Estimates of central moments for one kind of exponential-type operators

Abstract

Abstract：In this paper, the explicit estimates of central moments for one kind of exponential-type operators are derived. The estimates play an essential role in studying the explicit approximation properties of this family of operators. Using the proposed method, the results of Ditzian and Totik in 1987, Guo and Qi in 2007, and Mahmudov in 2010 can be improved respectively.In 1987, a new natural modulus of smoothness was introduced by Ditzian and Totik [1] for the investigation of exponential-type operators, such as Bernstein operators, Szász operators, Baskakov operators and Post-Widder operators. To investigate the approximation properties of these exponential-type operators or Bernstein-type operators, the estimates of central moments for these types of operators [1] were given firstly. However, the estimates of central moments for Meyer-König and Zeller operators, which belong to five kinds of Bernstein-type operators, are still open. Abel [2] derived the estimates of r-order moments for Meyer-König and Zeller operators, but due to the complicated expression of these operators, their central moments were not given. Recently, Guo et al [3] gave the estimates of 2p-order central moments for and Mahmudov [4] gave the estimates of m-order central moments for qBernstein operators in the case of 0 1 q < < . Based on these estimates, the approximation theorems for these qanalogue of operators and Bernstein operators were proposed [5][6][7][8][9][10][11] . In fact, all of these estimates mentioned above are only rough estimates, the values of constant C appearing in the estimates of upper bound of central moments are still unknown. In this paper, we will give explicit estimates of the central moments for one kind of exponential-type operators, which are connected with Szász operators and Baskakov operators directly, but not connected with Bernstein operators and Meyer-König and Zeller operators.

High‐efficiency separation of oil sands using ChCl‐based deep eutectic solvents

“…In the infrared spectrum of ChCl/U, the absorption peaks between 2750 and 3700 cm −1 implied the formation of intermolecular hydrogen bonds of NH…Cl and OH…O, combined with CH, CO, and CC absorption peaks indicated the general structures of U and Ch + , which proved the synthesis of the ChCl/U sample. [ 28 ] The OH peak (3290 cm −1 ) was a typical signal of EG (Figure 3B). In the infrared spectrum of ChCl/EG, the peak near 3343 cm −1 implied the formation of intermolecular hydrogen bonds of OH…O and OH…Cl, combined with the characteristic peaks between 1300 and 500 cm −1 and the CC absorption peak indicated the general structures of EG and Ch + , which proved the synthesis of the ChCl/EG sample.…”

“…, which proved the synthesis of the ChCl/U sample. [28] The O H peak (3290 cm À1 ) was a typical signal of EG (Figure 3B). In the infrared spectrum of ChCl/EG, the peak near 3343 cm À1 implied the formation of intermolecular hydrogen bonds of O H…O and O H…Cl, combined with the characteristic peaks between 1300 and 500 cm À1 and the C C absorption peak indicated the general structures of EG and Ch + , which proved the synthesis of the ChCl/EG sample.…”

Business Strategy Analysis in Retail Economy: A Case Study of 7-Eleven

“…This is not just a matter of delivery time. It also includes the fastest way to receive order information from various stores through the information network and the most efficient collection of goods for each specific order [5]. Every product at 7-Eleven is "convenient, daily, and indispensable" to the target customer [6].…”