Voronovskaya type asymptotic approximation by modified Gamma operators
Abstract In this paper we deal with the following modified Gamma operators A n ( f ; x ) = ∫ 0 ∞ K n ( x , t ) f ( t ) dt , defined for any f for which the last integral is convergent, where K n ( x , t ) = ( 2 n + 3 ) ! n ! ( n + 2 ) ! · t n · x n + 3 ( x + t ) 2 n + 4 , x , t ∈ ( 0 , ∞ ) . We study some direct results in asymptotic approximation.
A note on the paper "Voronovskaya type asymptotic approximation by modified Gamma operators"
In the present paper we studied the Voronovskaya type theorem for modified Gamma operators presented by A. Izgi in the paper ''Voronovskaya type asymptotic approximation by modified Gamma operators'' [Applied Mathematics and Computation 217 (2011) 8061-8067]. We show that this theorem is not true. We present a correct form of this theorem and its proof. Furthermore, we obtain an error estimate for modified Gamma operators.
Approximation properties of Gamma operators
Abstract In this paper the approximation properties of Gamma operators G n are studied to the locally bounded functions and the absolutely continuous functions, respectively. Firstly, in Section 2 of the paper a quantitative form of the central limit theorem in probability theory is used to derive an asymptotic formula on approximation of Gamma operators G n for sign function. And then, this asymptotic formula combining with a metric form Ω x ( f , λ ) is used to derive an asymptotic estimate on the rate of convergence of Gamma operators G n for the locally bounded functions. Next, in Section 3 of the paper the optimal estimate on the first order absolute moment of the Gamma operators G n ( | t − x | , x ) is obtained by direct computations. And then, this estimate and Bojanic–Khan–Cheng's method combining with analysis techniques are used to derive an asymptotically optimal estimate on the rate of convergence of Gamma operators G n for the absolutely continuous functions.