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Approximation by a complex q-Baskakov-Stancu operator in compact disks
Journal of Inequalities and Applications volume 2014, Article number: 249 (2014)
Abstract
In this paper, we consider a complex q-Baskakov-Stancu operator and study some approximation properties. We give a quantitative estimate of the convergence, Voronovskaja-type result and exact order of approximation in compact disks.
MSC:30E10, 41A25, 41A28.
1 Introduction
Recently complex approximation operators have been studied intensively. For this approach, we refer to the book of Gal [1], where he considers approximation properties of several complex operators such as Bernstein, q-Bernstein, Favard-Szasz-Mirakjan, Baskakov and some others. Also we refer to the useful book of Aral, Gupta and Agarwal [2] who consider many applications of q-calculus in approximation theory. Now, for the construction of the new operators, we give some notations on q-analysis [3, 4].
Let . The q-integer and the q-factorial are defined by
and
respectively. For integers , the q-binomial coefficient is defined as
The q-derivative of is denoted by and defined as
also
q-Pochhammer formula is given by
with , . The q-derivative of the product and the quotient of two functions f and g are
and
respectively (see in [3]). Moreover, we have
where denotes the divided difference of the function f on the knots (see [4] also [5]).
In [6], Aral and Gupta constructed the q-Baskakov operator as
where , and f is a real-valued continuous function on . The authors studied the rate of convergence in a polynomial weighted norm and gave a theorem related to monotonic convergence of the sequence of operators with respect to n. Not only they proved a kind of monotonicity by means of q-derivative but also they expressed the operator in terms of divided differences as follows:
, similar to the case of classical Baskakov operators in the sense of Lupaş in [7]. That is to say, for and , so they proved that
where stands for q-divided differences given by ,
for .
A different type of the q-Baskakov operator was also given by Aral and Gupta in [8]. In [9] Finta and Gupta studied the q-Baskakov operator for . Using the second-order Ditzian-Totik modulus of smoothness, they gave direct estimates. They also introduced the limit q-Baskakov operator.
In [10] Gupta and Radu introduced a q-analogue of Baskakov-Kantorovich operators and studied weighted statistical approximation properties of them for . They also obtained some direct estimations for error with the help of weighted modulus of smoothness. Moreover, Durrmeyer-type modifications of q-Baskakov operators were studied in [11] and [12]. In [13], Söylemez, Tunca and Aral defined a complex form of q-Baskakov operators by
for , , replacing x by z in the operator given by (1.2). They obtained a quantitative estimate for simultaneous approximation, Voronovskaja-type result and degree of simultaneous approximation in compact disks.
In recent years, a Stancu-type generalization of the operators has been studied. Büyükyazıcı and Atakut considered a Stancu-type generalization of the real Baskakov operators in [14]. Also in [15], q-Baskakov-Beta-Stancu operators were introduced. In [16] Gupta-Verma studied the Stancu-type generalization of complex Favard-Szasz-Mirakjan operators and established some approximation results in the complex domain. In [17] Gal, Gupta, Verma and Agrawal introduced complex Baskakov-Stancu operators and studied Voronovskaja-type results with quantitative estimates for these operators attached to analytic functions on compact disks.
Now we define a new type of the complex q-Baskakov-Stancu operator
where ; for , we take . We suppose that f is analytic on the disk , and has exponential growth in the compact disk with all derivatives bounded in by the same constant.
Note that taking , reduces to the complex q-Baskakov operator given in (1.4).
In this work, for such f and , we study some approximation properties of the complex q-Baskakov-Stancu operator which is defined by forward differences.
2 Auxiliary results
In this section, we give some results which we shall use in the proof of theorems.
Lemma 1 Let us define , , and denotes the set of all nonnegative integers. Then, for all , and , we have the following recurrence formula:
Hence
for all .
Proof Now we can write
Using relation (1.1) and taking , and , we obtain
using this in we reach
□
Lemma 2 Let α and β satisfy . Denoting and by given in (1.4), for all , we have the following recursive relation for the images of monomials under in terms of , :
Proof We can use mathematical induction with respect to k. For , equality (2.4) holds. Let it be true for , namely
Using (2.1), we have
Taking into account the recurrence relation for the complex q-Baskakov operator in Lemma 2 in [13], we get
which implies
which proves the lemma. □
3 Approximation by a complex q-Baskakov-Stancu operator
In this section, we give quantitative estimates concerning approximation with the following theorem.
Theorem 1 For , let
be a function with all its derivatives bounded in by the same positive constant, analytic in , namely for all and suppose that there exist and , with the property for all (which implies for all ).
Let , and be arbitrary but fixed. Then, for all and , we have
with
Proof Using (2.1), one can obtain
Moreover, we have
Now from (3.1) and the Bernstein inequality (see [1]), we have
where is the standard maximum norm over . Passing to modulus for all and , we have that
In order to get an estimate for in (3.2), we use the following fact:
for . Taking into account Lemma 1 in [13] for , , and (1.3), we have
Now, considering (3.3) in (3.2), for all , , with and ,
Using the above inequalities beginning from and using the mathematical induction with respect to k, we arrive at
Also we obtain the following: for it is not difficult to see that
Now, taking into account the proof of Theorem 1 in [13], we can write, for , , , that
which implies
Here from the analyticity of f we have and . Also from the hypotheses of the theorem, one can get
for all , , . □
Theorem 2 Let , and . Under the hypotheses of Theorem 1, for all and , the following Voronovskaja-type result
holds with
Proof For all , let us consider
Using the fact that , we get
From Theorem 2 in [13], we have
Furthermore, in order to estimate the second sum, using Lemma 2, we obtain
Also it is clear that
which implies
Now from (3.3) we obtain
Also, we need to prove the following inequality:
Moreover, taking in Theorem 1, we have
Writing (3.8), (3.6), (3.9) and (3.10) in (3.7), we have
Thus the proof is completed. □
Now, let us give a lower estimate for the exact degree in approximation by .
