Difference of Some Positive Linear Approximation Operators for Higher-Order Derivatives

Hari Mohan Srivastava,Vijay Gupta,Ana Maria Acu

doi:10.3390/sym12060915

Hari Mohan Srivastava, Vijay Gupta

Open Access

https://doi.org/10.3390/sym12060915

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Abstract

In the present paper, we deal with some general estimates for the difference of operators which are associated with different fundamental functions. In order to exemplify the theoretical results presented in (for example) Theorem 2, we provide the estimates of the differences between some of the most representative operators used in Approximation Theory in especially the difference between the Baskakov and the Szász–Mirakyan operators, the difference between the Baskakov and the Szász–Mirakyan–Baskakov operators, the difference of two genuine-Durrmeyer type operators, and the difference of the Durrmeyer operators and the Lupaş–Durrmeyer operators. By means of illustrative numerical examples, we show that, for particular cases, our result improves the estimates obtained by using the classical result of Shisha and Mond. We also provide the symmetry aspects of some of these approximations operators which we have studied in this paper.

Highlights

The main goal of this paper is to extend the above result for the positive linear operators that have different fundamental functions
The studies of the differences of positive linear operators has become an interesting area of research in Approximation Theory
The results of the type which we have presented here were obtained for a class of positive linear operators constructed with the same fundamental functions

Summary

Applications of Theorem 2

As applications of the Theorem 2, we give estimates of the differences between some of the most used positive linear operators in Approximation Theory. The considered examples involve the Baskakov type operators, the Szász–Mirakyan type operators, and the Durrmeyer type operators. We show for the Durrmeyer type operators that, in some particular cases, our result improves the estimates obtained by using the classical result of Shisha and Mond [26]

Difference Between the Baskakov and the Szász–Mirakyan Operators

Difference Between the Baskakov and the Szász–Mirakyan–Baskakov Operators

Difference Between the Baskakov and the Szász–Mirakyan–Kantorovich Operators

Difference of Two Genuine-Durrmeyer Type Operators

Difference of the Durrmeyer Operators and the Lupaş–Durrmeyer Operators

Conclusions