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

and Applied Analysis 3 Throughout this paper, M denotes a positive constant independent of x, n, and f which may be different in different places. It is worth mentioning that for λ 1, we recover the results of 6 . 2. Auxiliary Lemmas To prove the theorems, we need some lemmas. By simple computation, we have V ′′ n ( f ;x ) n n 1 ∞ ∑ k 0 vn 2,k x ( f ( k 2 n ) − 2f ( k 1 n ) f ( k n )) 2.1


Introduction and Main Results
Let f be a function defined on the interval 0, ∞ .The operators V n f; x are defined as follows: 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: 2α , where h > 0 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 f w wf ∞ .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 w x x a 1 x −b , 0 < a < 1, b > 0, 0 < α < 1, and C B 0, ∞ is the set of bounded continuous functions on 0, ∞ .
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 C B 0, ∞ denote the set of bounded continuous functions on the interval 0, ∞ , and let where ϕ x x 1 x , w x x a 1 x −b , x ∈ 0, ∞ , 0 ≤ a < λ ≤ 1, and b ≥ 0.Moreover, the K-functional is given by where We are now in a position to state our main results.
Then the following statements are equivalent: Abstract and Applied Analysis 3 Throughout this paper, M denotes a positive constant independent of x, n, and f which may be different in different places.It is worth mentioning that for λ 1, we recover the results of 6 .

Auxiliary Lemmas
To prove the theorems, we need some lemmas.By simple computation, we have Proof.We notice 7

2.4
For c 0, d 0, the result of 2.3 is obvious.For c > 0, d / 0, there exists m ∈ Z, such that 0 < −2d/m < 1.Using H ölder's inequality, we have For c > 0, d 0 or c 0, d / 0, the proof is similar to that of 2.5 .Thus, this proof is completed.
Proof.By Lemma 2.1, we get

2.10
Note that for x ∈ E n , one has the following inequality 7

2.11
Applying H ölder's inequality and Lemma 2.1, we have

2.17
The proof is completed.

2.24
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 λ 1, a 0, 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 Ω t be an increasing positive function on 0, a , the inequality r > α holds true for h, t ∈ 0, a .Then one has

Proof of Theorem 1.2
Proof.First, we prove it as follows. i

3.2
Abstract and Applied Analysis 9 The Proof of 3.1 In fact, i for k 0, since x ∈ E c n , we have
The proof of 3.2 If b − 2 λ ≤ 0, by 9.5.10 and 9.6.3 of 7 , using the Cauchy-Schwarz inequality and the H ölder inequality, we obtain

3.7
If b − 2 λ > 0, by 2.3 , we get V n 1 t b−2 λ ; x ≤ M 1 x b−2 λ , and using the Cauchy-Schwarz inequality and the H ölder inequality, we have
Next, we prove Theorem 1.2.For g ∈ D, if x ∈ E c n , by 3.1 , we have

3.9
Abstract and Applied Analysis 11 If x ∈ E n , by 3.2 , we get

3.10
Therefore, for f ∈ C 0 a,b,λ , g ∈ D, by Lemma 2.2 and 3.9 , 3.10 , and the definition of

3.11
Taking the infimum on the right-hand side over all g ∈ D, we get

3.12
This completes the proof of Theorem 1.2.

3.14
Using Lemma 2.2 and Lemma 2.3, we have

3.15
Taking the infimum on the right-hand side over all g ∈ D, we get

3.18
This completes the proof of Theorem 1.3.