Abstract
The purpose of this paper is to introduce a modification of q-Dunkl generalization of exponential functions. These types of operators enable better error estimation on the interval [frac{1}{2},infty) than the classical ones. We obtain some approximation results via a well-known Korovkin-type theorem and a weighted Korovkin-type theorem. Further, we obtain the rate of convergence of the operators for functions belonging to the Lipschitz class.
Highlights
IntroductionIntroduction and preliminariesIn , Bernstein [ ] introduced the following sequence of operators Bn : C[ , ] → C[ , ] defined by n Bn(f ; x) =n xk( – x)n–kf k , x ∈ [ , ] k n ( . ) k=for n ∈ N and f ∈ C[ , ]
Introduction and preliminariesIn, Bernstein [ ] introduced the following sequence of operators Bn : C[, ] → C[, ] defined by n Bn(f ; x) =n xk( – x)n–kf k, x ∈ [, ] k n ( . ) k=for n ∈ N and f ∈ C[, ]
The first q-analogue of well-known Bernstein polynomials was introduced by Lupaş by applying the idea of q-integers [ ]
Summary
Introduction and preliminariesIn , Bernstein [ ] introduced the following sequence of operators Bn : C[ , ] → C[ , ] defined by n Bn(f ; x) =n xk( – x)n–kf k , x ∈ [ , ] k n ( . ) k=for n ∈ N and f ∈ C[ , ]. In , Bernstein [ ] introduced the following sequence of operators Bn : C[ , ] → C[ , ] defined by n Bn(f ; x) =. In , for x ≥ , Szász [ ] introduced the operators Sn(f ; x) = e–nx ∞ (nx)k f k!. In the field of approximation theory, the application of q-calculus emerged as a new area. The first q-analogue of well-known Bernstein polynomials was introduced by Lupaş by applying the idea of q-integers [ ]. In , Phillips [ ] considered another q-analogue of the classical Bernstein polynomials. Many authors introduced q-generalizations of various operators and investigated several approximation properties [ – ]
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