The revision of some results for Bernstein-Stancu type operators

Abstract

In the present paper the author revises some known results, respectively establishes new results for Bernstein-Stancu operators and for a particular case of the same operators, introduced first by L. Lupas¸ and A. Lupas¸.

Generalized Positive Linear Operators Based on PED and IPED

“…These Stirling numbers of second kind are very useful in calculating the moments of linear positive operators especially for higher order moments. In [17,18], Dan obtained higher order moments for Bernstein type operators by using these numbers. In what follows, we shall find the moments of L (α) n…”

Bézier-Summation-Integral-Type Operators That Include Pólya–Eggenberger Distribution

“…Later, Miclȃuş [4,5] presented an interesting work on these operators using the idea of convex functions and divided differences including Voronovskaja-type theorem while the classical Voronovskaja theorem [6] is stated as follows.…”

On Approximation Properties of Generalized Lupas Type Operators Based on Polya Distribution with Pochhammer k-Symbol

“…Now, we begin our study by giving moments, central moments and rate of convergence for the operators p13q . We first give Lemma 2.1 , Lemma 2.3 and Corollary 2.2 without proof, which follows from the results given in the paper [21] for α " k n , k ě 0. Throughout this paper, N denotes the set of positive integers and N 0 " N Y t0u and let us denote the monomials e j ptq " t j for j P N 0 .…”

“…where psq m is a rising factorial also known as the Pochhammer symbol namely, psq m " " s ps `1q ps `2q ... ps `m ´1q f or m P N 1 f or m " 0, s ‰ 0 (5) where s is a real or complex number. In 2012, Miclaus [21] reconsidered the operators p4q and in this work, some of the properties of the operators such as moments, the remainder term and the monotocity properties, were recalculated with a different technique and also asymptotic behaviour of the p4q was discussed. Up to now, many operators which are based on Polya distribution have been extensively studied.…”

“…For more detail see [13,[17][18][19]23,24]. Our present study is motivated essentially by the theory of Polya distribution of the operators, considered by Lupaş and Lupaş [20], presented in [21] and the Pochhammer k-symbol given by [13]. Here, we define some slight modifications of Polya distribution for the special case α " k n , k nonnegative real number, applying the notion of Pochhammer k-symbol in the definition of Polya distribution.…”