Abstract and Applied AnalysisVolume 2011 (2011), Article ID 101852, 13 pagesdoi:10.1155/2011/101852

Research Article

Direct and Inverse Approximation Theorems for Baskakov Operators with the Jacobi-Type Weight

Guo Feng1,2

¹School of Mathematics and Information Engineering, Taizhou University, Zhejiang, Taizhou 317000, China²School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received 2 May 2011; Accepted 10 September 2011

Academic Editor: Ruediger Landes

Copyright © 2011 Guo Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce a new norm and a new K-functional . Using this K-functional, direct and inverse approximation theorems for the Baskakov operators with the Jacobi-type weight are obtained in this paper.

1. Introduction and Main Results

Let be a function defined on the interval . The operators are defined as follows: where which were introduced by Baskakov in 1957 [1]. Becker [2] and Ditzian [3] had studied these operators and obtained direct and converse theorems. In [4, 5] Totik gave a result: if , then if and only if , where and k is a positive constant. We may formulate the following question: do the Baskakov operators have similar property in the case of weighted approximation with the Jacobi weights? It is well known that the weighted approximation is not a simple extension, because the Baskakov operators are unbounded for the usual weighted norm . Xun and Zhou [6] introduced the norm and have discussed the rate of convergence for the Baskakov operators with the Jacobi weights and obtained where , and is the set of bounded continuous functions on .

In this paper, we introduce a new norm and a new K-functional, using the K-functional, and we get direct and inverse approximation theorems for the Baskakov operators with the Jacobi-type weight.

First, we introduce some useful definitions and notations.

Definition 1.1. Let denote the set of bounded continuous functions on the interval , and let where , and .Moreover, the K-functional is given by where .We are now in a position to state our main results.

Theorem 1.2. If , then

Theorem 1.3. Suppose . Then the following statements are equivalent:

Throughout this paper, denotes a positive constant independent of , and which may be different in different places. It is worth mentioning that for , we recover the results of [6].

2. Auxiliary Lemmas

To prove the theorems, we need some lemmas. By simple computation, we have or

Lemma 2.1. Let . Then

Proof. We notice [7] For , the result of (2.3) is obvious. For , there exists , such that . Using Hölder's inequality, we have For or , the proof is similar to that of (2.5). Thus, this proof is completed.

Lemma 2.2. Let . Then

Proof. By Lemma 2.1, we get

Lemma 2.3. Let . Then

Proof. For ; using (2.1) and Lemma 2.1, we have For , by (2.2), we get Note that for , one has the following inequality [7] Applying Hölder’s inequality and Lemma 2.1, we have Note that for , one has . Hence, Combining (2.9)–(2.14), we get Thus, The proof is completed.

Lemma 2.4. Let , and . Then

Proof. (1) For the case or , if , using (2.1) and Lemma 2.1, we have (i)If , (ii)If , Combining (2.19)–(2.21), we have Thus, If , we have By using the method similar to that of (2.19)–(2.23), it is not difficult to obtain the same inequality as (2.23).(2) For the case , the proof is similar to that of case (1) and even simpler. Therefore the proof is completed.

Lemma 2.5 (see [8, page 200]). Let be an increasing positive function on , the inequality holds true for . Then one has

3. Proofs of Theorems

3.1. Proof of Theorem 1.2

Proof. First, we prove it as follows.(i)If , then (ii)If , then The Proof of (3.1)In fact, (i) for , since , we have If , we get If , we have () If , since , we have Combining (3.4), (3.5) and (3.6), we obtain (3.1). The proof of (3.2)If , by (9.5.10) and (9.6.3) of [7], using the Cauchy-Schwarz inequality and the Hölder inequality, we obtain If , by (2.3), we get , and using the Cauchy-Schwarz inequality and the Hölder inequality, we have Combining (3.7) and (3.8), we obtain (3.2).Next, we prove Theorem 1.2. For , if , by (3.1), we have If , by (3.2), we get Therefore, for , by Lemma 2.2 and (3.9), (3.10), and the definition of , we obtain Taking the infimum on the right-hand side over all , we get This completes the proof of Theorem 1.2.

3.2. Proof of Theorem 1.3

Proof. By Theorem 1.2, we know . Now,we will prove . In view of (1), we get By the definition of K-functional, we may choose to satisfy Using Lemma 2.2 and Lemma 2.3, we have Taking the infimum on the right-hand side over all , we get By Lemma 2.4, we get Leting , we get This completes the proof of Theorem 1.3.

Acknowledgments

The author would like to thank Professor Ruediger Landes and the anonymous referees for their valuable comments, remarks, and suggestions which greatly help us to improve the presentation of this paper and make it more readable. The project is supported by the Natural Science Foundation of China (Grant no. 10671019).

References

V. A. Baskakov, “An example of a sequence of linear positive operators in the spaces of continuous functions,” Doklady Akademii Nauk SSSR, vol. 112, pp. 249–251, 1957.

M. Becker, “Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces,” Indiana University Mathematics Journal, vol. 27, no. 1, pp. 127–142, 1978.

Z. Ditzian, “On global inverse theorems of Szász and Baskakov operators,” Canadian Journal of Mathematics, vol. 31, no. 2, pp. 255–263, 1979.

V. Totik, “Uniform approximation by Baskakov and Meyer-König and Zeller operators,” Periodica Mathematica Hungarica, vol. 14, no. 3-4, pp. 209–228, 1983.

V. Totik, “Uniform approximation by exponential-type operators,” Journal of Mathematical Analysis and Applications, vol. 132, no. 1, pp. 238–246, 1988.

P. C. Xun and D. X. Zhou, “Rate of Convergence for Baskakov Operators with Jacobi-Weights,” Acta Mathematics Applicatae Sinica, vol. 18, pp. 127–139, 1995 (Chinese).

Z. Ditzian and V. Totik, Moduli of Smoothness, vol. 9, Springer, New York, NY, USA, 1987.

W. Chen, Approximation Theory of Operators, Xiamen University Publishing House, Xiamen, China, 1989.