Theorem 3 Suppose that and suppose that the hypotheses on f and on the constants R, M, A in the statement of Theorem 1 hold, and let , . If f is not a polynomial of degree ≤0, then the lower estimate
holds for all n, where the constant depends on f, α, β, q and r.
Proof For all and , we get
We set by
Passing to the norm and using the inequality
we get
Since f is not a polynomial of degree ≤0 in , we have
It can also be seen in [[1], pp.75-76]. Now, from Theorem 2 it follows that
Since as , for , there exists an depending on f, r, α, β and q such that for all ,
which implies
for all . Now, for , we have
with
which finally implies
for all with
This proves the theorem. □
Combining now Theorem 3 with Theorem 1, we immediately get the following equivalence result.
Remark 1 Suppose that , and that the hypotheses on f and on the constants R, M, A in the statement of Theorem 1 hold, and let be fixed. If f is not a polynomial of degree ≤0, then we have the following equivalence:
for all n, where the constants in the equivalence depend on f, α, β, q and r.
Concerning the approximation by the derivatives of complex q-Baskakov-Stancu operators, we can state the following theorem.
Theorem 4 Suppose that and that the hypotheses on f and on the constants R, M, A in the statement of Theorem 1 hold, and let , and be fixed. If f is not a polynomial of degree , then we have the following equivalence:
for all n, where the constants in the equivalence depend on f (that is, on M, A), r, and p.
Proof Denote by Γ the circle of radius with centered at 0. Since and , we have and from Cauchy’s formulas and Theorem 1 we obtain, for all and , that
which proves one of the inequalities in the equivalence.
Now we need to prove the lower estimate. From Cauchy’s formula we get
Furthermore, using (3.11) one can have
for all and . Applications of Cauchy’s formula imply
Now passing to the norm we obtain
and from Theorem 2 we have
Since f is not a polynomial of degree ≤0 in , we have
(see [[1], pp.77-78]). The rest of the proof is obtained similarly to that of Theorem 3. □
Remark 2 Note that if we take , then Theorems 1, 2, 3 and 4 reduce to the results in [13].
References
Gal SG Series on Concrete and Applicable Mathematics 8. In Approximation by Complex Bernstein and Convolution Type Operators. World Scientific, Hackensack; 2009.
Aral A, Gupta V, Agarwal RP: Applications of q-Calculus in Operator Theory. Springer, Berlin; 2013.
Ernst, T: The history of q-calculus and a new method. U.U.U.D.M Report 2000, 16, ISSN 1101–3591, Department of Mathematics, Upsala University (2000)
Phillips GM CMS Books in Mathematics 14. In Interpolation and Approximation by Polynomials. Springer, New York; 2003. [Ouvrages de Mathématiques de la SMC]
Stancu, DD: Course in Numerical Analysis. Faculty of Mathematics and Mechanics, Babes-Bolyai University Press, Cluj (1997)
Aral A, Gupta V: Generalized q -Baskakov operators. Math. Slovaca 2011,61(4):619–634. 10.2478/s12175-011-0032-3
Lupaş A: Some properties of the linear positive operators, II. Mathematica 1967,9(32):295–298.
Aral A, Gupta V: On q -Baskakov type operators. Demonstr. Math. 2009,42(1):109–122.
Finta Z, Gupta V: Approximation properties of q -Baskakov operators. Cent. Eur. J. Math. 2010,8(1):199–211. 10.2478/s11533-009-0061-0
Gupta V, Radu C: Statistical approximation properties of q -Baskakov-Kantorovich operators. Cent. Eur. J. Math. 2009,7(4):809–818. 10.2478/s11533-009-0055-y
Aral A, Gupta V: On the Durrmeyer type modification of the q -Baskakov type operators. Nonlinear Anal. 2010,72(3–4):1171–1180. 10.1016/j.na.2009.07.052
Gupta V, Aral A: Some approximation properties of q -Baskakov-Durrmeyer operators. Appl. Math. Comput. 2011,218(3):783–788. 10.1016/j.amc.2011.01.057
Söylemez D, Tunca GB, Aral A: Approximation by complex q -Baskakov operators in compact disks. An. Univ. Oradea, Fasc. Mat. 2014,XXI(1):167–181.
Büyükyazıcı İ, Atakut Ç: On Stancu type generalization of q -Baskakov operators. Math. Comput. Model. 2010, 52: 752–759. 10.1016/j.mcm.2010.05.004
Maheshwari P, Sharma D: Approximation by q -Baskakov-Beta-Stancu operators. Rend. Circ. Mat. Palermo 2012, 61: 297–305. 10.1007/s12215-012-0090-6
Gupta V, Verma DK: Approximation by complex Favard-Szasz-Mirakjan-Stancu operators in compact disks. Math. Sci. 2012., 6: Article ID 25
Gal SG, Gupta V, Verma DK, Agrawal PN: Approximation by complex Baskakov-Stancu operators in compact disks. Rend. Circ. Mat. Palermo 2012, 61: 153–165. 10.1007/s12215-012-0082-6
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Özden, D.S., Arı, D.A. Approximation by a complex q-Baskakov-Stancu operator in compact disks. J Inequal Appl 2014, 249 (2014). https://doi.org/10.1186/1029-242X-2014-249
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DOI: https://doi.org/10.1186/1029-242X-2014-249