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

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.

Statistical convergence of martingale difference sequence via deferred weighted mean and Korovkin-type theorems

“…Moreover, if one considers the positive linear operators of the types Baskakov and Szász-Mirakyan [6], and Beta Szász-Mirakjan [16] in place of Bernstein polynomial B m ( f ; ϑ) in Example 3, then with the same algebraic test functions it will also satisfy the conclusion of Korovkin-type approximation theorem via our purposed mean for martingale difference sequences of random variables. Consequently, these operators are also valid for Theorem 4; however, it will not satisfy Theorem 3.…”

Modified Operators Interpolating at Endpoints

“…This section deals with estimates of the differences between classical operators and modified operators. Very recently, estimates of the differences of certain positive linear operators were obtained in [14][15][16][17][18].…”

Blending‐type approximation by Lupaş–Durrmeyer‐type operators involving Pólya distribution

“…Agrawal et al 18,19 constructed the Bézier variant and Generalized Boolean Sum of Lupaş–Durrmeyer type operators () involving Pólya distribution (also see previous studies 20,21 ). In Gupta et al, 22 general estimates of the differences of the linear positive operators with different fundamental basis and their derivatives were obtained. Srivastava et al 23 gave the Dunkl‐type modification of the q ‐Szász‐beta type operators and established uniform convergence on weighted spaces.…”