnd again using Lemma 2 we getFinally, from the above equality, we obtain Therefore, from the hypothesis of the theorem, we getwhen and .
Hence, the proof is completed by the two dimensional Korovkin theorem.
Theorem 4 (see ). There exists a sequence of positive operators , acting from to , satisfying the conditions and there exists a function for which where
Now, consider the following operator:
Theorem 5. Let for any where , , , and have the same conditions as in Theorem 3.
Proof. For all , there exist sufficiently large positive real numbers and such thatwhen and .
Let , be sufficiently large so that By Theorem 3, and for the proof of the second term we haveFinally, since , the term is bounded. Furthermore, because of the fact thatin case Lemma 2 is used, the term is bounded for sufficiently large and . Hence, we get by (25) that Since is arbitrary, then . This completes the proof.
Now, consider the subspace of which is defined by
Theorem 6. Let the sequences , , , and satisfy the same properties as in Theorem 3. Then for all , we obtain
Proof. For all , observe that Therefore, for all , we can find sufficiently large numbers and such that for and and there exist natural numbers and such that for all and .
Hence, for large and , we haveBy Theorem 3, it is sufficient to show that as .
Using (33) and (34), we getwhereBy Lemma 2, it is clear that there exist independent of and such that Therefore, for and we have This completes the proof.
4. Order of Convergence
In this section, we compute the rate of convergence of the operators in terms of the full modulus of continuity and partial modulus of continuities.
Let and . Then the definition of the modulus of continuity of is given byIt is known that  for any we know that Also, for any we have
Theorem 7. For any , the following inequalities are satisfied, where
Proof. We directly haveBy linearity and positivity of the operators, we getUsing Lemma 1 and Cauchy-Schwartz inequality, we haveThen, we getwhere we choose as in (46).
In the same way, we obtainwhere is given in (45). Combining (50) and (51), we obtain (43).
Now, by using linearity and the monotonicity of the operators, and taking into account (40), we have Using (41) and the Cauchy-Schwartz inequality, we get (44).
5. Generalization of the Bivariate Chlodowsky Variant of Bernstein-Schurer Operators Based on -Integers
In this section, we introduce generalization of bivariate Chlodowsky variant of Bernstein-Schurer operators based on -integers. The generalized operators help us to approximate continuous functions defined on more general weighted spaces. Note that this kind of generalization was considered earlier for the Chlodowsky-Bernstein polynomials .
For , consider any continuous function and define Let us consider the generalization of the as follows:where and and have the same properties of bivariate Chlodowsky variant of Bernstein-Schurer operators based on -integers.
Theorem 8. For all continuous functions satisfying , , and , we have where