Approximation by a kind of complex modified q-Durrmeyer type operators in compact disks
© Ren and Zeng; licensee Springer 2012
Received: 25 March 2012
Accepted: 7 September 2012
Published: 2 October 2012
In this paper, in order to make the convergence faster to a function being approximated, we introduce a kind of complex modified q-Durrmeyer type operators which can reproduce constant and linear functions. We study the approximation properties of these operators. We obtain the order of simultaneous approximation and a Voronovskaja-type result with a quantitative estimate for these complex modified q-Durrmeyer type operators attached to analytic functions on compact disks. More important, our results show the overconvergence phenomenon for these complex operators.
Agarwal and Gupta  have extended the operators which were given by (1.1) to a complex space and have studied the approximation properties of these complex operators. They have obtained the order of approximation and a Voronovskaja-type result with a quantitative estimate for these complex operators attached to analytic functions on compact disks.
The moments of the operators were obtained as follows (see ):
By simple computation, we get the moments of the operators .
where , , and , .
2 Auxiliary results
In the sequel, we shall need the following auxiliary results.
Considering the definition of the , for any , applying the principle of mathematical induction, we immediately obtain the desired conclusion. □
which implies that we get the desired conclusion. □
Corollary 1 Denote , let and . Then for all and , we have .
In view of and , by simple calculation, we can get the recurrence in the statement. □
Proof Using formula (2.1), by simple calculation, we can easily get the recurrence (2.2), the proof is omitted here. □
Proof The proof is easy by using the Bernstein inequality and the complex mean value theorem, the proof is omitted here. □
3 Main results
The first main result is expressed by the following upper estimates.
- (i)For all and , we have
- (ii)(Simultaneous approximation) If are arbitrary fixed, then for all and , we have
where is defined as in (i) above.
- (i)By Lemma 4, Lemma 5 and Corollary 1, for all , we get
- (ii)Denoting by Γ the circle of radius and center 0, since for any and we have , by the Cauchy’s formulas it follows that for all and , we have
which proves the theorem. □
Remark 3 Let be fixed. Since we have as , by passing to limit with in the estimates in Theorem 1, we do not obtain the convergence of to , . But this situation can be improved by choosing with as . Indeed, since in this case as (see Videnskii , formula (2.7)), from Theorem 1 we get that , for , uniformly for , for any .
The following Voronovskaja-type result with a quantitative estimate holds.
where and .
for all , , .
where is a polynomial of degree 3 in k defined as , is expressed in the above.
As and the series is absolutely convergent in , it easily follows that , which implies that . This completes the proof of the theorem. □
In the following theorem, we will obtain the exact order in approximation.
where and the constant depends on f, r and on the sequence but is independent of n.
Indeed, supposing the contrary, it follows that for all , that is for all . Thus, f is a polynomial of degree ≤1, a contradiction to the hypothesis.
where , this completes the proof. □
Combining Theorem 3 with Theorem 1, we get the following result.
where and the constants in the equivalence depend on f, r and on the sequence but are independent of n.
Considering the derivatives of complex modified q-Durrmeyer type operators, we can prove the following result.
where and the constants in the equivalence depend on f, r, , p and on the sequence but are independent of n.
Proof Taking into account the upper estimate in Theorem 1, it remains to prove the lower estimate only.
Indeed, supposing the contrary, it follows that , that is, is a polynomial of degree . Let and , then the analyticity of f obviously implies that f is a polynomial of degree , a contradiction.
Now let , then the analyticity of f obviously implies that f is a polynomial of degree , a contradiction to the hypothesis.
In conclusion, , and in continuation reasoning exactly as in the proof of Theorem 3, we can get the desired conclusion. □
This work is supported by the National Natural Science Foundation of China (Grant No. 61170324), the Natural Science Foundation of Fujian Province of China (Grant No. 2010J01012), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant No. JA12324), and the National Defense Basic Scientific Research Program of China (Grant No. B1420110155).
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