The -Chlodowsky and -Szasz-Durrmeyer Hybrid Operators on Weighted Spaces

Abstract

The main aim of this article is to introduce a new type of -Chlodowsky and -Szasz-Durrmeyer hybrid operators on weighted spaces. To this end, we give approximation properties of the modified new -Hybrid operators. Moreover, in the weighted spaces, we examine the rate of convergence of the modified new -Hybrid operators by means of moduli of continuity. In addition, we derive Voronovskaja’s type asymptotic formula for the related operators.

1. Introduction

Polynomial approach and the classical approximation theory constitute a basic research area in applied mathematics. The development of the approximation theory plays an important role in the numerical solution of partial differential equations, data processing sciences, and many other disciplines. For example, it is widely used in geometric modelling in the aerospace and automotive industries to calculate approximate values with basic functions. Works in this field go back to the 18th century and still continue as a powerful tool in scientific calculations. Furthermore, it is used in civil engineering projects to analyze the energy efficiency and earthquake resistance data of different types of buildings in thermography calculations and earthquake engineering.

Many generalized versions of these polynomials have been studied by many authors. Some of these studies are [1–13]. During the studies of approximation theory, many new operators and their generalizations were introduced with different spaces and variables. For example, on the interval , etc., analog type, Petree type, King’s type, and Weighted space type operators are introduced.

In this study, the analog type operator is defined. The studies regarding the analog type operator are as follows. First, the Bernstein polynomials were produced by Phillips [14]. When is used, the results are the same as for classical operators. However, new operators with different properties are obtained for . Gupta [15] introduced and analyzed the approach characteristics of -Durrmeyer operators. Gupta and Heping [16] identified the -Durrmeyer operators and estimated the rate of convergence for continuous functions with the help of the moduli of continuity. In [2, 17], Mahmudov defined the King-type -Szász operators. He obtained the rate of convergence on weighted spaces and a Voronovskaya-type theorem for these operators. Some other studies based on classical theory are [17–24].

In light of the above studies, the following result has been obtained. For a real-valued function, defined on the interval , the operatorsfor the general operator kernelshave been studied by many authors. On the contrary, the operatorhas not been studied yet.

Therefore, we introduced the following operator:which is -Chlodowsky and -Szasz-Durrmeyer hybrid operators on weighted spaces. Here, and are defined as in (2)-(3), , , , and is an increasing and positive sequence with properties and . In this article, we intend to study the approximation properties of the operator . We produced our study by making use of [25].

The important terms of q-analysis which are used in this paper are given below, see [26, 27],

Given value of and , we define the -integer by

The -factorial is defined byfor ; we define the -binomial coefficients by

The -binomial can be written in the following forms:

Exponential function has two -analogs, see [26, 27]:where . It is easily observed that

The definite Jackson integrals and the improper integrals of the function are defined by

The series on the right-hand side in (12) is guaranteed to be convergent if the function has the property in a right neighborhood of for , .

The Gamma functions are defined in two ways. These are

For every and , one haswhere . Especially, for any positive integer ,see [28].

The present note deals with the study of the -Chlodowsky and -Szasz-Durrmeyer hybrid operators on weighted spaces. Firstly, we estimate the moments for the operators. We also study the rate of convergence for these operators . Furthermore, definitions and some properties for weighted spaces are given. We guess the order of approximation by weighted Voronovskaya-type theorem.

2. Estimation Moments

Here, we will prove for . By the definition of the Gamma function in (15), we havewhere, for ,for ,and for ,

Lemma 1. For , , and , we have

Proof. Using (19), we obtainUsing (20) and , we obtainUsing (20) and , we obtainwhich completes the proof.

Lemma 2. Let , , and , and we have

Proof. By Lemma 1, we havewhich completes the proof.

By simple calculations, we obtainhere; since will be and , the following equation is written:where appears to be constrained in the above inequality, and this result is available in [25]. Likewise, A. İzgi obtained the following result in [25]:

3. Approximation of in Weighted Spaces

In this section, we use Gadjiev’s Korovkin-type theorems on the weighted spaces [4, 29]. Let be the set of all functions over the real line, , where such thatwhere is a positive constant depending on the function . Now, let

It is clear that , where is the linear normed space with the norm

Let be an increasing and positive sequence with features

Lemma 3. defined in (5) is a sequence of positive and linear operators that move to . So, there isfor on .

Proof. Taking advantage of equations (22) and (24), we haveTherefore, is uniformly bounded on , sinceunder the condition in (36) that completes the proof.

Theorem 1. Let . Then,

Proof. From Lemma 1, we haveaccording to (36), which completes the proof.

4. Main Results

Here, we estimate the rate of approximation of the hybrid operators. The following theorem gives the rate of approximation of the sequence of operators in terms of moduli of continuity of a function . For , the moduli of continuity is defined as follows:where .

The weighted moduli of continuity is defined as follows:

It is seen that they provide the characteristics of the continuity module. In what follows,for every . A similar definition can be found in [5, 25].

Theorem 2. Let , and we havewhere is the positive constant for and .

Proof. From the well-known features of modulus of continuity, we haveThen, is a linear-positive operator, and we writefrom Lemmas 1-2, (31), and (32), we obtainand using Cauchy–Schwartz inequalityWe can write by (32)-(33) thatwhere .
As a resultwherewhich completes the proof.
Now, we prove a Voronovskaja-type result for the operators.

Theorem 3. Let , , and be fixed, and we have

Proof. By Taylor formula for and ,where andBy implementing L’Hospital’s rule twice,Then, and . As a result, we can writeNow, let us use the Cauchy–Schwarz inequality in the last term of the last equation:Since and as , applying Lemma 2 and (31) and (32),As a resultfrom (33), and we havewhich completes the proof.

5. Conclusion

In this paper, the approximation properties and rate of convergence of -Chlodowsky and -Szasz-Durrmeyer hybrid operators in weighted spaces are investigated.

For further research in this topic, it would be interesting to study whether the quality of the approximation pythagorean fuzzy set operators, -statistical convergence operators, and -complex operators directly influence the quality of the approximation of the characteristics. Some studies can be used for future research, such as [30–34].

Data Availability

All data generated or analyzed during this study are included in this published article. They are cited at relevant places within the text as references.

Conflicts of Interest

The authors do not have any conflicts of interest.

Authors’ Contributions

All authors contributed equally to this paper.