Chauhan Ruchi, Ispir Nurhayat, Agrawal P N
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667 India.
Department of Mathematics, Faculty of Sciences, Gazi University, Ankara, 06500 Turkey.
J Inequal Appl. 2017;2017(1):50. doi: 10.1186/s13660-017-1298-y. Epub 2017 Feb 27.
Agrawal (Boll. Unione Mat. Ital. 8:169-180, 2015) introduced a Stancu-type Kantorovich modification of the operators proposed by Ren and Zeng (Bull. Korean Math. Soc. 50(4):1145-1156, 2013) and studied a basic convergence theorem by using the Bohman-Korovokin criterion, the rate of convergence involving the modulus of continuity, and the Lipschitz function. The concern of this paper is to obtain Voronoskaja-type asymptotic result by calculating an estimate of fourth order central moment for these operators and discuss the rate of convergence for the bivariate case by using the complete and partial moduli of continuity and the degree of approximation by means of a Lipschitz-type function and the Peetre -functional. Also, we consider the associated GBS (generalized Boolean sum) operators and estimate the rate of convergence for these operators with the help of a mixed modulus of smoothness. Furthermore, we show the rate of convergence of these operators (univariate case) to certain functions with the help of the illustrations using Maple algorithms and in the bivariate case, the rate of convergence of these operators is compared with the associated GBS operators by illustrative graphics.
阿格拉瓦尔（《意大利数学联合会通报》8:169 - 180，2015年）引入了任和曾（《韩国数学会通报》50(4):1145 - 1156，2013年）所提出算子的一种斯坦库型坎托罗维奇修正，并利用博赫曼 - 科罗夫金准则、涉及连续性模的收敛速度以及利普希茨函数研究了一个基本收敛定理。本文的关注点是通过计算这些算子的四阶中心矩估计来获得沃罗诺斯卡娅型渐近结果，并利用完全和部分连续性模以及借助利普希茨型函数和彼得雷泛函的逼近度来讨论二元情形下的收敛速度。此外，我们考虑相关的广义布尔和（GBS）算子，并借助混合光滑模估计这些算子的收敛速度。再者，我们借助Maple算法的图示展示这些算子（一元情形）对某些函数的收敛速度，在二元情形下，通过图示将这些算子的收敛速度与相关的GBS算子进行比较。
