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Approximation properties of bivariate complex q-Balàzs-Szabados operators of tensor product kind
Journal of Inequalities and Applications volume 2014, Article number: 20 (2014)
In this study, we consider the bivariate complex q-Balàzs-Szabados operators of the tensor product kind. Approximation properties of these operators attached to analytic functions on compact polydisks are investigated by using the results in the univariate case obtained for q-Balàzs-Szabados operators in (İspir and Yıldız Özkan in J. Inequal. Appl. 2013:361, 2013). In this sense, the upper estimate, the Voronovskaja-type theorem, and the lower estimate are obtained. The exact degree of its approximation is also given.
The approximation properties of the q-analogue operators in compact disks have recently been an active area of the research in the field of the approximation theory [1–8]. Details of the q-calculus can be found in [9–11].
Balázs  defined the Bernstein-type rational functions. She gave an estimate for the order of its convergence and proved an asymptotic approximation theorem and a convergence theorem concerning the derivative of these operators. In , Balázs and Szabados obtained the best possible estimate under more restrictive conditions, in which both the weight and the order of convergence would be better than . They applied their results to the approximation of certain improper integrals by quadrature sums of positive coefficients based on a finite number of equidistant nodes. The q-form of these operators was given by Doğru. He investigated Korovkin-type statistical approximation properties of these operators for the functions of one and two variables . Atakut and Ispir  defined the bivariate real Bernstein-type rational functions of the Bernstein-type rational functions given by Balázs in  and proved the approximation theorems for these functions. Ispir and Gupta  studied the Bézier variant of generalized Kantrovich-type Balazs operators.
Approximation properties of the rational Balázs-Szabados operators on compact disks in the complex plane were investigated by Gal . He proved the upper estimate in an approximation of these operators. Also, he obtained the exact degree of its approximation by using a Voronovskaja-type result. In , the approximation properties given by Gal in the complex plane was extended to the bivariate case.
The complex q-Balázs-Szabados operators was defined in  as follows:
where is uniformly continuous and bounded on with for , , , , , , , and for .
We consider the following complex bivariate q-Balázs-Szabados operators of the tensor product kind:
where is a uniformly continuous function bounded on , , , , for , , .
for all , and with and .
The complex bivariate q-Balázs-Szabados operators of the tensor product kind are well defined and linear, and these operators are analytic for all , , and .
The aim of this paper is to obtain the exact degree of approximation of the complex bivariate q-Balázs-Szabados operators of the tensor product kind. The Voronovskaja-type theorem in the bivariate case is very different from the univariate case, so the exact degree of approximation of these operators can be obtained for .
Throughout this paper, we denote by the uniform norm of the function f in the space of continuous functions on and by the norm of the function f in the space of bounded functions on , where for .
The convergence results will be obtained under the condition that is analytic in for and , which ensures the representation , where for all .
2 Auxiliary results
Let be a sequence satisfying
We need the following lemmas in order to prove the main results for the operators (1).
Lemma 1 Let , , and . If is a uniformly continuous function bounded on and analytic in then we have the form
for all , where with , for .
Proof For any , we define
From the hypothesis on f, it is clear that each is bounded on , which implies that
where is a constant depending on , so all are well defined for all , , , , and .
It is clear that each is bounded on and
From the linearity of , we have
It suffices to prove that
for any fixed , , , and . Since
we can write
for and .
In equation (3), taking the limit as and using , we get the result. □
Lemma 2 Let , , and . For all , , , and the following inequality holds:
Proof Using Lemma 4 in , the lemma is easily proved, so we omit the proof of the lemma. □
3 Main results
Let us denote by the space of all uniformly continuous complex valued functions defined on , bounded on and analytic in and for which there exist , and with for all (which implies for all ).
We have the following upper estimate.
Theorem 1 Let and be sequences satisfying the conditions given in equation (2) and let , , and . If , then for all , , and the following inequality holds:
and also the series , and are convergent.
Proof Using Lemma 1, we can write
Taking into account Lemma 4 in  and the estimate given in the proof of Theorem 2 in , for all and , we obtain
Applying equation (5) in equation (4), we get
Choosing and as given in the theorem, we reach the desired result. □
For , we define the parametric extensions of the Voronovskaja formula by
where for given in Lemma 6 in .
Their product (composition) gives
After a simple calculation, we obtain the commutativity property,
In the following a Voronovskaja-type result for the operators (1) is presented. It will be the product of the parametric extensions generated by Voronovskaja’s formula in the univariate case.
Theorem 2 Let and be sequences satisfying the conditions given in equation (2) and let , , and . If , then for all , , and the following inequality holds:
where , , and are fixed constants,
Proof From the analyticity of f in , since all partial derivatives of f are analytic in , using Lemma 1, we can write
Applying now to equation (7) with respect to and Lemma 1 in , we obtain
In equation (8), passing now to absolute value for and and taking into account the Lemma 4 in  and the estimate given in the proof of Theorem 3 in , it follows that
for and .
Similarly, using the estimate given in the proof of Theorem 3 in  for and we have
we can write
Considering Lemma 6 in  and the estimate given in the proof of Theorem 3 in , for and , we obtain
and also, using
we can write
Taking into account Lemma 6 in  and the estimate given in the proof of Theorem 3 in , for and we get
for and . Using equations (9)-(12), we get
If we estimate , then by reason of the symmetry we get a similar order of approximation, simply interchanging above the places of n with m and with .
In conclusion, using the commutativity property, we reach the result. □
Let us denote by the space of all complex valued functions where they and their first and second partial derivatives are uniformly continuous on , bounded on and analytic in , and there exist , , with (which implies for all ).
Theorems 1 and 2 will be used to find the exact degree in the approximation of . In this sense, we have the following lower estimate.
Theorem 3 Let and be sequences satisfying the conditions given in equation (2) and let , , and . If and f is not a solution of the complex partial differential equation
then for all we have
Proof From equation (6), we can write
Considering Theorems 2 and 3 in , we get
Under the conditions of the theorem, since , , for , it is also clear that
From Theorem 2, we obtain
Using for and , we get
Similarly, it follows that
From equations (14) and (15), we can write
In equation (13), taking into account the inequalities
and equation (16), it follows that
for all with depending only f, and . We used that by hypothesis we have . □
Combining Theorem 2 with Theorem 3, we immediately obtain the following result giving the exact degree of the operators (1).
Corollary 1 Suppose that the hypothesis in the statement of Theorem 3 holds. If the Taylor series of f contains at least one term of the form with and or of the form with and , then for all we have
Proof It suffices to prove that, under the hypothesis on f, it cannot be a solution of the complex partial differential equation
Indeed, suppose the contrary. Since a simple calculation gives
by setting equal to zero and by the identification of the coefficients, from the terms under the first and fifth sign ∑, we immediately get , for all and , which contradicts the hypothesis on f. Therefore the hypothesis and the lower estimate in Theorem 3 are satisfied, which completes the proof. □
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İspir N, Yıldız Özkan E: Approximation properties of complex q -Balazs-Szabados operators in compact disks. J. Inequal. Appl. 2013., 2013: Article ID 361
The author is grateful to the editor and the reviewers for making valuable suggestions, leading to a better presentation of the work.
I declare that I have no competing interests.
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Yıldız Özkan, E. Approximation properties of bivariate complex q-Balàzs-Szabados operators of tensor product kind. J Inequal Appl 2014, 20 (2014). https://doi.org/10.1186/1029-242X-2014-20
- complex approximation
- q-Balàzs-Szabados operators
- order of convergence
- Voronovskaja-type theorem
- exact degree of approximation