一类修正的Szász型算子的逼近性质

Approximation Properties of a Modified Szász Type Operators

摘要: 为了提高对函数的逼近程度，人们采取各种方法，构造King型算子就是其中的一种。本文构造了保持函数1和e^−μx(μ>0)的King-Szász型算子，对各阶矩的展开式应用Mathematica软件计算，得到了此类算子在[0,∞)区间上的一致逼近定理以及逼近误差的正定理。借助Taylor展式及连续模，得到其Voronovskaja型渐近估计。本文证明了该类算子统计逼近的Korovkin型定理，在此基础上，进一步研究了该类算子的统计逼近性质。

Abstract: There are many ways to improve the approximation degree of function, and the construction of King type operators is one of them. In this paper, we construct King-Szász type operators which preserve the functions 1 and e^−μx(μ>0). The expansions of the moments are calculated by Mathematica software. The uniform approximation theorem on the interval [0,∞) and the positive theorem of approximation error of this kind of operator are obtained. By means of Taylor expansion and continuous modulus, the Voronovskaja type asymptotic estimation is obtained. The Korovkin type theorem of statistical approximation of this kind of operator is also proved. Finally, the statistical approximation properties are further studied.