Weighted Simultaneous Approximation of the Linear Combinations of Baskakov Operators

Abstract

In this paper, the approximation property of the linear combinations of Baskakov operators is investigated by using a Jacobi weight function. Moreover, both the positive and inverse theorems for the weighted simultaneous approximation of the linear combinations of Baskakov operators are proved.

1. Introduction

The functional analysis methods are widely used to study the approximation theory since the 20th century. Due to the combination of functional analysis methods and classical analysis techniques, such approximation theories developed quickly and had formed a theoretical system.

It is well known that the positive and inverse theories are the most significant problems in the operator approximation theorem. In 1972, some pioneering works had been done by H. Berens and G. Lorentz [1], which lead to a hot topic in related research fields. In the meantime, many kinds of approximation tools had been proposed and widely used in practice, such as smooth modulus and K-functional. In particular, the smooth modulus introduced by Ditzian [2] in 1994 contains the related results by using the classic smooth modulus and the Ditzian-Totik modulus. Moreover, such kinds of smooth modulus are usually used to construct an optimal polynomial to approximate a complicated function [3–5].

For any , the corresponding Baskakov operator [6] is defined aswhere , . Linear combination of the Baskakov operator [7] is defined as follows:where and are related constants which satisfy the following conditions:

Let be the supremum or infimum norm, ,where

Hence, the weighted K-functional can be defined as follows:and the corrected weighted K-functional is defined as

According to [8], one can obtain the relationship between the smooth modulus and K-functional

Note that and , and we have

Let and , and introducing an auxiliary operatorthen the linear combination of the auxiliary operator can be defined as follows:where and are related constants which satisfy (3). Note that is a bounded operator, and . Assuming , we have

The Baskakov operator has been studied in many research studies by using a lot of deep methods [9–13]. In previous work [14, 15], some pointwise results of the Baskakov operator with weighted approximate were obtained.

From the literature [16], some preexisting results are given as follows.

Theorem 1. Let , , then we have

Theorem 2. Let , then we have

According to [17], the following theorems hold.

Theorem 3. Let , , then we obtain

Theorem 4. Let , then we have

According to [18], we get the following theorems.

Theorem 5. Let , , then we have

Theorem 6. Let , then we get

Although there are many research studies about Baskakov operator approximation [19], we can further improve these theories. On the basis of the aforementioned research literature, we firstly applied the equivalence relation between the weighted smooth modulus and K-functional to explore the simultaneous approximation of the linear combination of the Baskakov operator with Jacobi weight. And we obtained both the positive and inverse theorems for the weighted simultaneous approximation of the linear combinations of Baskakov operators. Thereby, we unite and expand the results about the existing smooth modulus , , and , where C is a constant.

Then, we will introduce some concepts and properties in Section 2. In Section 3, we will prove the pros-theorem of simultaneous approximation of the linear combination of the Baskakov operator. The proof of the cons-theorem will be given in Section 4. Finally, we will analyze the equivalence theorem of approximation.

2. Concepts and Properties

The weight function that we used has the following properties.

Property 1. Let then we haveMoreover, weighted smooth modulus and weighted main part smooth modulus have the following connection.

Property 2. .In order to prove the cons-theorem, we need to introduce a new K-functional and put some symbols firstly. Let , thenA new K-functional is defined: and a new smooth modulus is defined:

3. Pros-Theorem of Simultaneous Approximation of the Linear Combination of the Baskakov Operator

3.1. Auxiliary Lemmas

Lemma 1 (see [18]). Let then we have

Lemma 2 (see [15]). If , then

According to Lemma 2, we obtain the following corollaries.

Corollary 1. When , we have

Lemma 3 (see [17]). If , then

Corollary 2. If , then

Proof. It follows from (30) that we get

Lemma 4. (1)If , then(2)If then

Proof. (1)When , the conclusion is established obviously. For the case of , let , then we have For , , so when x in , f increases. Hence, we have ; for , we obtain , and then , combined with inequality ; then, we get(2)Due to , thereby we haveand combined with inequality , then we getand then combined with inequality , the proposition is established.

Lemma 5. Let , , , then we have

Proof. (1)If , and using Lemma 4, we get In the proof process above, we used the Hőlder inequality and the conclusion of formula (26). i.e., Similarly, we obtain(2)When , by using the Hőlder inequality, (26), (28), and (29) and the conclusion of [15], we can get Therefore, and the proposition is established.

3.2. Positive Theorem

Theorem 7. Let , , then we get

Proof. For , let Taylor expand in x,It follows from (25) that we obtainThen, for and all of we have

Theorem 8. Let , and if , then we have

Proof. From Theorem 7, we obtain

4. Cons-Theorem of Simultaneous Approximation of the Linear Combination of the Baskakov Operator

4.1. Base Lemmas

Lemma 6. If , we have

Proof. (1)When , we have , and then Firstly, considering the first term of , the following part is similar, and by using the methods of [17] and the Hǒlder inequality, we get Therefore,(2)When , similar to the proof of [8], we havewhere is the situation when , and is the last situations. Let , s.t. , by the Hǒlder inequality, then we haveThen, we obtain ; therefore,

Lemma 7. Supposing that , we get

Proof. Supposing that satisfies , and then by reference [8], we havewhere , and when , we getWhen , using the method of [8], we can obtainSo, using the Hőlder inequality and Lemma 2, we getbut , and thereby the conclusion is established.

Lemma 8 (see [20]). Supposing , then we have

4.2. Cons-Theorem

Theorem 9. Supposing , then we have

Proof. By the definition of , selecting a suitable , which satisfies the following formulaby the known conditionwe haveBy using Lemmas 6 and 7, we obtainby using the Berens-Lorentz lemma [8], we get . Especially, when , we can obtain , i.e., . By contrast, when , for , by using Lemma 8, we getThen, , and thereby

5. Equivalence Theorem

Theorem 10. Supposing then the following propositions are equivalent:

Proof. It follows from Theorem 8, and we know (71)(72) is established. And from Theorem 9, (72)(71) is also established. Finally, by reference [8], we know (72)(73) is established, too.

Data Availability

Mathematical deduction is the main method in this paper, and there are no experimental data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by the Science and Technology Project of Department of Education of Jiangxi Province (No. GJJ190551, GJJ180565, and GJJ180583), the National Natural Science Foundation of China (No. 41662018), and the Doctoral Research Startup Project of Jinggangshan University (No. JZB17002).