On the order of approximation by modified summation-integral-type operators based on two parameters

Abstract

In this article, we the study generalized family of positive linear operators based on two parameters, which are a broad family of many well-known linear positive operators, e.g., Baskakov-Durrmeyer, Baskakov-Szász, Szász-Beta, Lupaş-Beta, Lupaş-Szász, genuine Bernstein-Durrmeyer, and Pǎltǎnea. We first find direct estimates in terms of the second-order modulus of continuity, then we find an estimate of error in the Ditzian-Totik modulus of smoothness. Then we discuss the rate of approximation for functions in the Lipschitz class. Furthermore, we study the pointwise Grüss-Voronovskaja-type result and also establish the Grüss-Voronovskaja-type asymptotic formula in quantitative form.

Jain's operator: A new construction and applications in approximation theory

“…After the operators are introduced, various generalizations of these operators are also discussed. In recent years, integral-type generalizations known as Durrmeyer-type and Kantorovich-type generalizations have gained a lot of importance in literature [10][11][12][13]. Moreover, in 1972, Jain [14] introduced the following class of positive linear operators with the aid of a Poisson-type distribution for the function 𝑓 ∈ C(R + ), where R + ∶= (0, ∞) as…”

“…After the operators are introduced, various generalizations of these operators are also discussed. In recent years, integral‐type generalizations known as Durrmeyer‐type and Kantorovich‐type generalizations have gained a lot of importance in literature [10–13].…”

On the Approximation by Bivariate Szász–Jakimovski–Leviatan-Type Operators of Unbounded Sequences of Positive Numbers

“…There are many published articles related to these works, for example, those by Kajla et al [7], Mursaleen et al [8][9][10][11], Mohiuddine et al [12][13][14][15][16], Nasiruzzaman et al [6,[17][18][19][20][21][22], Özger et al [23]. For studies on Bernstein and Szász types operators involving the idea of Chlodowsky and Charlier polynomials, we refer to [24][25][26][27][28].…”

On a Family of Parameter-Based Bernstein Type Operators with Shape-Preserving Properties

“…Some of the most famous cases are the Schurer polynomials, Kantorovich polynomials, Stancu polynomials, q-Bernstein polynomials, Durrmeyer polynomials, Favard-Szász-Mirakyan operators, Baskakov operators, and numerous others [3][4][5][6]. Some of the recent advances could be traced in [7][8][9][10][11].…”