Rate of Approximation for Modified Lupaş-Jain-Beta Operators

The main intent of this paper is to innovate a new construction of modified Lupaş-Jain operators with weights of some Beta basis functions whose construction depends on σ such that σ0=0 and infx∈0,∞σ′x≥1. Primarily, for the sequence of operators, the convergence is discussed for functions belong to weighted spaces. Further, to prove pointwise convergence Voronovskaya type theorem is taken into consideration. Finally, quantitative estimates for the local approximation are discussed.


Introduction
In 1972, Jain [1] with the help of Poisson distribution introduced a famous linear positive operators as follows: where m ≥ 1, f defined on ½0, ∞Þ, and If we put α = 0 in (1), then it becomes Szász-Mirakyantype.
In 1995, Lupaş [2] introduced a sequence of linear positive operators; later on in 1999, it was modified by Agratini [3] as follows: and also discussed the Kantorovich and Durrmeyer variant of operator (1). In 2018, Tunca et al. [4] modified operator (3) in such a way that in the construction, authors take the negative subscript -1 of the Pochhammer symbol into consideration; due to this, the calculations become simpler in a remarkable degree just as In order to approximate Lebesgue integrable functions, the most important modifications are Kantorovich and Durrmeyer integral operators. The Durrmeyer variant of operator (1) is introduced by Tarabie [5] and Mishra and Patel [6] with some beta basis functions.
In 2011, Cárdenas-Morales et al. [7] defined the Bernstein-type operators by B m ð f oσ −1 Þoσ and also presents a better degree of approximation depending on σ. This type of approximation operators generalizes the Korovkin set from fe 0 , e 1 , e 2 g to fe 0 , σ, σ 2 g. The Durrmeyer variant of B m ð f oσ −1 Þoσ is defined in [8]. In 2014, Aral et al. [9] defined a similar modification of Szász-Mirakyan-type operators by using a suitable function σ.
Motivated by the above mentioned work very recently, Bodur [10] introduced a new modification of operator (4) by using a suitable function σ, which satisfies the following properties: ðσ 1 Þσ be a continuously differentiable function on ½0, ∞Þ ðσ 2 Þσð0Þ = 0 and inf x∈½0,∞Þ The new formulated operators are defined as for m ≥ 1, x ≥ 0, and suitable functions f defined on ½0, ∞Þ.
As we know, in order to approximate Lebesgue integrable functions, the most important modifications are Kantorovich and Durrmeyer integral operators. Motivated by the above mentioned Durrmeyer type generalizations of various operators and also from [11][12][13][14][15][16][17][18][19][20][21][22][23], in this paper, Durrmeyer-type modification of generalized Lupaş-Jain operators (5) by taking weights of some beta basis function is defined as follows: where m ∈ ℕ and b m,j ðϑÞ is defined as where βðm + 1, jÞ is the beta basis function and σ is a function satisfying the conditions ðσ 1 Þ and ðσ 2 Þ given above. The rest of the work is organized as follows: in the second section, moments and central moments for D α m,σ are calculated. In the third section, we study convergence properties of D α m,σ in the light of weighted space. In the fourth section, we obtain the order of approximation of new constructed operators associated with the weighted modulus of continuity. In the fifth section, we shall prove Voronovskaya-type theorem in quantitative form. These kinds of results are very useful to describe the rate of point-wise convergence. Finally, in the last section, we obtain some local approximation results related to K-functional.

Basic Results
In this section, we prove some lemmas for D α m,σ which are required to prove our main results.
m,σ be given by (6). Then for each x ≥ 0, m ∈ ℕ, and 0 ≤ α < 1, we have By using beta function and Lemma 2.1 in [4], it can be proved. So we omit it. Now, from the linearity of the operators D α m,σ , we can state Lemma 2.

Lemma 2.
For operators D α m,σ , we have the following properties: Here, we prove the convergence of D α m,σ by using weight function. Let λðxÞ be a function satisfying the conditions ðσ 1 Þ and ðσ 2 Þ given above. Also, let λðxÞ = 1 + σ 2 ðxÞ be a weight function and the weighted space is defined as follows: where M f is a constant which depends only on f , with the norm Journal of Function Spaces Also, we mention some subspaces of B λ ½0, ∞Þ as It is obvious that C * λ ½0,∞Þ ⊂ U λ ½0,∞Þ ⊂ C λ ½0,∞Þ ⊂ B λ ½0,∞Þ: We have the following results for the weighted Korovkintype theorems due to Gadjiev [24] Lemma 3. [24].
The positive linear operators T m , m ≥ 1 act from C λ ½0, ∞Þ to B λ ½0, ∞Þ if and only if the inequality holds, where M m > 0 is a constant depending on m.
Theorem 4 (see [24]). Let the sequence of positive linear operators T m , m ≥ 1 acting from C λ ½0, ∞Þ to B λ ½0, ∞Þ and satisfying Then, for each f ∈ C * λ ½0,∞Þ, we have Remark 5. Examining Lemma 1 based on the famous Korovkin theorem [25], it is clear that ðD α m,σ Þ m≥1 does not form an approximation process. Now, in order to obtain convergence properties, we replace the constant α by α m ∈ ½0, 1Þ such that lim m→∞ α m = 0: Theorem 6. Let 0 ≤ α m < 1 such that lim m→∞ α m = 0, and also, let D α m,σ be the sequence of positive linear operators. Then, for each function f ∈ C * λ ½0,∞Þ, we have Proof. From Lemma 1, we obtain Hence, by Theorem 4, we deduce

Rate of Convergence
In this part, we would like to determine the rate of convergence for D α m,σ by weighted modulus of continuity ω σ ð f ; δÞ which was introduced by Holhos [26] in 2008, as follows: where f ∈ C λ ½0,∞Þ, with the following properties: Theorem 7 (see [26]). Let T m : C λ ½0,∞Þ → B λ ½0,∞Þ be a sequence of positive linear operators with where the sequences a m , b m , c m , and d m converge to zero as m ⟶ ∞. Then for all f ∈ C λ ½0,∞Þ, where Theorem 8. Let 0 ≤ α < 1 such that lim m→∞ α m = 0, and also, let D α m,σ be the sequence of positive linear operators. Then for all f ∈ C λ ½0,∞Þ, we have

Journal of Function Spaces
Proof. We should calculate the sequences ða m Þ, ðb m Þ, ðc m Þ, and ðd m Þ, in order to apply Theorem 7. In light of Lemma 1, clearly, we have Finally, Thus, Theorem 7 is satisfied. Hence, we have the desired result.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
We declare that there is no conflict of interest.