Bernstein-type operators that reproduce exponential functions

Abstract

In this paper we recover a generalization of the classical Bernstein operators introduced by Morigi and Neamtu in 2000. Specifically, we focus on a sequence of operators that reproduce the exponential functions exp(μt) and exp(2μt) , μ > 0. We study its convergence, this including qualitative and quantitative theorems, an asymptotic formula and saturation results. We also show their shape preserving properties by considering generalized convexity. Finally, a comparison is stated, that shows that in a certain sense and for certain family of illustrative functions the new sequence approximates better than the classical Bernstein polynomials. Mathematics subject classification (2010): 41A36, 41A25, 41A40.

On approximation properties of Baskakov-Schurer-Szász operators preserving exponential functions

“…Immediately, we desire to demonstrate that our modified operators approximate better than classical Baskakov-Schurer-Szász operators. This part, we take into consideration of article which is Aral et al [3]. Ultimate theorem which would like to be given as below: Proof.…”

On Sequences of J. P. King-Type Operators

“…For papers inspired by [38,39] we refer the readers to [40][41][42] and [43][44][45][46], respectively.…”

“…In this section we review some results contained in [5,27,43], where the authors deal with different modifications of the Bernstein operators based on King's idea.…”

“…In this section we discuss the operators defined in [43]. From now on, set ( ) fl ( / − 1)/( / − 1) and recall that exp ( ) fl ( > 0).…”

Approximation by Baskakov-Szász-Stancu Operators Preserving Exponential Functions

“…Now, we desire to demonstrate that our modified operators approximate better than classical Baskakov-Szász-Stancu operators. This part, we take into consideration of article which is Aral et al [3]. Last theorem which would like to be given as below:…”