一类指数型算子的一致逼近

Uniform Approximation by a Family of Exponential—Type Operators

由文献[4]我们知道,当P(x)不同时,由齐次偏微分方程(α/αx×w(n,x,u)=n/p(x)×w(n,x,y)·(μ-x)及规范化条件integral from -∞=1 to ∞×w(n,x,u)du=1确定出的指数型算子integral from -∞=1 to ∞×w(n,x,u)f(u)·du亦不同。文[1]讨论了p(x)是至多二次的多项式时指数型算子的一致逼近问题,本文将就P(x)的更一般的情形给出一致逼近的正定理及饱和类。

Let Ln（f;x）=integral from n=-∞ to∞ W（n,x,u）f（u）du is the exponcmiai opreator which is determined by the homogeneous partial differential equation W（n,x,u）=n/p（x）)W（n,x,u）（u—x）and the normalization condiuon integral from n=-∞ to∞ W（n,x,u）du=1.In this paper, suppose thd function ψ（x）=1/2p（x） satisfies conditions of the so—called weight function, we obtained some results as follows: ① for every f∈C{A,B},we have |Ln（f）—f|≤Kωψ（f;1/（1/2n）） ② 啊 n} is globally saturated on {A,B} with order {1/n} and with saturation class {f|f′loeally absolutely continuous and |ψ2f'|≤Kf} ③ for any f∈{A,B} the following statements are equivaicnt to each other: （ⅰ） Ln（f）—f=0（1） uniformly in （A,B）, （ⅱ） if h→0 then f（x+hψ（x））—f（x） tends to zero uniformly in （A,B）. （ⅲ） the function f(g-1（x）) is uniformly continuous on （p,q）, where p,q and g are defined in （7）.