Some Approximation Properties of -Baskakov-Beta-Stancu Type Operators
This paper deals with new type -Baskakov-Beta-Stancu operators defined in the paper. First, we have used the properties of -integral to establish the moments of these operators. We also obtain some approximation properties and asymptotic formulae for these operators. In the end we have also presented better error estimations for the -operators.
In the recent years, the quantum calculus (-calculus) has attracted a great deal of interest because of its potential applications in mathematics, mechanics, and physics. Due to the applications of -calculus in the area of approximation theory, -generalization of some positive operators has attracted much interest, and a great number of interesting results related to these operators have been obtained (see, for instance, [1–3]). In this direction, several authors have proposed the -analogues of different linear positive operators and studied their approximation behaviors. Also, Aral and Gupta  defined -generalization of the Baskakov operators and investigated some approximation properties of these operators. Subsequently, Finta and Gupta  obtained global direct error estimates for these operators using the second-order Ditzian Totik modulus of smoothness. To approximate Lebesgue integrable functions on the interval , modified Beta operators  are defined as where and .
The discrete -Beta operators are defined as
Recently, Maheshwari and Sharma  introduced the -analogue of the Baskakov-Beta-Stancu operators and studied the rate of approximation and weighted approximation of these operators. Motivated by the Stancu type generalization of -Baskakov operators, we propose the -analogue of the operators , recently introduced and studied for special values by Gupta and Kim  as where and .
We know that and . We mention that (see ).
Very recently, Gupta et al.  introduced some direct results in simultaneous approximation for Baskakov-Durrmeyer-Stancu operators. The aim of this paper is to study the approximation properties of a new generalization of the Baskakov type Beta Stancu operators based on -integers. We estimate moments for these operators. Also, we study asymptotic formula for these operators. Finally, we give better error estimations for the operator (3). First, we recall some definitions and notations of -calculus. Such notations can be found in [10, 11]. We consider as a real number satisfying . For , The -binomial coefficients are given by The -derivative of a function is given by The -analogues of product and quotient rules are defined as The -Jackson integrals and the -improper integrals are defined as [12, 13] provided that the sums converge absolutely. Using (9), De Sole and Kac  defined the -analogue of Beta functions of second kind as follows: where . This function is -constant in ; that is, . It was observed in  that is independent of ; this is because from the integral and the term cancels out. In particular for any positive integer , we have Also, we have In , Gupta and Kim obtained recurrence formula for the moments of the operators as follows.
Theorem 1 (see ). If one defined the central moments as then, for , one has the following recurrence relation:
2. Moment Estimates
Lemma 2 (see ). The following equalities hold.(i).(ii).(iii), for .
Lemma 3. The following equalities hold.(i).(ii), for .(iii), for .
Proof. The operators are well defined on function , , . By Lemma 2, for every and , we have Similarly,
Remark 4. For all , , we have the following recursive relation for the images of the monomials under in terms of , , as
Remark 5. If we put and , we get the moments of the modified Beta operators  as
Remark 6. From Lemma 3, we have
3. Direct Result and Asymptotic Formula
Let the space of all real-valued continuous bounded functions be endowed with the norm . Further, let us consider the following -functional: where and . By [15, page 177, Theorem 2.4], there exists an absolute constant such that where is the second-order modulus of smoothness of . Also we set
Theorem 7. Let and such that as . Then for all and , there exists an absolute constant such that
Proof. We are introducing the auxiliary operators as follows: From (25) and Lemma 3, we have Let and . Using Taylor’s formula applying , and by (26), we get On the other hand, since We conclude by Remark 6 that From (25), we can write that Now, taking into account the boundedness of and from (31), we get Now, taking infimum on the right-hand side over all and from (21), we get where . This proves the theorem.
Our next result in this section is an asymptotic formula.
Theorem 8. Let be bounded and integrable function on the interval ; the second derivative of exists at a fixed point and such that as . Consider
Proof. Using Taylor’s expansion of , we can write
where is bounded and . Applying the operator to the above relation, we get where and are defined in Remark 6.
Using Cauchy-Schwarz inequality, we have Using Theorem 1 with the help of Remark 4, we can easily find that Also, since Thus, which completes the proof.
4. Better Estimation
It is well known that the operators preserve constant as well as linear functions. To make the convergence faster, King  proposed an approach to modify the classical Bernstein polynomials, so that this sequence preserves two test functions: and . After this, several researchers have studied that many approximating operators, , possess these properties; that is, , where or , for example, Bernstein, Baskakov, and Baskakov-Durrmeyer-Stancu operators (see [4, 5, 17–19]).
As the operators introduced in (3) preserve only the constant functions, further modification of these operators is proposed to be made so that the modified operators preserve the constant as well as linear functions. For this purpose, the modification of is as follows: where
Lemma 9. For each , one has
Lemma 10. For each , the following equalities hold:
Theorem 11. Let and such that as . Then for all and , there exists an absolute constant such that
Proof. Let and . Using Taylor’s formula, we get
Applying , we get
Obviously, we have
Since , thus
Now, taking infimum on the right-hand side over all and from (21), we get which proves the theorem.
Theorem 12. Let be bounded and integrable function on the interval ; the second derivative of exists at a fixed point and such that as ; then
The proof follows along the same lines of Theorem 8.
Conflict of Interests
The authors declare that there is no conflict of interests.
The authors thank the anonymous learned reviewers for their valuable suggestions, which substantially improved the standard of the paper. Special thanks are due to Professor Adam Kowalewski, for kind cooperation and smooth behavior during communication and for his efforts to send the reports of the paper timely.
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