Research Article | Open Access
Direct and Inverse Approximation Theorems for Baskakov Operators with the Jacobi-Type Weight
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 . Becker  and Ditzian  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  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 .
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  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  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 , 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.
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).
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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.