On approximation properties of a new construction of Baskakov operators

Abstract

The purpose of this research is to construct sequences of Baskakov operators such that their construction consists of a function σ by use of two function sequences, $\xi _{n} $ ξ n and $\eta _{n} $ η n . In these operators, σ not only features the sequences of operators but also features the Korovkin function set $\lbrace 1,\sigma ,\sigma ^{2} \rbrace $ { 1 , σ , σ 2 } in a weighted function space such that the operators fix exactly two functions from the set. Thereafter, weighted uniform approximation on an unbounded interval, the degree of approximation with regards to a weighted modulus of continuity, and an asymptotic formula of the new operators are presented. Finally, some illustrative results are provided in order to observe the approximation properties of the newly defined Baskakov operators. The results demonstrate that the introduced operators provide better results in terms of the rate of convergence according to the selection of σ.

A Note on New Construction of Meyer-König and Zeller Operators and Its Approximation Properties

“…The following inequality can be deduced readily if one uses the relation (19) in (25), that is to say,…”

“…Hereby, the most crucial feature of the defined Bernstein operator is that it fixes the set of {1, τ, τ 2 } instead of the standard Korovkin's test functions, {1, t, t 2 }. In this direction, in [13] Gamma type operators by Erençin et al,in [14] Bernstein-Durrmeyer type operators by Acar et al,in [15,16] Szász-Mirakyan types operators by Aral et al,and in [17] Lupaş type operators by Qasim et al,in [18] Bernsrtein-Chlodowksi types operators, in [19] Balazs types operators and in [20] Baskakov type operators by Usta have been introduced.…”

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].…”

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

“…In addition to these, Cárdenas‐Morales et al [23] showed that a new class of Bernstein‐type operators, which defined based on , preserves instead of classical Korovkin test functions, . With this motivation, different types of generalizations have been presented for another type of positive linear operators such as Bersntein–Durmeyyer‐type operators in Acar et al [24], Szász–Mirakyan‐type operators in Aral et al [25], Bernstein–Chlodowsky‐type operators in Usta [26], Balázs‐type operators in Usta [27], Baskakov‐type operators in Usta [28], Meyer–König and Zeller‐type operators in Cai et al [29], and Gamma‐type operators in Erençin and Raşa [30].…”