Weighted Simultaneous Approximation of the Linear Combinations of Baskakov Operators

0e 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 f ∈ C[0, +∞], the corresponding Baskakov operator [6] is defined as


Introduction
e 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][4][5].
Linear combination of the Baskakov operator [7] is defined as follows: where n i and C i (n) are related constants which satisfy the following conditions: (a) n � n 0 < n 1 < . . . < n r− 1 ≤ Cn, (3) 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 f (s) ∈ C[0, ∞) and s ∈ N, and we have Let g ∈ C[0, ∞) and s ∈ N, and introducing an auxiliary operator then the linear combination of the auxiliary operator can be defined as follows: where n i and C i (x) are related constants which satisfy (3). Note that V n,s is a bounded operator, and V n, e Baskakov operator has been studied in many research studies by using a lot of deep methods [9][10][11][12][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.
According to [17], the following theorems hold.
According to [18], we get the following theorems.
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. ereby, we unite and expand the results about the existing smooth modulus 2 Complexity en, 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. e proof of the cons-theorem will be given in Section 4. Finally, we will analyze the equivalence theorem of approximation.

Concepts and Properties
e weight function w(x) that we used has the following properties.
(20) Moreover, weighted smooth modulus and weighted main part smooth modulus have the following connection.
In order to prove the cons-theorem, we need to introduce a new K-functional and put some symbols firstly. Let 0 < λ < 1, 0 < α < s, then A new K-functional is defined: and a new smooth modulus is defined:
According to Lemma 2, we obtain the following corollaries.
Proof. It follows from (30) that we get

Base Lemmas
Firstly, considering the first term of r j�0 (− 1) r r j f((k + r − j)/n + s i�1 u i )du 1 . . . du s , the following part is similar, and by using the methods of [17] and the Hǒlder inequality, we get where I 1 is the situation when k � 0, and I 2 is the last situations. Let p ∈ N, s.t. i/2p < 1 − (α − s)(1 − λ)/2, by the Hǒlder inequality, then we have