Abstract and Applied Analysis

Volume 2011 (2011), Article ID 101852, 13 pages

http://dx.doi.org/10.1155/2011/101852

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

^{1}School of Mathematics and Information Engineering, Taizhou University, Zhejiang, Taizhou 317000, China^{2}School 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*

*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. Proofs of Theorems**3.1. Proof of Theorem 1.2*

*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*

*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*

*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*

*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. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Ditzian, “On global inverse theorems of Szász and Baskakov operators,”
*Canadian Journal of Mathematics*, vol. 31, no. 2, pp. 255–263, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Totik, “Uniform approximation by Baskakov and Meyer-König and Zeller operators,”
*Periodica Mathematica Hungarica*, vol. 14, no. 3-4, pp. 209–228, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Totik, “Uniform approximation by exponential-type operators,”
*Journal of Mathematical Analysis and Applications*, vol. 132, no. 1, pp. 238–246, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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). View at Google Scholar - 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.

*
*