关于无界连续函数逼近的渐近估计

On the asymptotic formulae of approximation of unbounded continuous functions

"扩展乘数法"是研究无界连续函数,特别是大范围无界连续函数的逼近理论的方法。为了研究线性算子逼近满足某一类增长阶要求的无界连续函数时的误差估计,在"扩展乘数法"中引入经典试探函数组"1,x,x2",得到了满足某些条件的线性正算子改造为逼近此类无界函数的渐近估计,给出了具有一般性的、实用的渐近公式。并以此作为实例,研究了Landau积分型算子逼近无界函数的渐近估计式,可以很容易地得到许多有价值的结论。因此,这种结合既有理论价值又有实际意义。

As is well known, it is very important to approximate unbounded continuous functions, especially those defined on large range. An effective method called the method of multiplier-enlargement has been put forward by Hsu Lizhi and Wang Renhong. In order to estimate the error of approximation of unbounded continuous functions by linear operators, this paper applies the classical appropriate function'1,x,x^2' to the method of multiplier-enlargement to get the asymptotic estimation of approximation of these unbounded functions with reformed positive linear operators, and gives general asymptotic formulae. As an example, the asymptotic formulae of approximation of unbounded continuous functions with Landau operators are discussed, and therefore many important conclusions can be obtained with ease. The example shows that this combination is worth popularizing in theory and practice.