Abstract
The objective of this paper is to construct univariate and bivariate blending type -Schurer–Kantorovich operators depending on two parameters and to approximate a class of measurable functions on . We present some auxiliary results and obtain the rate of convergence of these operators. Next, we study the global and local approximation properties in terms of first- and second-order modulus of smoothness, weight functions, and by Peetre’s -functional in different function spaces. Moreover, we present some study on numerical and graphical analysis for our operators.
1. Introduction
In 1912, Bernstein [1] introduced a polynomial aswhere . For the operators given by (1), he showed that converges to uniformly where . In 1962, Schurer [2] modified operators given in (1) as: for , a real numberwhere . One can note that for , the polynomials presented in (2) reduces to polynomials given by (1). The operators are introduced in (1) and (2) are restricted for continuous functions only and are different in respect to the domain of function . Additionally, several researchers, ., Mursaleen et al. [3], Acar et al. [4, 5], Mohiuddine et al. [6], Acu et al. [7], İçöz and Çekim [8, 9], and Kajla and Micláus [10, 11] constructed new sequences of linear positive operators to investigate the rapidity of convergence and order of approximation in different functional spaces in terms of several generating functions. Some other researchers developed many other useful operators [6, 12–30] in the same field. In the recent past, for and , Chen et al. [31] constructed a sequence of new linear positive operators aswhere , , and
The operators defined in (3) are named as -Berstein operator of order m.
Remark 1. One can note that for , the relation (3) reduces to classical Bernstein operators [1].
Later, Aral and Erbay [32] introduced a parametric extension of Baskakov operators. Recently, Özger et al. [33] constructed a sequence -Bernstein–Schurer operators to approximate a class of continuous functions as: For every where stands for the continuous and bounded function aswhereand .
Now, we construct the -Bernstein–Schurer–Kantorovich operators and their moments.Motivated by the above development, we introduce positive linear operators to discuss the approximation properties in Lebesgue measurable space asIn the remaining part, we calculate basic Lemmas and order of approximation. Moreover, results of global and local approximation in terms of continuity modulus, weight functions, Lipschitz class and Lipschitz maximal function, Peetre’s -functional, and second-order smoothness modulus were analyzed. Furthermore, -bivariate Schurer–Kantorovich operators are constructed and their pointwise and uniform approximation results are investigated.
2. Basic Estimates
Lemma 2 (see [33]). Let , . For the operator defined in (5), one has
Lemma 3. For the operators discussed in (7), we obtain
Lemma 4. For the operator given by (8), we obtain
Proof. By Lemma 3, it easily demonstrated Lemma 4.
Lemma 5. Let , , represent the central moments of introduced in (8). Then, we have
Proof. Using Lemmas 3 and 4, it can be easily be proved.
3. Convergence Behaviour of
Definition 6 (see [6]). For , the modulus of continuity for a uniformly continuous function is defined asIn represent continuous function that is uniformaly and . Then, one get
Theorem 7. Let represent the set of operators provided by (8). On each bounded subset of ; then, converges uniformly to where .
Proof. For the proof of this result, it is enough to thatBy Lemma 3, it is obvious that for as . The proof of Theorem 7 is complete.
Example 1. As one can see, for the following set of parameters , , and , the operator converges uniformly to the function as increases which is illustrated in Figure 1.
Figure 2 and Table 1 also demonstrated our analytical results.
Theorem 8 (see [34]). Let be a linear and positive operator and let be the function defined by
If for any and any , the operator verifies
Theorem 9. Let the sequence of operators introduced by (8) and , we obtainwhere .
Proof. In the light of Theorem 8 and Lemmas 3 and 4, we haveOn choosing, .
Hence, it completes the proof of this result.
4. Pointwise Approximation Results
Here, we consider the Lipschitz type space [35] aswhere is a real valued constant number, , , and .
Theorem 10. For , one yieldwhere and .
Proof. For , one hasSince for all , we getFor , this outcome is valid and by Hölder’s inequality with and , we getSince for all , we obtainHence, completes the proof.
5. Global Approximation
From [36], we recall some notation to prove the global approximation results.
In terms of the weight function and , we have
, the constant depends on }.
endowed with the norm space of continuous functions
and
Theorem 11. Let the be the operators given by (8) and . Then, we havewhere .
Proof. To prove this result, it is sufficient to prove thatFrom Lemma 3, we getFor ,which implies that as .
Similarly, we see that as .
6. Bivariate Case of Operators and Their Approximation Behaviour
Take and is the class of all continuous functions on equipped with the norm . Then, for all and , we construct a bivariate sequence. Bivariate generalized baskakov operators are as follows:where
Lemma 12. Let represent the moments. Then, for the operators (equation (31)), one has
Proof. From Lemma 3 and linearity property, we havewhich proves Lemma 12.
Lemma 13. Let represent central moments functions. Then, the operators defined by (Section 6.1) satisfies the following identities:
Proof. From Lemma 12 and linearity property, we havewhich proves Lemma 13.
7. Degree of Convergence
For any and modulus of continuity of the second order is given bywith defined by the partial modulus of continuity as
Theorem 14. For any , we have
Proof. For the proof of Theorem 14, generally, we use the well-known Cauchy–Schwarz inequality. Thus, we see thatIf we choose and , then we can simply achieve our objectives.
Here, we analyze convergence in terms of the Lipschitz class for bivariate functions. For and , maximal Lipschitz function space on is given bywhere is continuous and bounded on , and
Theorem 15. Let , then for any , there exists such thatwhere and are defined by Theorem 14.
Proof. Take and . For any , and , we let . Thus, we can write hereApply , we obtainFor all, and , the inequality , thusTherefore,On applying Hölder inequality on , we getThus, we can obtainWe have completed the proof.
Example 2. It is observed in this example that for the different set of parameters , , and , the operator converges uniformly to the function (blue) as (green) and (red) increases which is shown in Figure 3.
8. Conclusion
In this paper, we construct the univariate and bivariate blending type -Schurer–Kantorovich operators depending on two parameters and to approximate a class of measurable functions on . Along with the auxiliary results, we obtain the rate of convergence of these operators. Also, we study the global and local approximation properties in terms of first- and second-order modulus of smoothness, weight functions, and by Peetre’s -functional in different function spaces. Additionally, we give some examples on numerical and graphical analysis for our operators.
Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
Disclosure
A preprint Rao et al. [37] has previously been published.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors have equal contributions.