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Bernstein-Type Operators That Reproduce Exponential Functions
(Element, 2018) Aral, Ali; Cardenas-Morales, Daniel; Garrancho, Pedro
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(mu t) and exp(2 mu t), mu > 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.
Szasz-Mirakyan Type Operators Which Fix Exponentials
(Springer Basel Ag, 2017) Acar, Tuncer; Aral, Ali; Cardenas-Morales, Daniel; Garrancho, Pedro
In this paper, we construct a new general class of operators which have the classical Szasz Mirakyan ones as a basis, and fix the functions and with . The convergence of the corresponding sequences is discussed in exponential weighted spaces, and a Voronovskaya type result is given. Also we define a new weighted modulus of smoothness and determine the approximation order of the constructed operators. Finally, we study the goodness of the estimates of our new operators via saturation results.