Abstract
Because of their simple form and high quality, linear arithmetical operators, particularly linear positive operators, are very popular. The number of in-depth studies on linear operator approximation is extensive, and the majority of the published material falls into four categories: types and structures of operators, approximation of operators, order of approximation and the converse theorem of operators, and operator saturation. The Baskakov operator will be transformed in the same way to explore its approximation characteristics as well as the approximation theorem and converse theorem.
1. Introduction and Background
1.1. Background
The famous Korovkin theorem [1], back in 1952, was a great boost to the studies of the positive linear arithmetic approximation. The results of the studies focusing on the problems with the order of approximation and the saturation of linear arithmetic were concluded in Devore’s book in 1972 [2]. In that same year, Berens and Lorentz [3] solved the converse theorem of the approximation problem of the Bernstein operator, and the converse theorem and its equivalence theorems have become a research hotspot since then. The subsequent development of the modulus of smoothness and the Korovkin-functionals, as well as some other qualified linear arithmetic, drew a large number of research studies and applications. Ditzian and Totik’s paper from 1987 [4] detailed the key findings.
Many academics have developed tweaks and revisions to various well-known operators to speed up the approximation process [5]. Integral transform, generalisation, weighting, combinatorics, and applying probability approaches are some of the methods for making such alterations and corrections [6].
We know that the Bernstein operator is linear, so there is
Here, . However, the Bernstein Operator does not have its retention with quadratic function,
In order to keep the quadratic function constant, a transformation was applied to the Bernstein operator in the reference literature [7].
Here,
Thus,
Other literature [8, 9] have done similar transformations to the Meyer-König and Zeller operators; we will perform the same transformation to the Baskakov operator to study its approximation properties, the approximation theorem, and the converse theorem.
1.2. Modulus of Smoothness and the K-Functional
Assume (asquare-integrable function on , written as ), , , so the Rth order modulus of smoothness is defined as
With
When in specific, it is called the classical modulus of smoothness; and when 1, it is called the Ditzian–Totik modulus of smoothness. The Rth order K-functional of is defined as
Here, is defined to be a sum set of absolute continuous functions on . Based on our reference literature [10], we know is equivalent to , which means there exists a constant , so that when :
Note: when , we use ; ; for convenience.
1.3. Baskakov Operator
For , the famous Baskakov operator is defined as
Where , = . Calculation shows
This demonstrates that the Baskakov operator preserves the function’s linearity but not its quadratic function [11]. There have been several in-depth studies on the approximation of the Baskakov operator, the results of which are documented in the literature [12–19]. When we apply the procedure described in [2] to the Baskakov operator, we can easily get an approximation result using the first-order classical modulus of smoothness as follows:
Theorem 1. If , then
Literature [19] used the generalized modulus of smoothness to study the approximation theorem and its reverse theorem of and has gotten
Theorem 2. Let , , then
Theorem 3. Let , ; , then
Here, Theorems 2 and 3 have unified the results of the classic modulus of smoothness and the Ditzian–Totik modulus correspondingly.
1.4. The Structure of the Modified Baskakov Operator
Literature [4] has addressed the linear combination to increase the order of approximation of the Baskakov operator, and this thesis will execute the following adjustment to the Baskakov operator:where keeps the same,
Obviously, Here we know that is linear positive operator on .
This means this operator remains constant to function 1 and .
1.5. Conclusion
For the modified Baskakov operator , we have the following conclusions:
Theorem 4. Assume ,and there exists t, which let
So,where , (any closed interval).
Theorem 5. Assume , , so
Where is the same with expression (16)
Theorem 6. Assume , , so
Theorem 7. Assume , , , so
The approximation features of the modified Baskakov operator will be discussed in Chapter II (Theorems 4 and 5). The approximation theorem and its reverse theorem will be discussed in Chapter III (Theorems 6 and 7).
Let us restate the important notations that will be used:(1) means there is constant C that makes ;(2);(3) =: , ;(4) means the positive integer set, ;(5) is defined as an absolute continuous function space on ;(6) means a constant, it refers to different values according to the situation.
2. The Approximation of the Modified Baskakov Operator
First of all, let us introduce the Korovkin theorem built by Ditzian for the local convergence of linear positive operators. Let or , and . Written as follows:where is a positive constant related to .
Theorem 8 (see [20]). Let be a linear positive operator sequence on , and applies to the following conditions:(1)For , there’s ,(2).
Then, for every , there is
Apply this theorem to operator , and we can get
Theorem 9. Let , and there exists that makes: ,
Thus, there iswhere , (any closed interval).
Proof. Make , then there isTherefore, a recurrence relation existsTherefore,Written asSo,When , there isSubstitute this back into , we getTherefore,where is a positive constant dependent on . Please note thatSo for , there isTake the following values: ( is any determinate, nonnegative integer), then for any , based on the Schwarz inequality, we can get:Thus,From Theorem 8, we know that for and , there isTheorem proven.
Now assume , then for any positive integer , there isNote that , thus we have
Theorem 10. Assume , , so
Note: because
Based on this, we can tell the operator has a better performance in the order of approximation than the classic Baskakov operator . In fact,is equivalent withbecausewhich is
Thus, we can get
3. Theorem and the Converse Theorem of the Modified Baskakov Operator
3.1. Theorem of Approximation
To build the approximation theorem of the operator, we need to introduce a few following theorems first:
Lemma 1. Assume , then .
Proof. We note thatWhen , there iswhen , there isLemma proven.
Lemma 2. Assume g , thenwhere = .
Proof. Fromwe can getNow we estimate
When , there isWhen , there isTherefore,The first item within the bracket in the above equation is as follows:The second item within the bracket in the above equation is as follows:andWhen ,When ,Thus, we haveSubstitute back to (58), we getSubstitute equation (62) and the result of Lemma 1 into (57), we getSubstitute this equation and (56) back into (55) we getThus, Lemma 2 is proven.
With Lemma 1 and Lemma 2, we can build the approximation theorem of operator under the generalized modulus of smoothness as follows:
Theorem 11. Assume , , then
Proof. Based on the definition of , we can select the that makesApply Lemma 2, we getTheorem proven.
3.2. The Reverse Theorem of Approximation
Let us now construct the opposite theorem of approximation of the operator’s equivalent form . First and foremost, certain important theorems must be introduced in order to prepare.
Lemma 3. Assume , , then
Proof. When ,Thus, there isNow we estimate
When ,When ,So we can getFor the summarized equation in the square bracket above, based on the Holder inequality, we getFor the second item in the above square bracket, because 1, so there isFrom equations (73), (74), and (75), we getBecause when , , so,Thus,From Lemma 2, we haveWhen ,For any ,So when , there isWhen , based on equation (77), there isAfter all, we getIn summary,Lemma 3 proven.
Lemma 4. Assume , ,
Proof. UseWe can getThe last inequality has used the result of (84), and now Lemma 4 is proven.
Lemma 5. For , , , there is
Proof. When , it is clear that the equation holds. Now we move to the proof for . For , the Holder inequality gives us the following:
When , , there isso
Theorem 12. Assume , , ,
Proof. Theorem 11 gives us its sufficiency directly, so now we try to prove its necessity. Since is a convex function, we can getWe can select a that satisfiesApply, Lemmas 3–5From equations (93) and (95), we haveFor any fixed and all , use a sufficiently great that makes the following equation stand:Now we haveThus,Use lemma Lorenz-Hermann (literature [20] P199), then we can getTheorem 12 proven.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.