A new construction of Lupaş operators and its approximation properties

Abstract

The aim of this paper is to study a new generalization of Lupaş-type operators whose construction depends on a real-valued function ρ by using two sequences $u_{m} $ u m and $v_{m}$ v m of functions. We prove that the new operators provide better weighted uniform approximation over $[0,\infty )$ [ 0 , ∞ ) . In terms of weighted moduli of smoothness, we obtain degrees of approximation associated with the function ρ. Also, we prove Voronovskaya-type theorem, quantitative estimates for the local approximation.

On approximation properties of a new construction of Baskakov operators

“…However, the most significant feature of this newly introduced Bernstein operator is that it preserves the set of {1, τ , τ 2 } instead of the classical Korovkin's test functions {1, x, x 2 }. Accordingly, in [1] Bernstein-Durrmeyer operators by Acar et al, in [4,5] Szász-Mirakyan operators by Aral et al, in [26] Bernstein-Chlodowksi operators by Usta, in [27] Balázs operators by Usta, and in [24] Lupaş operators by Qasim et al have been introduced. In this respect, Patel et al made a similar study for the Baskakov operators and obtained the following operator:…”

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

“…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.…”